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sthal Numbers and Associated Hessenberg Matrices</a>, J. Int. Seq., Vol. 21 (2018), Article 18.2.5.
%H A000045 Arzu Özkoç, <a href="http://link.springer.com/article/10.1186/s13662-015-0486-7/fulltext.html">Some algebraic identities on quadra Fibona-Pell integer sequence</a>, Advances in Difference Equations, 2015, 2015:148.
%H A000045 Ram Krishna Pandey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pandey/pandey7.html">On Some Magnified Fibonacci Numbers Modulo a Lucas Number</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.1.7.
%H A000045 J. Patterson, <a href="http://www.bath.ac.uk/~ma1jmp/link.html">The Fibonacci Sequence</a>. [broken link]
%H A000045 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H A000045 Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
%H A000045 T. K. Petersen and B. E. Tenner, <a href="http://arxiv.org/abs/1202.4765">The depth of a permutation</a>, arXiv:1202.4765v1 [math.CO], 2012-2014.
%H A000045 Ivars Peterson, <a href="http://www.sciencenews.org/articles/20060603/mathtrek.asp">Fibonacci's Missing Flowers</a>.
%H A000045 Akos Pinter and Volker Ziegler, <a href="http://arxiv.org/abs/1005.3624">On Arithmetic Progressions in Recurrences - A new characterization of the Fibonacci sequence</a>, arXiv:1005.3624 [math.NT], 2010.
%H A000045 Simon Plouffe, Project Gutenberg, <a href="http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt">The First 1001 Fibonacci Numbers</a> [broken link]
%H A000045 Project Nayuki, <a href="http://www.nayuki.io/page/fast-fibonacci-algorithms">Fast Fibonacci algorithms</a> (fast doubling is faster than matrix multiplication).
%H A000045 Phakhinkon Phunphayap, Prapanpong Pongsriiam, <a href="https://www.emis.de/journals/JIS/VOL21/Munarini/muna8.html">Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients</a>, J. Int. Seq. 21 (2018), #18.3.1.
%H A000045 S. Rabinowitz, <a href="http://www.mathpropress.com/stan/bibliography/algorithmicFib.pdf">Algorithmic Manipulation of Fibonacci Identities</a> (1996).
%H A000045 A. Radhakrishnan, L. Solus, and C. Uhler, <a href="https://doi.org/10.1016/j.dam.2018.03.015">Counting Markov equivalence classes for DAG models on trees</a>, Discrete Applied Mathematics 244 (2018): 170-185.
%H A000045 Arulalan Rajan, R. Vittal Rao, Ashok Rao and H. S. Jamadagni, <a href="http://arxiv.org/abs/1205.5398">Fibonacci Sequence, Recurrence Relations, Discrete Probability Distributions and Linear Convolution</a>, arXiv preprint arXiv:1205.5398 [math.PR], 2012. - From _N. J. A. Sloane_, Oct 23 2012
%H A000045 Aayush Rajasekaran, <a href="https://uwspace.uwaterloo.ca/bitstream/handle/10012/13202/Rajasekaran_Aaayush.pdf?sequence=3">Using Automata Theory to Solve Problems in Additive Number Theory</a>, MS thesis, University of Waterloo, 2018.
%H A000045 Marc Renault, <a href="http://www.math.temple.edu/~renault/fibonacci/thesis.html">Properties of the Fibonacci sequence under various moduli</a>, MSc Thesis, Wake Forest U, 1996.
%H A000045 N. Renton, <a href="http://www.users.bigpond.net.au/renton/903.htm">The fibonacci Series</a>
%H A000045 B. Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.pdf">On the Average Growth of Random Fibonacci Sequences</a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
%H A000045 Rosetta Code, <a href="http://rosettacode.org/wiki/Fibonacci_sequence">A collection of codes to compute fibonacci numbers with different computer languages</a>
%H A000045 E. S. Rowland, <a href="http://people.hofstra.edu/Eric_Rowland/investigations/fibonacci.html">Fibonacci Sequence Calculator up to n=1474</a>
%H A000045 Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv preprint arXiv:1508.07894, 2015
%H A000045 Shiva Samieinia, <a href="http://www.math.su.se/reports/2007/6/">Digital straight line segments and curves</a>. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
%H A000045 A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Tsikouras/tsikouras67.html">On the Dominance Partial Ordering of Dyck Paths</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.5.
%H A000045 D. Schweizer, <a href="http://math.holycross.edu/~davids/fibonacci/fibonacci.html">First 500 Fibonacci Numbers in blocks of 100.</a> [broken link]
%H A000045 Mark A. Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/j5/j5.Abstract.html">Tiling proofs of some formulas for the Pell numbers of odd index</a>, Integers, 9 (2009), 53-64.
%H A000045 Mark A. Shattuck and Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Shattuck/shattuck56.html">Periodicity and Parity Theorems for a Statistic on r-Mino Arrangements</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.6.
%H A000045 S. Silvia, <a href="http://arttech.about.com/library/weekly/aa060900a_fibonacci_sequence.htm">Fibonacci sequence</a> [broken link]
%H A000045 Parmanand Singh, <a href="http://dx.doi.org/10.1016/0315-0860(85)90021-7">The so-called Fibonacci numbers in ancient and medieval India</a>, Historia Mathematica, Volume 12 (3), 1985, 229-244.
%H A000045 Jaap Spies, <a href="http://www.jaapspies.nl/oeis/a000045.sage">Sage program for computing A000045</a>
%H A000045 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A000045 Z.-H. Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences For Fibonacci Numbers</a>
%H A000045 Asep K. Supriatna, Ema Carnia, Meksianis Z. Ndii, <a href="https://doi.org/10.1016/j.heliyon.2019.e01130">Fibonacci numbers: A population dynamics perspective</a>, Heliyon (2019) Vol. 5, Issue 1, e01130.
%H A000045 Roberto Tauraso, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Tauraso/tauraso3.html">A New Domino Tiling Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.3.
%H A000045 Thesaurus.Maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/3788">Fibonacci sequence</a> [broken link]
%H A000045 K. Tognetti, <a href="/A000045/a000045.pdf">Letter to N. J. A. Sloane (with attachments), May 25 1994</a>
%H A000045 K. Tognetti, <a href="/A000045/a000045_2.pdf">The Search for the Golden Sequence</a>, Draft Manuscript, May 25 1994.
%H A000045 K. Tognetti, <a href="http://www.austms.org.au/Modules/Fib">Fibonacci-His Rabbits and His Numbers and Kepler</a>
%H A000045 Tony van Ravenstein, <a href="/A000045/a000045_1.pdf">The three gap theorem (Steinhaus conjecture)</a>, Journal of the Australian Mathematical Society (Series A) 45.03 (1988): 360-370. [Annotated scanned copy]
%H A000045 C. Vila, <a href="http://www.boingboing.net/2010/03/22/dreamlike-animation.html">Nature by numbers</a> (animation).
%H A000045 Christobal Vila, <a href="http://wimp.com/naturenumbers/">Nature Numbers</a> (Video related to Fibonacci numbers)
%H A000045 N. N. Vorob'ev, <a href="http://eom.springer.de/F/f040020.htm">Fibonacci numbers</a>, Springer's Encyclopaedia of Mathematics.
%H A000045 A. Y. Z. Wang, P. Wen, <a href="https://doi.org/10.1186/s13660-015-0595-6">On the partial finite sums of the reciprocals of the Fibonacci numbers</a>, Journal of Inequalities and Applications, 2015.
%H A000045 Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Wagner/wagner3.html">Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000045 Robert Walker, <a href="http://www.youtube.com/watch?v=Wx4ZfuMl-FI&amp;NR=1">Inharmonic "Golden Rhythmicon" - Fibonacci Sequence in Pairs Approaching Golden Ratio - With Bounce</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Double-Free Set</a>, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>, <a href="http://mathworld.wolfram.com/ResistorNetwork.html">Resistor Network</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HosoyaIndex.html">Hosoya Index</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H A000045 Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A000045 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities">Cassini and Catalan identities</a>
%H A000045 Willem's Fibonacci site, <a href="http://home.zonnet.nl/LeonardEuler/fiboe.htm">Fibonacci</a>
%H A000045 Mike Winkler, <a href="http://arxiv.org/abs/1412.0519">On the structure and the behaviour of Collatz 3n + 1 sequences - Finite subsequences and the role of the Fibonacci sequence</a>, arXiv:1412.0519 [math.GM], 2014.
%H A000045 Roman Witula, Damian Slota and Edyta Hetmaniok, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from255to263.pdf">Bridges between different known integer sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.
%H A000045 R. Yanco, <a href="/A007380/a007380.pdf">Letter and Email to N. J. A. Sloane, 1994</a>
%H A000045 R. Yanco and A. Bagchi, <a href="/A007380/a007380_1.pdf">K-th order maximal independent sets in path and cycle graphs</a>, Unpublished manuscript, 1994. (Annotated scanned copy)
%H A000045 Donovan Young, <a href="https://www.emis.de/journals/JIS/VOL21/Young/young2.html">The Number of Domino Matchings in the Game of Memory</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.1.
%H A000045 Aimei Yu and Xuezheng Lv, <a href="http://dx.doi.org/10.1007/s10910-006-9088-7">The Merrifield-Simmons indices and Hosoya indices of trees with k pendant vertices</a>, J. Math. Chem., Vol. 41 (2007), pp. 33-43. See page 35.
%H A000045 Tianping Zhang and Yuankui Ma, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Zhang/zhang56.html">On Generalized Fibonacci Polynomials and Bernoulli Numbers</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.3.
%H A000045 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000045 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A000045 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000045 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H A000045 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A000045 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000045 G.f.: x / (1 - x - x^2).
%F A000045 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x). - _Paul D. Hanna_, Oct 26 2013
%F A000045 F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)).
%F A000045 Alternatively, F(n) = ((1/2+sqrt(5)/2)^n - (1/2-sqrt(5)/2)^n)/sqrt(5).
%F A000045 F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).
%F A000045 F(n) = round(phi^n/sqrt(5)).
%F A000045 F(n+1) = Sum_{j=0..floor(n/2)} binomial(n-j, j).
%F A000045 A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Jan 03 2017
%F A000045 E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - _Len Smiley_, Nov 30 2001
%F A000045 [0 1; 1 1]^n [0 1] = [F(n); F(n+1)]
%F A000045 x | F(n) ==> x | F(kn).
%F A000045 A sufficient condition for F(m) to be divisible by a prime p is (p - 1) divides m, if p == 1 or 4 (mod 5); (p + 1) divides m, if p == 2 or 3 (mod 5); or 5 divides m, if p = 5. (This is essentially Theorem 180 in Hardy and Wright.) - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 29 2001
%F A000045 a(n)=F(n) has the property: F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1). - _Miklos Kristof_, Nov 13 2003
%F A000045 From _Kurmang. Aziz. Rashid_, Feb 21 2004: (Start)
%F A000045 Conjecture 1: for n >= 2, sqrt(F(2n+1) + F(2n+2) + F(2n+3) + F(2n+4) + 2*(-1)^n) = (F(2n+1) + 2*(-1)^n)/F(n-1). [For a proof see Comments section.]
%F A000045 Conjecture 2: for n >= 0, (F(n+2)*F(n+3)) - (F(n+1)*F(n+4)) + (-1)^n = 0.
%F A000045 [Two more conjectures removed by _Peter Luschny_, Nov 17 2017 ]
%F A000045 Theorem 1: for n >= 0, (F(n+3)^ 2 - F(n+1)^ 2)/F(n+2) = (F(n+3)+ F(n+1)).
%F A000045 Theorem 2: for n >= 0, F(n+10) = 11*F(n+5) + F(n).
%F A000045 Theorem 3: for n >= 6, F(n) = 4*F(n-3) + F(n-6). (End)
%F A000045 Conjecture 2 of Rashid is actually a special case of the general law F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1) (take n <- n+1 and m <- -(n+4) in this law). - Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 22 2005
%F A000045 Conjecture 2 of Rashid Kurmang simplified: F(n)*F(n+3) = F(n+1)*F(n+2)-(-1)^n. Follows from d'Ocagne's identity: m=n+2. - _Alex Ratushnyak_, May 06 2012
%F A000045 Conjecture: for all c such that 2-Phi <= c < 2*(2-Phi) we have F(n) = floor(Phi*a(n-1)+c) for n > 2. - _Gerald McGarvey_, Jul 21 2004
%F A000045 |2*F(n) - 9*F(n+1)| = 4*A000032(n) + A000032(n+1). - _Creighton Dement_, Aug 13 2004
%F A000045 For x > Phi, Sum_{n>=0} F(n)/x^n = x/(x^2 - x - 1) - _Gerald McGarvey_, Oct 27 2004
%F A000045 F(n+1) = exponent of the n-th term in the series f(x, 1) determined by the equation f(x, y) = xy + f(xy, x). - _Jonathan Sondow_, Dec 19 2004
%F A000045 a(n-1) = Sum_{k=0..n} (-1)^k*binomial(n-ceiling(k/2), floor(k/2)). - _Benoit Cloitre_, May 05 2005
%F A000045 F(n+1) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - _Paul Barry_, Aug 28 2005
%F A000045 Fibonacci(n) = Product_{j=1..ceiling(n/2)-1} (1 + 4(cos(j*Pi/n))^2). [Bicknell and Hoggatt, pp. 47-48.] - _Emeric Deutsch_, Oct 15 2006
%F A000045 F(n) = 2^-(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)*5^k. - _Hieronymus Fischer_, Feb 07 2006
%F A000045 a(n) = (b(n+1) + b(n-1))/n where {b(n)} is the sequence A001629. - _Sergio Falcon_, Nov 22 2006
%F A000045 F(n*m) = Sum_{k = 0..m} binomial(m,k)*F(n-1)^k*F(n)^(m-k)*F(m-k). The generating function of F(n*m) (n fixed, m = 0,1,2,...) is G(x) = F(n)*x / ((1 - F(n-1)*x)^2 - F(n)*x*(1 - F(n-1)*x) - (F(n)*x)^2). E.g., F(15) = 610 = F(5*3) = binomial(3,0)* F(4)^0*F(5)^3*F(3) + binomial(3,1)* F(4)^1*F(5)^2*F(2) + binomial(3,2)* F(4)^2*F(5)^1*F(1) + binomial(3,3)* F(4)^3*F(5)^0*F(0) = 1*1*125*2 + 3*3*25*1 + 3*9*5*1 + 1*27*1*0 = 250 + 225 + 135 + 0 = 610. - _Miklos Kristof_, Feb 12 2007
%F A000045 From _Miklos Kristof_, Mar 19 2007: (Start)
%F A000045   Let L(n) = A000032(n) = Lucas numbers. Then:
%F A000045   For a >= b and odd b,  F(a+b) + F(a-b) = L(a)*F(b).
%F A000045   For a >= b and even b, F(a+b) + F(a-b) = F(a)*L(b).
%F A000045   For a >= b and odd b,  F(a+b) - F(a-b) = F(a)*L(b).
%F A000045   For a >= b and even b, F(a+b) - F(a-b) = L(a)*F(b).
%F A000045   F(n+m) + (-1)^m*F(n-m) = F(n)*L(m);
%F A000045   F(n+m) - (-1)^m*F(n-m) = L(n)*F(m);
%F A000045   F(n+m+k) + (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = F(n)*L(m)*L(k);
%F A000045   F(n+m+k) - (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = L(n)*L(m)*F(k);
%F A000045   F(n+m+k) + (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = L(n)*F(m)*L(k);
%F A000045   F(n+m+k) - (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = 5*F(n)*F(m)*F(k). (End)
%F A000045 A corollary to Kristof 2007 is 2*F(a+b) = F(a)*L(b) + L(a)*F(b). - _Graeme McRae_, Apr 24 2014
%F A000045 For n > m, the sum of the 2m consecutive Fibonacci numbers F(n-m-1) thru F(n+m-2) is F(n)*L(m) if m is odd, and L(n)*F(m) if m is even (see the McRae link). - _Graeme McRae_, Apr 24 2014.
%F A000045 F(n) = b(n) + (p-1)*Sum_{k=2..n-1} floor(b(k)/p)*F(n-k+1) where b(k) is the digital sum analog of the Fibonacci recurrence, defined by b(k) = ds_p(b(k-1)) + ds_p(b(k-2)), b(0)=0, b(1)=1, ds_p=digital sum base p. Example for base p=10: F(n) = A010077(n) + 9*Sum_{k=2..n-1} A059995(A010077(k))*F(n-k+1). - _Hieronymus Fischer_, Jul 01 2007
%F A000045 F(n) = b(n)+p*Sum_{k=2..n-1} floor(b(k)/p)*F(n-k+1) where b(k) is the digital product analog of the Fonacci recurrence, defined by b(k) = dp_p(b(k-1)) + dp_p(b(k-2)), b(0)=0, b(1)=1, dp_p=digital product base p. Example for base p=10: F(n) = A074867(n) + 10*Sum_{k=2..n-1} A059995(A074867(k))*F(n-k+1). - _Hieronymus Fischer_, Jul 01 2007
%F A000045 a(n) = denominator of continued fraction [1,1,1,...] (with n ones); e.g., 2/3 = continued fraction [1,1,1]; where barover[1] = [1,1,1,...] = 0.6180339.... - _Gary W. Adamson_, Nov 29 2007
%F A000045 F(n + 3) = 2F(n + 2) - F(n), F(n + 4) = 3F(n + 2) - F(n), F(n + 8) = 7F(n + 4) - F(n), F(n + 12) = 18F(n + 6) - F(n). - _Paul Curtz_, Feb 01 2008
%F A000045 1 = 1/(1*2) + 1/(1*3) + 1/(2*5) + 1/(3*8) + 1/(5*13) + ... = 1/2 + 1/3 + 1/10 + 1/24 + 1/65 + 1/168 + ...; where A059929 = (0, 2, 3, 10, 24, 65, 168, ...). - _Gary W. Adamson_, Mar 16 2008
%F A000045 a(2^n) = Product_{i=0..n-2} B(i) where B(i) is A001566. Example 3*7*47 = F(16). - _Kenneth J Ramsey_, Apr 23 2008
%F A000045 F(n) = (1/(n-1)!) * (n^(n-1) - (C(n-2,0) + 4*C(n-2,1) + 3*C(n-2,2))*n^(n-2) + (10*C(n-3,0) + 49*C(n-3,1) + 95*C(n-3,2) + 83*C(n-3,3) + 27*C(n-3,4))*n^(n-3) - (90*C(n-4,0) + 740*C(n-4,1) + 2415*C(n-4,2) + 4110*C(n-4,3) + 3890*C(n-4,4) + 1950*C(n-4,5) + 405*C(n-4,6))*n^(n-4) + ... ). - _André F. Labossière_, Nov 24 2004
%F A000045 a(n+1) = Sum_{k=0..n} A109466(n,k)*(-1)^(n-k). -_Philippe Deléham_, Oct 26 2008
%F A000045 a(n) = Sum_{l_1=0..n+1} Sum_{l_2=0..n}...Sum_{l_i=0..n-i}... Sum_{l_n=0..1} delta(l_1,l_2,...,l_i,...,l_n), where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i + l_(i+1) >= 2 for i=1..n-1 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise. - _Thomas Wieder_,