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### -- Licenses: Artistic-2.0
### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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%F A212804 a(n) = Sum_{k=0..n} (C(k,n-k) - C(k,n-k-1)). - _Peter Luschny_, Oct 01 2014
%F A212804 a(n) = (2^(-1-n)*((1-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(1+sqrt(5))^n))/sqrt(5). - _Colin Barker_, Sep 25 2016
%F A212804 a(n) = A000045(n-1), n>=1. - _R. J. Mathar_, Apr 14 2018
%e A212804 From _Petros Hadjicostas_, Jan 08 2019: (Start)
%e A212804 For n=6, we have a(6) = 5. The binary necklaces of length n+1 = 7 with exactly one occurrence of 00 are as follows: 0011111, 0010111, 0011011, 0011101, and 0010101.
%e A212804 The corresponding cyclic compositions of n+1 = 7 with exactly one 1 (under MacMahon's bijection) are as follows: 1+6, 1+2+4, 1+3+3, 1+4+2, 1+2+2+2.
%e A212804 Of course, removing the 1 from the cyclic composition, we get a (linear) composition of n=6 with parts >=2 (as stated above by Joerg Arndt): 6, 2+4, 3+3, 4+2, 2+2+2. (For linear compositions, 2+4 is not the same as 4+2.)
%e A212804 (End)
%t A212804 Table[Fibonacci[n-1], {n, 0, 40}] (* _Vladimir Reshetnikov_, Sep 24 2016 *)
%o A212804 (MAGMA) [Fibonacci(n + 1) - Fibonacci(n): n in [0..50]]; // _Vincenzo Librandi_, Dec 09 2012
%Y A212804 Cf. A000045, A105809 (alternating row sums).
%Y A212804 Column k=1 of A320341.
%K A212804 nonn,easy
%O A212804 0,5
%A A212804 _N. J. A. Sloane_, May 27 2012, following a suggestion from _R. K. Guy_.

%I A104763
%S A104763 1,1,1,1,1,2,1,1,2,3,1,1,2,3,5,1,1,2,3,5,8,1,1,2,3,5,8,13,1,1,2,3,5,8,
%T A104763 13,21,1,1,2,3,5,8,13,21,34,1,1,2,3,5,8,13,21,34,55,1,1,2,3,5,8,13,21,
%U A104763 34,55,89,1,1,2,3,5,8,13,21,34,55,89,144,1,1,2,3,5,8,13,21,34,55,89,144
%N A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.
%C A104763 Triangle of A104762, Fibonacci sequence in each row starts from the right.
%C A104763 The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - _Johannes W. Meijer_, Sep 22 2010
%C A104763 Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - _Boris Putievskiy_, Dec 13 2012
%H A104763 Reinhard Zumkeller, <a href="/A104763/b104763.txt">Rows n = 1..100 of table, flattened</a>
%H A104763 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO]
%F A104763 F(1) through F(n) starting from the left in n-th row.
%F A104763 T(n,k)=A000045(k), 1<=k<=n. - _R. J. Mathar_, May 02 2008
%F A104763 a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 13 2012
%e A104763 First few rows of the triangle are:
%e A104763 1;
%e A104763 1, 1;
%e A104763 1, 1, 2;
%e A104763 1, 1, 2, 3;
%e A104763 1, 1, 2, 3, 5;
%e A104763 1, 1, 2, 3, 5, 8;
%e A104763 1, 1, 2, 3, 5, 8, 13;
%e A104763 ...
%o A104763 (Haskell)
%o A104763 a104763 n k = a104763_tabl !! (n-1) !! (k-1)
%o A104763 a104763_row n = a104763_tabl !! (n-1)
%o A104763 a104763_tabl = map (flip take $ tail a000045_list) [1..]
%o A104763 -- _Reinhard Zumkeller_, Aug 15 2013
%Y A104763 Cf. A104762, A000045.
%Y A104763 Cf. A000071 (row sums). - _R. J. Mathar_, Jul 22 2009
%Y A104763 Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - _Johannes W. Meijer_, Sep 22 2010
%K A104763 nonn,tabl,easy
%O A104763 1,6
%A A104763 _Gary W. Adamson_, Mar 23 2005
%E A104763 Edited by _R. J. Mathar_, May 02 2008
%E A104763 Extended by _R. J. Mathar_, Aug 27 2008

%I A240733
%S A240733 1,1,2,3,5,8,13,21,32,50,78,121,187,289,448,693,1072,1658,2564,3966,
%T A240733 6134,9487,14673,22695,35101,54288,83964,129862,200850,310643,480452,
%U A240733 743085,1149282,1777523,2749182,4251987,6576279,10171116,15731022,24330178,37629950
%N A240733 Floor(6^n/(2+2*cos(Pi/9))^n).
%C A240733 a(n) is the perimeter (rounded down) of a nonaflake after n iterations, let a(0) = 1. The total number of sides is 9*A000400(n). The total number of holes is A002452(n). 2*cos(Pi/9) = 1.87938524... = diagonal b of nonagon (see comments in A123609).
%H A240733 Vincenzo Librandi, <a href="/A240733/b240733.txt">Table of n, a(n) for n = 0..1000</a>
%H A240733 Kival Ngaokrajang, <a href="/A240733/a240733_1.pdf">Illustration of nonaflake for n = 0..3</a>
%H A240733 Wikipedia, <a href="http://en.wikipedia.org/wiki/N-flake">n-flake</a>
%p A240733 A240733:=n->floor(6^n/(2+2*cos(Pi/9))^n); seq(A240733(n), n=0..50); # _Wesley Ivan Hurt_, Apr 12 2014
%t A240733 Table[Floor[6^n/(2 + 2*Cos[Pi/9])^n], {n, 0, 50}] (* _Wesley Ivan Hurt_, Apr 12 2014 *)
%o A240733 (PARI) {a(n)=floor(6^n/(2+2*cos(Pi/9))^n)}
%o A240733        for (n=0, 100, print1(a(n), ", "))
%Y A240733 Cf. A000400, A002452, A123609, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).
%K A240733 nonn,easy
%O A240733 0,3
%A A240733 _Kival Ngaokrajang_, Apr 11 2014

%I A177194
%S A177194 1,1,2,3,5,8,13,21,34,55,89,144,233,377,987,1597,2584,4181,6765,17711,
%T A177194 28657,46368,121393,196418,317811,514229,1346269,3524578,9227465,
%U A177194 24157817,63245986,267914296,433494437,53316291173,86267571272
%N A177194 Fibonacci numbers whose decimal expansion does not contain any digit 0.
%C A177194 The probability that fib(n) contains no 0's decreases to zero as n goes to infinity. Its maximum value seems to be F(184) having 39 digits, including no zeros.
%F A177194 a(n) = A000045(A076564(n)). [From _R. J. Mathar_, Oct 18 2010]
%e A177194 a(7)=13 since fib(7) does not contain the digit 0.
%Y A177194 Cf. A176253, A000045
%K A177194 nonn,base
%O A177194 1,3
%A A177194 _Carmine Suriano_, May 04 2010

%I A005181 M0693
%S A005181 1,1,2,3,5,8,13,21,34,55,91,149,245,404,666,1097,1809,2981,4915,8104,
%T A005181 13360,22027,36316,59875,98716,162755,268338,442414,729417,1202605,
%U A005181 1982760,3269018,5389699,8886111,14650720,24154953,39824785,65659970,108254988,178482301
%N A005181 a(n) = ceiling(exp((n-1)/2)).
%C A005181 This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - _Alonso del Arte_, Mar 18 2013
%D A005181 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005181 I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.
%H A005181 Vladimir Pletser, <a href="/A005181/b005181.txt">Table of n, a(n) for n = 0..1000</a>
%H A005181 R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H A005181 R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.
%H A005181 R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H A005181 I. Stewart, <a href="http://www.whydomath.org/Reading_Room_Material/ian_stewart/9505.html">Fibonacci Forgeries</a>
%H A005181 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>
%H A005181 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers.</a>
%F A005181 Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - _Alois P. Heinz_, Feb 19 2019
%p A005181 seq(round(ceil(exp((n-1)/2))), n=0..50); # _Vladimir Pletser_, Sep 15 2013;
%t A005181 Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* _Alonso del Arte_, Mar 18 2013 *)
%o A005181 (Python)
%o A005181 import math
%o A005181 for n in range(99):
%o A005181   print str(int(math.ceil(math.e**((n-1)*0.5))))+',',
%o A005181 # _Alex Ratushnyak_, Mar 18 2013
%Y A005181 Cf. A000045, A019774.
%K A005181 nonn
%O A005181 0,3
%A A005181 _N. J. A. Sloane_, _R. K. Guy_
%E A005181 A few more terms from _Alonso del Arte_, Mar 18 2013

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 11-20 of 92

%I A177246
%S A177246 1,1,2,3,5,8,13,21,55,89,233,377,610,987,1597,6765,17711,28657,75025,
%T A177246 121393,317811,2178309,5702887,39088169,1836311903,2971215073,
%U A177246 12586269025,32951280099,53316291173,86267571272,591286729879
%N A177246 Fibonacci numbers whose decimal expansion does not contain any digit 4.
%C A177246 Probability that Fib(n) contains no 4's goes to zero as n grows to infinity. I suppose that the maximum number is Fib(114) having 24 digits, none of them being a "4".
%H A177246 Robert Israel, <a href="/A177246/b177246.txt">Table of n, a(n) for n = 1..41</a> (conjectured to be complete)
%e A177246 a(9)=55 is the 9th Fibonacci having no digit 4's.
%p A177246 remove(t -> has(convert(t,base,10),4), map(combinat:-fibonacci, [$1..1000])); # _Robert Israel_, Dec 13 2018
%t A177246 Select[Fibonacci@Range@114, !MemberQ[IntegerDigits[#], 4] &] (* _Amiram Eldar_, Dec 13 2018 *)
%Y A177246 Cf. A000045, A177194, A177195, A177231, A177245, A176253.
%K A177246 nonn,base
%O A177246 1,3
%A A177246 _Carmine Suriano_, May 06 2010

%I A261575
%S A261575 0,1,1,2,3,5,8,13,21,34,55,29,1,24,2,53,3,17,6,10,10,27,16,37,26,4,43,
%T A261575 41,9,1,45,52,1,26,2,3,11,55,4,37,57,7,48,52,12,25,50,20,13,43,33,38,
%U A261575 33,54,51,16,28,1,29,50,22,2,20,7,51,3,49,57,13,6,9
%N A261575 Table of Fibonacci numbers in base 60 representation: row n contains the sexagesimal digits of A000045(n) in reversed order.
%C A261575 A261585(n) = length of n-th row;
%C A261575 T(n,0) = A261606(n) = in base 60: last sexagesimal digit of A000045(n);
%C A261575 T(n,A261607(n)-1) = A261607(n) = in base 60: initial  sexagesimal digit of A000045(n);
%C A261575 A000045(n) = sum(T(n,k)*60^k : k = 0..A261585(n)-1).
%H A261575 Reinhard Zumkeller, <a href="/A261575/b261575.txt">Rows n = 0..1000 of triangle, flattened</a>
%H A261575 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261575 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%e A261575 A000045(42) = 20*60^4 + 40*60^3 + 20*60^2 + 38*60^1 + 16*60^0 = 267914296.
%e A261575 . ----------------------------------------------------------------------
%e A261575 .   n | T(n,*)       n | T(n,*)             n | T(n,*)
%e A261575 . ----+---------   ----+---------------   ----+-------------------------
%e A261575 .   0 | [0]         21 | [26,2,3]          42 | [16,38,20,40,20]
%e A261575 .   1 | [1]         22 | [11,55,4]         43 | [17,7,55,26,33]
%e A261575 .   2 | [1]         23 | [37,57,7]         44 | [33,45,15,7,54]
%e A261575 .   3 | [2]         24 | [48,52,12]        45 | [50,52,10,34,27,1]
%e A261575 .   4 | [3]         25 | [25,50,20]        46 | [23,38,26,41,21,2]
%e A261575 .   5 | [5]         26 | [13,43,33]        47 | [13,31,37,15,49,3]
%e A261575 .   6 | [8]         27 | [38,33,54]        48 | [36,9,4,57,10,6]
%e A261575 .   7 | [13]        28 | [51,16,28,1]      49 | [49,40,41,12,0,10]
%e A261575 .   8 | [21]        29 | [29,50,22,2]      50 | [25,50,45,9,11,16]
%e A261575 .   9 | [34]        30 | [20,7,51,3]       51 | [14,31,27,22,11,26]
%e A261575 .  10 | [55]        31 | [49,57,13,6]      52 | [39,21,13,32,22,42]
%e A261575 .  11 | [29,1]      32 | [9,5,5,10]        53 | [53,52,40,54,33,8,1]
%e A261575 .  12 | [24,2]      33 | [58,2,19,16]      54 | [32,14,54,26,56,50,1]
%e A261575 .  13 | [53,3]      34 | [7,8,24,26]       55 | [25,7,35,21,30,59,2]
%e A261575 .  14 | [17,6]      35 | [5,11,43,42]      56 | [57,21,29,48,26,50,4]
%e A261575 .  15 | [10,10]     36 | [12,19,7,9,1]     57 | [22,29,4,10,57,49,7]
%e A261575 .  16 | [27,16]     37 | [17,30,50,51,1]   58 | [19,51,33,58,23,40,12]
%e A261575 .  17 | [37,26]     38 | [29,49,57,0,3]    59 | [41,20,38,8,21,30,20]
%e A261575 .  18 | [4,43]      39 | [46,19,48,52,4]   60 | [0,12,12,7,45,10,33]
%e A261575 .  19 | [41,9,1]    40 | [15,9,46,53,7]    61 | [41,32,50,15,6,41,53]
%e A261575 .  20 | [45,52,1]   41 | [1,29,34,46,12]   62 | [41,44,2,23,51,51,26,1]
%o A261575 (Haskell)
%o A261575 a261575 n k = a261575_tabf !! n !! k
%o A261575 a261575_row n = a261575_tabf !! n
%o A261575 a261575_tabf = [0] : [1] :
%o A261575    zipWith (add 0) (tail a261575_tabf) a261575_tabf where
%o A261575    add c (a:as) (b:bs) = y : add c' as bs where (c', y) = divMod (a+b+c) 60
%o A261575    add c (a:as) [] = y : add c' as [] where (c', y) = divMod (a+c) 60
%o A261575    add 1 _ _ = [1]
%o A261575    add _ _ _ = []
%Y A261575 Cf. A000045, A261585 (row lengths), A261587 (row sums), A261598 (row products), A261606 (left edge), A261607 (right edge).
%K A261575 nonn,tabf,base
%O A261575 0,4
%A A261575 _Reinhard Zumkeller_, Sep 09 2015

%I A000044 M0691 N0255
%S A000044 1,1,1,2,3,5,8,13,21,34,55,89,144,232,375,606,979,1582,2556,4130,6673,
%T A000044 10782,17421,28148,45480,73484,118732,191841,309967,500829,809214,
%U A000044 1307487,2112571,3413385,5515174,8911138,14398164,23263822,37588502,60733592,98130253,158553878,256183302,413927966,668803781,1080619176,1746009572,2821113574,4558212008
%N A0000