A134504 -id:A134504 - 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

A134494 a(n) = Fibonacci(6n+2).

Original entry on oeis.org

1, 21, 377, 6765, 121393, 2178309, 39088169, 701408733, 12586269025, 225851433717, 4052739537881, 72723460248141, 1304969544928657, 23416728348467685, 420196140727489673, 7540113804746346429, 135301852344706746049, 2427893228399975082453

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, A001906, A001519, A015448, A014445, A033887-A033891, A049310, A049660, A099100, A102312, A103134, A134490 - A134504.

[Fibonacci(6*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

seq( combinat[fibonacci](6*n+2),n=0..10) ; # R. J. Mathar, Apr 17 2011

Table[Fibonacci[6n+2], {n, 0, 30}]
Table[ChebyshevU[3*n, 3/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n)=fibonacci(6*n+2) \\ Charles R Greathouse IV, Jun 11 2015

Vec((1+3*x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: ( 1+3*x ) / ( 1-18*x+x^2 ).
a(n) = 3*A049660(n)+A049660(n+1). (End)
a(n) = A000045(A016933(n)). - Michel Marcus, Nov 07 2013
a(n) = ((5-3*sqrt(5)+(5+3*sqrt(5))*(9+4*sqrt(5))^(2*n)))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = S(3*n, 3) = S(n,18) + 3*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

Index in definition corrected by T. D. Noe, Joerg Arndt, 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

A134501 a(n) = Fibonacci(7n + 3).

Original entry on oeis.org

2, 55, 1597, 46368, 1346269, 39088169, 1134903170, 32951280099, 956722026041, 27777890035288, 806515533049393, 23416728348467685, 679891637638612258, 19740274219868223167, 573147844013817084101, 16641027750620563662096

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-A134495, A103134, A134497 - A134504.

[Fibonacci(7*n+3): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[7n+3], {n, 0, 30}]
```

a(n)=fibonacci(7*n+3) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-2+3*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n+1) - 3*A049667(n). (End)
a(n) = A000045(A017017(n)). - Michel Marcus, Nov 07 2013

Offset changed to 0 by Vincenzo Librandi, Apr 16 2011

A134502 a(n) = Fibonacci(7n + 4).

Original entry on oeis.org

3, 89, 2584, 75025, 2178309, 63245986, 1836311903, 53316291173, 1548008755920, 44945570212853, 1304969544928657, 37889062373143906, 1100087778366101931, 31940434634990099905, 927372692193078999176, 26925748508234281076009

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-A134495, A103134, A134497-A134504.

[Fibonacci(7*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+4], {n, 0, 30}]
```

a(n)=fibonacci(7*n+4) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-3-2*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n) + 3*A049667(n+1). (End)
a(n) = A000045(A017029(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

A134503 a(n) = Fibonacci(7n + 5).

Original entry on oeis.org

5, 144, 4181, 121393, 3524578, 102334155, 2971215073, 86267571272, 2504730781961, 72723460248141, 2111485077978050, 61305790721611591, 1779979416004714189, 51680708854858323072, 1500520536206896083277

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, A134504.

[Fibonacci(7*n+5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n+5], {n, 0, 30}]
LinearRecurrence[{29,1},{5,144},20] (* Harvey P. Dale, Apr 24 2017 *)

a(n)=fibonacci(7*n+5) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Apr 17 2011: (Start)
G.f.: (-5+x) / (-1 + 29*x + x^2).
a(n) = 5*A049667(n+1) - A049667(n). (End)

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

A134489 a(n) = Fibonacci(5*n + 2).

Original entry on oeis.org

1, 13, 144, 1597, 17711, 196418, 2178309, 24157817, 267914296, 2971215073, 32951280099, 365435296162, 4052739537881, 44945570212853, 498454011879264, 5527939700884757, 61305790721611591, 679891637638612258

Offset: 0

Artur Jasinski, Oct 28 2007

The o.g.f. of {F(m*n + 2)}_{n>=0}, for m = 1, 2, ..., is

G(m,x) = (1 + F(m - 2)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and F(-1) = 1, and L = A000032. - Wolfdieter Lang, Feb 06 2023

Index entries for linear recurrences with constant coefficients, signature (11,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888-A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(5*n+2): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

Table[Fibonacci[5n + 2], {n, 0, 30}]
LinearRecurrence[{11,1},{1,13},20] (* Harvey P. Dale, May 05 2022 *)

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-1-2*x) / (-1 + 11*x + x^2).
a(n) = 2*A049666(n) + A049666(n+1). (End)
a(n) = A000045(A016873(n)). - Michel Marcus, Nov 05 2013

cp's OEIS Frontend

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

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

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A134494 a(n) = Fibonacci(6n+2).

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

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A134501 a(n) = Fibonacci(7n + 3).

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A134502 a(n) = Fibonacci(7n + 4).

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

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