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### -- Authors: Tobias Boege
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44 Dying rabbits: a(0) = 1; for 1 <= n <= 12, a(n) = Fibonacci(n); for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).
%C A000044 A107358 is a more satisfactory version, but I have left the present sequence unchanged (except for making the definition clearer) since it has been in the OEIS so long.
%C A000044 Number of compositions of n into parts 1, 3, 5, 7, 9, and 11. - _Joerg Arndt_, Sep 05 2014
%C A000044 If a(0) = 1 then it is not clear why a(2) = 1, it should be equal to a(1) + a(0) = 2. Does the first comment mean that a(0) is erroneous and should read a(0) = 0? In contrast to A107358, the term a(13) = 232 = 144 + 89 - 1 seems correct, since in this month the first and oldest pair of rabbits die. But a(14) should be equal to a(13) + a(12) = 232 + 144 because the first pair (which was also the only one present in month 2) has already died and there is no other pair aged 12 months. In general, the number of pairs which die in month n because they are aged exactly 12 months, equals a(n-14): this is the number of newborn pairs in month n - 12, viz. a(n-12) = a(n-13) [those from preceding month] + a(n-14) [the newborn ones] - #(those which die). - _M. F. Hasler_, Oct 06 2017
%D A000044 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000044 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000044 Harvey P. Dale, <a href="/A000044/b000044.txt">Table of n, a(n) for n = 0..1000</a>
%H A000044 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A000044 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A000044 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
%F A000044 G.f.: 1/(1 - z - z^3 - z^5 - z^7 - z^9 -z^11).
%F A000044 G.f. A(x) = 1 / (1 - x / (1 - x^2 / (1 + x^10 / (1 + x^2 / (1 - x^2 / (1 + x^6 / (1 + x^2 / (1 - x^2 / (1 + x^2))))))))). - _Michael Somos_, Jan 04 2013
%F A000044 For n >= 11, a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-11). - _Eric M. Schmidt_, Sep 04 2014
%e A000044 G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 + 21*x^8 + 34*x^9 + ...
%p A000044 with(combinat); f:=proc(n) option remember; if n=0 then RETURN(1); fi; if n <= 12 then RETURN(fibonacci(n)); fi; f(n-1)+f(n-2)-f(n-13); end;
%t A000044 CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t A000044 LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},{1,1,1,2,3,5,8,13,21,34,55,89,144},100] (* _Harvey P. Dale_, Mar 24 2012 *)
%o A000044 (MAGMA) [ n eq 1 select 1 else n le 13 select Fibonacci(n-1) else Self(n-1)+Self(n-2)-Self(n-13): n in [1..50] ]; // _Klaus Brockhaus_, Dec 21 2010
%o A000044 (PARI) Vec(1/(1-z-z^3-z^5-z^7-z^9-z^11)+O(z^50)) \\ _Charles R Greathouse IV_, Jun 10 2011
%Y A000044 Cf. A107358. See A000045 for the Fibonacci numbers.
%K A000044 nonn,easy
%O A000044 0,4
%A A000044 _N. J. A. Sloane_; entry revised May 25 2005
%E A000044 G.f. corrected by _Charles R Greathouse IV_, Jun 10 2011

%I A048817
%S A048817 1,1,2,3,5,8,13,21,29,37,66,95,161,227,293,520,813,1106,1919,2732,
%T A048817 3545,4358,7903,12261,20164,28067,48231,68395,88559,156954,245513,
%U A048817 334072,579585,825098,1070611,1316124,1561637,2877761,4439398,6001035
%N A048817 Numerators of convergents to A058914.
%D A048817 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 441-443.
%H A048817 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [Broken link]
%H A048817 Steven R. Finch, <a href="http://web.archive.org/web/20010208143502/http://www.mathsoft.com/asolve/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [From the Wayback machine]
%H A048817 <a href="/index/Me#MinkowskiQ">Index entries for Minkowski's question mark function</a>
%H A048817 <a href="/index/Me#MinkowskiQ">Index entries for sequences related to Minkowski's question mark function</a>
%e A048817 f(0)=0, f(1/8)=1/4, f(1/4)=1/3, f(3/8)=2/5, f(1/2)=1/2, f(5/8)=3/5...
%e A048817 1/2, 1/3, 2/5, 3/7, 5/12, 8/19, ...
%Y A048817 Denominators in A048818. Cf. A048819-A048822.
%Y A048817 Associated continued fraction in A058914. Denominators in A048818.
%K A048817 nonn,frac
%O A048817 0,3
%A A048817 _Christian G. Bower_, Apr 15 1999

%I A105471
%S A105471 0,1,1,2,3,5,8,13,21,34,55,89,44,33,77,10,87,97,84,81,65,46,11,57,68,
%T A105471 25,93,18,11,29,40,69,9,78,87,65,52,17,69,86,55,41,96,37,33,70,3,73,
%U A105471 76,49,25,74,99,73,72,45,17,62,79,41,20,61,81,42,23,65,88,53,41,94,35,29,64
%N A105471 a(n) = Fibonacci(n) mod 100.
%C A105471 a(n) = A105472(n)*10 + A003893(n);
%C A105471 the sequence is periodic with period 300; all blocks of 60 successive terms contain 20 even and 40 odd numbers, see A003893.
%H A105471 Vincenzo Librandi, <a href="/A105471/b105471.txt">Table of n, a(n) for n = 0..1000</a>
%H A105471 <a href="/index/Rec#order_300">Index entries for linear recurrences with constant coefficients</a>, order 300.
%F A105471 a(n) = (a(n-1) + a(n-2)) mod 100 for n>1, a(0) = 0, a(1) = 1.
%t A105471 Mod[Fibonacci[Range[0,100]],100] (* _Harvey P. Dale_, Jun 12 2014 *)
%o A105471 (MAGMA) [Fibonacci(n) mod(100): n in [0..75]]; // _Vincenzo Librandi_, Jul 23 2014
%o A105471 (Haskell)
%o A105471 a105471 n = a105471_list !! n
%o A105471 a105471_list = 0 : 1 :
%o A105471    zipWith ((flip mod 100 .) . (+)) a105471_list (tail a105471_list)
%o A105471 -- _Reinhard Zumkeller_, Aug 06 2014
%o A105471 (PARI) a(n)=fibonacci(n%300)%100 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A105471 Cf. A000045.
%K A105471 nonn,easy
%O A105471 0,4
%A A105471 _Reinhard Zumkeller_, Apr 09 2005

%I A243063
%S A243063 1,1,2,3,5,8,13,21,34,55,89,144,233,377,61,438,499,937,1436,2373,389,
%T A243063 2762,3151,5913,964,6877,7841,14718,22559,37277,59836,97113,156949,
%U A243063 25462,182411,27873,21284,49157,7441,56598,6439,6337,12776,19113,31889,512,3241
%N A243063 Numbers generated by a Fibonacci-like sequence in which zeros are suppressed.
%C A243063 Let x(1) = 1, x(2) = 1, then begin the sequence x(i) = no-zero(x(i-2) + x(i-1)), where the function no-zero(n) removes all zero digits from n.
%C A243063 The sequence behaves like a standard Fibonacci sequence until step 15, where x = no-zero(233 + 377) = no-zero(610) = 61. At step 16, x = 377 + 61 = 438. The sequence then proceeds until step 927, where x = no-zero(224 + 377) = no-zero(601) = 61. Therefore at step 928, x = 377 + 61 = 438 and the sequence repeats.
%H A243063 Anthony Sand, <a href="/A243063/b243063.txt">Table of n, a(n) for n = 1..927</a>
%F A243063 x(i) = no-zero(x(i-2) + x(i-1)). For example, no-zero(233 + 377) = no-zero(610) = 61.
%e A243063 x(3) = x(1) + x(2) = 1 + 1 = 2.
%e A243063 x(4) = x(2) + x(3) = 1 + 2 = 3.
%e A243063 x(15) = no-zero(x(13) + x(14)) = no-zero(233 + 377) = no-zero(610) = 61.
%e A243063 x(16) = 377 + 61 = 438.
%p A243063 noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end; # A004719
%p A243063 t1:=[1,1]; for n from 3 to 100 do t1:=[op(t1),noz(t1[n-1]+t1[n-2])]; od: t1; # _N. J. A. Sloane_, Jun 11 2014
%Y A243063 Cf. A000045, A004719, A242350, A243657, A243658, A306773.
%K A243063 nonn,base
%O A243063 1,3
%A A243063 _Anthony Sand_, Jun 09 2014

%I A185357
%S A185357 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4180,6763,
%T A185357 10942,17703,28642,46340,74974,121301,196254,317521,513720,831152,
%U A185357 1344728,2175647,3519998,5695035,9214046,14907484,24118947,39022252,63134437,102145749
%N A185357 Expansion of 1/(1 - x - x^2 + x^18 - x^20).
%C A185357 Limiting ratio is 1.61791..., the real root of -1 + x^2 - x^18 - x^19 + x^20. Signature in Mathematica is:
%C A185357 -CoefficientList[1 - x - x^2 + x^18 - x^20, x]
%C A185357 {-1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1}.
%C A185357 The sequence agrees with the Fibonacci numbers (A000045) for the first 18 terms.
%H A185357 G. C. Greubel, <a href="/A185357/b185357.txt">Table of n, a(n) for n = 0..1000</a>
%H A185357 David Terr and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PisotNumber.html">MathWorld: Pisot Number</a>
%H A185357 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).
%t A185357 CoefficientList[Series[1/(1 - x - x^2 + x^18 - x^20), {x, 0, 50}], x]
%o A185357 (PARI) Vec(1/(1-x-x^2+x^18-x^20) + O(x^50)) \\ _G. C. Greubel_, Nov 16 2016
%Y A185357 Cf. A117791, A107293, A204631.
%K A185357 nonn,easy
%O A185357 0,3
%A A185357 _Roger L. Bagula_, Jan 21 2012

%I A204026
%S A204026 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,3,3,2,1,1,2,3,5,3,2,1,1,2,3,5,5,3,
%T A204026 2,1,1,2,3,5,8,5,3,2,1,1,2,3,5,8,8,5,3,2,1,1,2,3,5,8,13,8,5,3,2,1,1,2,
%U A204026 3,5,8,13,13,8,5,3,2,1,1,2,3,5,8,13,21,13,8,5,3,2,1,1,2,3,5,8
%N A204026 Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.
%C A204026 A204026 represents the matrix M given by f(i,j)=min(F(i+1),F(j+1)) for i>=1 and j>=1.  See A204027 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
%e A204026 Northwest corner:
%e A204026 1 1 1 1 1 1
%e A204026 1 2 2 2 2 2
%e A204026 1 2 3 3 3 3
%e A204026 1 2 3 5 5 5
%e A204026 1 2 3 5 8 8
%e A204026 1 2 3 5 8 13
%t A204026 f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
%t A204026 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204026 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204026 Flatten[Table[f[i, n + 1 - i],
%t A204026   {n, 1, 15}, {i, 1, n}]]  (* A204026 *)
%t A204026 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204026 c[n_] := CoefficientList[p[n], x]
%t A204026 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204026 Table[c[n], {n, 1, 12}]
%t A204026 Flatten[%]                 (* A204027 *)
%t A204026 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204026 Cf. A204026, A204016, A202453.
%K A204026 nonn,tabl
%O A204026 1,5
%A A204026 _Clark Kimberling_, Jan 11 2012

%I A226999
%S A226999 1,0,1,1,2,3,5,8,13,21,35,55,93,149,248,403,671,1098,1827,3013,5013,
%T A226999 8313,13859,23063,38534,64341,107715,180355,302565,507784,853507,
%U A226999 1435415,2416941,4072272,6868062,11590807,19577555,33088481,55964327,94712212
%N A226999 Inverse Euler transform of A005169 (fountains of coins).
%C A226999 If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.
%H A226999 Alois P. Heinz, <a href="/A226999/b226999.txt">Table of n, a(n) for n = 1..4178</a>
%H A226999 R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H A226999 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A226999 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A226999 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%Y A226999 Cf. A005170.
%K A226999 nonn
%O A226999 1,5
%A A226999 _R. J. Mathar_, Jun 26 2013

%I A261587
%S A261587 0,1,1,2,3,5,8,13,21,34,55,30,26,56,23,20,43,63,47,51,98,31,70,101,
%T A261587 112,95,89,125,96,103,81,125,29,95,65,101,48,149,138,169,130,122,134,
%U A261587 138,154,174,151,148,122,152,156,131,169,241,233,179,235,178,236
%N A261587 Sum of sexagesimal digits of Fibonacci numbers in base 60 representation.
%C A261587 a(n) = sum of n-th row in table A261575.
%H A261587 Reinhard Zumkeller, <a href="/A261587/b261587.txt">Table of n, a(n) for n = 0..10000</a>
%H A261587 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261587 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261587 (Haskell)
%o A261587 a261587 = sum . a261575_row
%Y A261587 Cf. A261575, A000045, A004090, A261598.
%K A261587 nonn,base
%O A261587 0,4
%A A261587 _Reinhard Zumkeller_, Sep 09 2015

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Search: seq:1,1,2,3,5,8,13,21
Showing 21-30 of 92

%I A261598
%S A261598 0,1,1,2,3,5,8,13,21,34,55,29,48,159,102,100,432,962,172,369,2340,156,
%T A261598 2420,14763,29952,25000,18447,67716,22848,63800,21420,217854,2250,
%U A261598 35264,34944,99330,14364,1300500,0,8726016,2303910,544272,9728000,5615610,8419950
%N A261598 Product of sexagesimal digits of Fibonacci numbers in base 60 representation.
%C A261598 a(n) = product of n-th row in table A261575;
%C A261598 conjecture: a(n) = 0 for n > 3329 (empirically checked up to 36000).
%H A261598 Reinhard Zumkeller, <a href="/A261598/b261598.txt">Table of n, a(n) for n = 0..10000</a>
%H A261598  Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261598 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261598 (Haskell)
%o A261598 a261598 = product . a261575_row
%Y A261598  Cf. A261575, A000045, A246558, A261587.
%K A261598 nonn,base
%O A261598 0,4
%A A261598 _Reinhard Zumkeller_, Sep 09 2015

%I A261606
%S A261606 0,1,1,2,3,5,8,13,21,34,55,29,24,53,17,10,27,37,4,41,45,26,11,37,48,
%T A261606 25,13,38,51,29,20,49,9,58,7,5,12,17,29,46,15,1,16,17,33,50,23,13,36,
%U A261606 49,25,14,39,53,32,25,57,22,19,41,0,41,41,22,3