A134494 -id:A134494 - cp's OEIS frontend

A134492 a(n) = Fibonacci(6*n).

0, 8, 144, 2584, 46368, 832040, 14930352, 267914296, 4807526976, 86267571272, 1548008755920, 27777890035288, 498454011879264, 8944394323791464, 160500643816367088, 2880067194370816120, 51680708854858323072, 927372692193078999176, 16641027750620563662096

Offset: 0

Artur Jasinski, Oct 28 2007

All terms are divisible by 8. - Alonso del Arte, Jul 27 2013

Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - Felix Huber, Nov 03 2023

Colin Barker, Table of n, a(n) for n = 0..500
Hacène Belbachir, Soumeya Merwa Tebtoub and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A000032, A000045, A008588, A049660, A079343, A014445, A014448, A134493, A134494, A134495, A103134, A134497, A134498.

[Fibonacci(6*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n], {n, 0, 30}]
LinearRecurrence[{18,-1},{0,8},30] (* Harvey P. Dale, Aug 15 2017 *)

numlib::fibonacci(6*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(6*n) \\ Charles R Greathouse IV, Sep 16 2015

concat(0, Vec(8*x/(1-18*x+x^2) + O(x^20))) \\ Colin Barker, Jan 24 2016

[fibonacci(6*n) for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

a(n) = 18*a(n-1) - a(n-2) = 8*A049660(n). G.f.: 8*x/(1-18*x+x^2). - R. J. Mathar, Feb 16 2010
a(n) = A000045(A008588(n)). - Michel Marcus, Nov 08 2013
a(n) = ((-1+(9+4*sqrt(5))^(2*n)))/(sqrt(5)*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = L(2n-1) * F(2n+1)^2 + L(2n+1) * F(2n-1)^2, where F(n) = A000045(n) and L(n) = A000032(n). - Diego Rattaggi, Nov 12 2020
a(n) = Fibonacci(3*n) * Lucas(3*n) = A000045(3*n) * A000032(3*n) = A014445(n) * A014448(n). - Amiram Eldar, Jan 11 2022

Offset corrected by R. J. Mathar, Feb 16 2010

A103134 a(n) = Fibonacci(6n+4).

Original entry on oeis.org

3, 55, 987, 17711, 317811, 5702887, 102334155, 1836311903, 32951280099, 591286729879, 10610209857723, 190392490709135, 3416454622906707, 61305790721611591, 1100087778366101931, 19740274219868223167, 354224848179261915075, 6356306993006846248183

Offset: 0

Creighton Dement, Jan 24 2005

Gives those numbers which are Fibonacci numbers in A103135.
Generally, for any sequence where a(0)= Fibonacci(p), a(1) = F(p+q) and Lucas(q)*a(1) +- a(0) = F(p+2q), then a(n) = L(q)*a(n-1) +- a(n-2) generates the following Fibonacci sequence: a(n) = F(q(n)+p). So for this sequence, a(n) = 18*a(n-1) - a(n-2) = F(6n+4): q=6, because 18 is the 6th Lucas number (L(0) = 2, L(1)=1); F(4)=3, F(10)=55 and F(16)=987 (F(0)=0 and F(1)=1). See Lucas sequence A000032. This is a special case where a(0) and a(1) are increasing Fibonacci numbers and Lucas(m)*a(1) +- a(0) is another Fibonacci. - Bob Selcoe, Jul 08 2013
a(n) = x + y where x and y are solutions to x^2 = 5*y^2 - 1. (See related sequences with formula below.) - Richard R. Forberg, Sep 05 2013

Colin Barker, Table of n, a(n) for n = 0..750
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Subsequence of A033887.
Cf. A000032, A000045, A001906, A001519, A015448, A014445, A033888, A033889, A033890, A033891, A049310, A049660, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.
Cf. A103135.

[Fibonacci(6*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n+4], {n, 0, 30}]
LinearRecurrence[{18,-1},{3,55},20] (* Harvey P. Dale, Mar 29 2023 *)
Table[ChebyshevU[3*n+1, 3/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n)=fibonacci(6*n+4) \\ Charles R Greathouse IV, Feb 05 2013

G.f.: (x+3)/(x^2-18*x+1).
a(n) = 18*a(n-1) - a(n-2) for n>1; a(0)=3, a(1)=55. - Philippe Deléham, Nov 17 2008
a(n) = A007805(n) + A075796(n), as follows from comment above. - Richard R. Forberg, Sep 05 2013
a(n) = ((15-7*sqrt(5)+(9+4*sqrt(5))^(2*n)*(15+7*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = S(3*n+1, 3) = 3*S(n,18) + S(n-1,18), with the Chebyshev S polynomials (A049310), S(-1, x) = 0, and S(n, 18) = A049660(n+1). - Wolfdieter Lang, May 08 2023

Edited by N. J. A. Sloane, Aug 10 2010

A134497 a(n) = Fibonacci(6n+5).

Original entry on oeis.org

5, 89, 1597, 28657, 514229, 9227465, 165580141, 2971215073, 53316291173, 956722026041, 17167680177565, 308061521170129, 5527939700884757, 99194853094755497, 1779979416004714189, 31940434634990099905, 573147844013817084101, 10284720757613717413913

Offset: 0

Artur Jasinski, Oct 28 2007

Colin Barker, Table of n, a(n) for n = 0..750
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A049660, A101858, A103134, A134492, A134493, A134494, A134495.

[Fibonacci(6*n +5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n+5], {n, 0, 30}]
Take[Fibonacci[Range[100]],{5,-1,6}] (* Harvey P. Dale, Jun 18 2013 *)

a(n)=fibonacci(6*n+5) \\ Charles R Greathouse IV, Jun 11 2015

Vec((5-x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

G.f.: ( 5-x ) / ( 1-18*x+x^2 ). a(n) = 5*A049660(n+1)-A049660(n). - R. J. Mathar, Apr 17 2011
a(n) = A000045(A016969(n)). - Michel Marcus, Nov 08 2013
a(n) = ((25-11*sqrt(5)+(9+4*sqrt(5))^(2*n)*(25+11*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = 5*S(n, 18) - S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018
From Peter Bala, Aug 11 2022: (Start)
Let n ** m =  n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 5 ** 5 ** ... ** 5 (n+1 factors).
a(2*n+1) = a(n) ** a(n) = Fibonacci(12*n+11); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(18*n+17) and so on. (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A134493 a(n) = Fibonacci(6*n+1).

Original entry on oeis.org

1, 13, 233, 4181, 75025, 1346269, 24157817, 433494437, 7778742049, 139583862445, 2504730781961, 44945570212853, 806515533049393, 14472334024676221, 259695496911122585, 4660046610375530309, 83621143489848422977, 1500520536206896083277, 26925748508234281076009

Offset: 0

Artur Jasinski, Oct 28 2007

For positive n, a(n) equals (-1)^n times the permanent of the (6n)X(6n) tridiagonal matrix with ((-1)^(1/6))'s along the three central diagonals. - John M. Campbell, Jul 12 2011

a(n) = x + y where those two values are solutions to: x^2 = 5*y^2 + 1. (See related sequences with formula below). - Richard R. Forberg, Sep 05 2013

Colin Barker, Table of n, a(n) for n = 0..750
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000032, A000045, A049660, A134492, A134494, A134495, A103134, A134497.

[Fibonacci(6*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[6n+1], {n, 0, 30}]
```

a(n)=fibonacci(6*n+1) \\ Charles R Greathouse IV, Jul 15 2011

Vec((1-5*x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

From R. J. Mathar, Apr 17 2011: (Start)
G.f.: ( 1-5*x ) / ( 1-18*x+x^2 ).
a(n) = A049660(n+1) - 5*A049660(n). (End)
a(n) = Fibonacci(3*n+1)^2 + Fibonacci(3*n)^2. - Gary Detlefs, Oct 12 2011
a(n) = 18*a(n-1) - a(n-2). - Richard R. Forberg, Sep 05 2013
a(n) = A060645(n) + A023039(n), as derives from comment above. - Richard R. Forberg, Sep 05 2013
a(n) = ((5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n)))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
2*a(n) = Fibonacci(6*n) + Lucas(6*n). - Bruno Berselli, Oct 13 2017
a(n) = S(n, 18) - 5*S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018

Offset changed to 0 by Vincenzo Librandi, Apr 16 2011

A134495 a(n) = Fibonacci(6n + 3).

Original entry on oeis.org

2, 34, 610, 10946, 196418, 3524578, 63245986, 1134903170, 20365011074, 365435296162, 6557470319842, 117669030460994, 2111485077978050, 37889062373143906, 679891637638612258, 12200160415121876738

Offset: 0

Artur Jasinski, Oct 28 2007

From Tanya Khovanova, Jan 06 2023: (Start)
Fibonacci(6n+3) are divisible by 2 but not by 4.
These numbers are not divisible by 3. (End)

Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, 2*A007805, A049660, A134492, A134493, A134494, A103134.

Subsequence of A091999.

[Fibonacci(6*n +3): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n+3], {n, 0, 30}]
LinearRecurrence[{18,-1},{2,34},20] (* Harvey P. Dale, Jul 28 2018 *)

a(n)=fibonacci(6*n+3) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Apr 17 2011: (Start)
G.f.: (2-2*x) / (1 - 18*x + x^2).
a(n) = 2*A007805(n). (End)
a(n) = A000045(A016945(n)). - Michel Marcus, Nov 08 2013
a(n) = 2*(S(n, 18) - S(n-1, 18)), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018

Index in definition and offset corrected by R. J. Mathar, Apr 17 2011

A134504 a(n) = Fibonacci(7n + 6).

Original entry on oeis.org

8, 233, 6765, 196418, 5702887, 165580141, 4807526976, 139583862445, 4052739537881, 117669030460994, 3416454622906707, 99194853094755497, 2880067194370816120, 83621143489848422977, 2427893228399975082453

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n +6): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n+6], {n, 0, 30}]
LinearRecurrence[{29,1},{8,233},20] (* Harvey P. Dale, Jul 21 2021 *)

a(n)=fibonacci(7*n+6) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: (-8-x) / (-1 + 29*x + x^2). - R. J. Mathar, Jul 04 2011
a(n) = A000045(A017053(n)). - Michel Marcus, Nov 08 2013
a(n) = 29*a(n-1) + a(n-2). - Wesley Ivan Hurt, Mar 15 2023

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

cp's OEIS Frontend

A134492 a(n) = Fibonacci(6*n).

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A103134 a(n) = Fibonacci(6n+4).

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A134497 a(n) = Fibonacci(6n+5).

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A134493 a(n) = Fibonacci(6*n+1).

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A134495 a(n) = Fibonacci(6n + 3).

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A134504 a(n) = Fibonacci(7n + 6).

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