A134491 -id:A134491 - 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

A134490 a(n) = Fibonacci(5n + 3).

Original entry on oeis.org

2, 21, 233, 2584, 28657, 317811, 3524578, 39088169, 433494437, 4807526976, 53316291173, 591286729879, 6557470319842, 72723460248141, 806515533049393, 8944394323791464, 99194853094755497, 1100087778366101931

Offset: 0

Artur Jasinski, Oct 28 2007

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

Cf. A000045, A102312, A099100, A134491.

[Fibonacci(5*n+3): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[5n + 3], {n, 0, 30}]
```

a(n)=fibonacci(5*n+3) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = 11a(n-1) + a(n-2). - Paul Curtz, May 07 2008
From R. J. Mathar, Nov 02 2008: (Start)
G.f.: (2-x)/(1 - 11x - x^2).
a(n) = |A122574(n+2)| + A049666(n+1). (End)
a(n) = A000045(A016885(n)). - Michel Marcus, Nov 08 2013

Offset changed to 0 by R. J. Mathar, Jul 28 2008

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

A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 2, 0, -2, -1, 3, 2, -2, -3, 0, 5, 0, -5, -3, 3, 5, -5, -8, 0, 8, 0, -13, -8, 8, 13, -13, -21, 0, 21, 13, -34, -21, 21, 34, 0, -55, 0, 55, 34, -34, -55, 55, 89, 0, -89, 0, 144, 89, -89, -144, 144, 233, 0, -233, -144, 377, 233, -233, -377, 0, 610, 0, -610, -377, 377

Offset: 0

Paul Curtz, May 04 2008

Ignoring signs, the sequence contains A000045(2)=1 ten times and each of the following Fibonacci numbers A000045(i>2) four times.

			All 5 rows of the table T(d,n) are:
.0,.0,.0,.1,.3,.5,.5,..0,-13,-34,-55,-55,...0,.144,...
.0,.0,.1,.2,.2,.0,-5,-13,-21,-21,..0,.55,.144,.233,...
.0,.1,.1,.0,-2,-5,-8,.-8,..0,.21,.55,.89,..89,...0,...
.1,.0,-1,-2,-3,-3,.0,..8,.21,.34,.34,..0,.-89,-233,...
-1,-1,-1,-1,.0,.3,.8,.13,.13,..0,-34,-89,-144,-144,...

Cf. A102312 (A049666), A099100, A134489, A134490, A134491.

T(0,n)=A138112(n). T(d,n)= T(d-1,n+1)-T(d-1,n), d=1..4.
T(1,n)=A100334(n-1). T(2,n)=A103311(n). T(3,n) = -A138003(n-2). T(4,n)= -A105371(n).
sum_(d=0..4) T(d,n)=0 (columns sum to zero).

Edited by R. J. Mathar, Jul 04 2008

cp's OEIS Frontend

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

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A134490 a(n) = Fibonacci(5n + 3).

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

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A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

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