A033888 -id:A033888 - cp's OEIS frontend

A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 243, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3

Offset: 1

Benoit Cloitre, Sep 04 2002

nonn

If m == 1, 2 or 3 (mod 4) then F(m) is not divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A000203, A038500, A033888, A051064, A074724, A088838.

Table[3^IntegerExponent[Fibonacci[4n],3],{n,100}] (* Harvey P. Dale, Jun 03 2012 *)

a(n) = 3^valuation(fibonacci(4*n), 3); \\ Michel Marcus, May 13 2022

If k == 1 or 2 (mod 3) then a(3^m*k) = 3^(m+1) for m>=0.
a(n) = A038500(A033888(n)). - Amiram Eldar, May 13 2022
a(n) = 3^A051064(n) (conjectured). - Michel Marcus, May 17 2022
Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - Peter Bala, Jun 10 2022

A099922 a(n) = F(4n) - 2n, where F(n) = Fibonacci numbers A000045.

Original entry on oeis.org

1, 17, 138, 979, 6755, 46356, 317797, 2178293, 14930334, 102334135, 701408711, 4807526952, 32951280073, 225851433689, 1548008755890, 10610209857691, 72723460248107, 498454011879228, 3416454622906669, 23416728348467645

Offset: 1

Ralf Stephan, Nov 01 2004

nonn

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 54.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-16,9,-1)

Equals A033888(n) - 2n. Partial sums of A081071. Bisection of A054452.

Rest[CoefficientList[Series[x*(1+8x+x^2)/((1-x)^2*(1-7x+x^2)), {x, 0, 20}], x]] (* Georg Fischer, May 24 2019 *)

Vec(x*(1 + 8*x + x^2) / ((1 - x)^2*(1 - 7*x + x^2)) + O(x^25)) \\ Colin Barker, May 25 2019

G.f.: x*(1+8*x+x^2)/((1-x)^2 * (1-7*x+x^2)). [Corrected for offset by Georg Fischer, May 24 2019]
a(n) = Sum_{k=1..n} Lucas(2k-1)^2.
From Colin Barker, May 25 2019: (Start)
a(n) = (-((7-3*sqrt(5))/2)^n + ((7+3*sqrt(5))/2)^n)/sqrt(5) - 2*n.
a(n) = 9*a(n-1) - 16*a(n-2) + 9*a(n-3) - a(n-4) for n>4.
(End)

A269500 a(n) = Fibonacci(10*n).

Original entry on oeis.org

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605

Offset: 0

Ilya Gutkovskiy, Mar 03 2016

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (123,-1)

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).

Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]

a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016

concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

G.f.: 55*x/(1 - 123*x + x^2).
a(n) = 123*a(n-1) - a(n-2).
a(n) = A000045(10*n).
Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…

A081074 Fibonacci(4n)-3, or Fibonacci(2n-2)*Lucas(2n+2).

Original entry on oeis.org

0, 18, 141, 984, 6762, 46365, 317808, 2178306, 14930349, 102334152, 701408730, 4807526973, 32951280096, 225851433714, 1548008755917, 10610209857720, 72723460248138, 498454011879261, 3416454622906704, 23416728348467682

Offset: 1

R. K. Guy, Mar 04 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

[Fibonacci(4*n)-3: n in [1..50]]; // Vincenzo Librandi, Apr 20 2011

with(combinat): for n from 1 to 40 do printf(`%d,`,fibonacci(4*n)-3) od: # James Sellers, Mar 05 2003

Fibonacci[4Range[25]]-3 (* or *)
LinearRecurrence[{8,-8,1},{0,18,141},25] (* Paolo Xausa, Jan 07 2024 *)

a(n) = 8a(n-1) - 8a(n-2) + a(n-3).

G.f.: 3*x^2*(-6+x) / ( (x-1)*(x^2-7*x+1) ). a(n) = A033888(n)-3. - R. J. Mathar, Sep 03 2010

More terms from James Sellers, Mar 05 2003

A114182 F(4n) - 2n - 1 where F(n) = Fibonacci numbers. Also, the floor of the log base phi of sequence A090162 (phi = (1+Sqrt(5))/2).

Original entry on oeis.org

0, 16, 137, 978, 6754, 46355, 317796, 2178292, 14930333, 102334134, 701408710, 4807526951, 32951280072, 225851433688, 1548008755889, 10610209857690, 72723460248106, 498454011879227, 3416454622906668, 23416728348467644

Offset: 1

Greg Huber, Feb 04 2006

Index entries for linear recurrences with constant coefficients, signature (9,-16,9,-1).

Cf. A033888.

Table[Fibonacci[4n]-2n-1,{n,20}] (* or *) LinearRecurrence[{9,-16,9,-1},{0,16,137,978},20] (* Harvey P. Dale, May 28 2015 *)

G.f. x^2*(16-7*x+x^2) / ( (x^2-7*x+1)*(x-1)^2 ). - R. J. Mathar, Oct 19 2012

a(0)=0, a(1)=16, a(2)=137, a(3)=978, a(n)=9*a(n-1)-16*a(n-2)+ 9*a(n-3)- a(n-4). - Harvey P. Dale, May 28 2015

Corrected by R. J. Mathar, Oct 19 2012

A142880 a(n) = 7*a(n-3) - a(n-6).

Original entry on oeis.org

0, 1, 2, 3, 8, 13, 21, 55, 89, 144, 377, 610, 987, 2584, 4181, 6765, 17711, 28657, 46368, 121393, 196418, 317811, 832040, 1346269, 2178309, 5702887, 9227465, 14930352, 39088169, 63245986, 102334155, 267914296, 433494437, 701408733, 1836311903

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 28 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-1).

Cf. A119016, A002965, A002530, A048788.

a[0] = 0; a[1] = 1;
a[n_] := a[n] = If[Mod[n, 3] == 1, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 0, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]];
Table[a[n], {n, 0, 50}]
LinearRecurrence[{0,0,7,0,0,-1},{0,1,2,3,8,13},40] (* Harvey P. Dale, Jul 17 2021 *)

G.f.: -x*(1+x)*(x^3 - 2*x^2 - x - 1) / ( 1 - 7*x^3 + x^6 ).
a(3n) = A033888(n).
a(3n+1) = A033890(n).
a(3n+2)= A033891(n).
a(n) = 2*a(n-1) + a(n-2) if n == 1 (mod 3).
a(n) = a(n-1) + a(n-2) if n == 0 (mod 3).
a(n) = 2*a(n-1) - a(n-2) if n == 2 (mod 3).

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A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

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A099922 a(n) = F(4n) - 2n, where F(n) = Fibonacci numbers A000045.

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A269500 a(n) = Fibonacci(10*n).

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A081074 Fibonacci(4n)-3, or Fibonacci(2n-2)*Lucas(2n+2).

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A114182 F(4n) - 2n - 1 where F(n) = Fibonacci numbers. Also, the floor of the log base phi of sequence A090162 (phi = (1+Sqrt(5))/2).

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A142880 a(n) = 7*a(n-3) - a(n-6).

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