A134503 -id:A134503 - cp's OEIS frontend

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

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

A134498 a(n) = Fibonacci(7n).

Original entry on oeis.org

0, 13, 377, 10946, 317811, 9227465, 267914296, 7778742049, 225851433717, 6557470319842, 190392490709135, 5527939700884757, 160500643816367088, 4660046610375530309, 135301852344706746049, 3928413764606871165730

Offset: 0

Artur Jasinski, Oct 28 2007

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

Cf. A000045, A134499, A134500, A134501, A134502, A134503, A134504, A049667.

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

Table[Fibonacci[7n], {n, 0, 30}]
{a,b}={0,13};Do[Print[c={a,b}.{1,29}];a=b;b=c,{30}] (* Zak Seidov, Nov 02 2009 *)

numlib::fibonacci(7*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008

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

[fibonacci(7*n) for n in range(0, 16)] # Zerinvary Lajos, May 15 2009

G.f.: -13*x / ( -1+29*x+x^2 ). a(n) = 13*A049667(n). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A008589(n)). - Michel Marcus, Nov 08 2013

Offset changed from 1 to 0 by Vincenzo Librandi, 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

A134500 a(n) = Fibonacci(7n + 2).

Original entry on oeis.org

1, 34, 987, 28657, 832040, 24157817, 701408733, 20365011074, 591286729879, 17167680177565, 498454011879264, 14472334024676221, 420196140727489673, 12200160415121876738, 354224848179261915075, 10284720757613717413913

Offset: 0

Artur Jasinski, Oct 28 2007

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

Cf. A000045, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+2], {n, 0, 30}]
```

a(n)=fibonacci(7*n+2) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: (-1-5*x) / (-1 + 29*x + x^2). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A017005(n)). - Michel Marcus, Nov 07 2013

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

A134499 a(n) = Fibonacci(7*n+1).

Original entry on oeis.org

1, 21, 610, 17711, 514229, 14930352, 433494437, 12586269025, 365435296162, 10610209857723, 308061521170129, 8944394323791464, 259695496911122585, 7540113804746346429, 218922995834555169026, 6356306993006846248183

Offset: 0

Artur Jasinski, Oct 28 2007

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

Cf. A000032, A000045, A134498, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[7 n + 1], {n, 0, 30}]
```

a(n)=fibonacci(7*n+1) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: ( -1+8*x ) / ( -1+29*x+x^2 ). - R. J. Mathar, Apr 17 2011

2*a(n) = Fibonacci(7*n) + Lucas(7*n). - Bruno Berselli, Oct 13 2017

Offset corrected by Vincenzo Librandi, Apr 16 2011

cp's OEIS Frontend

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

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A134498 a(n) = Fibonacci(7n).

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

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A134500 a(n) = Fibonacci(7n + 2).

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A134499 a(n) = Fibonacci(7*n+1).

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