A134498 -id:A134498 - 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

A049667 a(n) = Fibonacci(7*n)/13.

Original entry on oeis.org

0, 1, 29, 842, 24447, 709805, 20608792, 598364773, 17373187209, 504420793834, 14645576208395, 425226130837289, 12346203370489776, 358465123875040793, 10407834795746672773, 302185674200528551210, 8773792386611074657863

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..650
Tanya Khovanova, Recursive Sequences
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (29, 1).

A column of array A028412.

Cf. A000045, A134498.

[Fibonacci(7*n)/13: n in [0..30]]; // G. C. Greubel, Dec 02 2017

a:= n-> (<<0|1>, <1|29>>^n)[1, 2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2017

Fibonacci[(7*Range[0,20])]/13 (* or *) LinearRecurrence[{29,1},{0,1},20] (* Harvey P. Dale, Sep 17 2017 *)

numlib::fibonacci(7*n)/13 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(7*n)/13 \\ Charles R Greathouse IV, Oct 07 2016

[fibonacci(7*n)/13 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1 - 29*x - x^2).
a(n) = A134498(n)/13.
a(n) = F(n, 29), the n-th Fibonacci polynomial evaluated at x=29. - T. D. Noe, Jan 19 2006
a(n) = 29*a(n-1) + a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Nov 22 2008
For n >= 1, a(n) equals the denominator of the continued fraction [29, 29, ..., 29] (with n copies of 29). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019
a(n) = ((-1)^n*7*F(n) + 14*5*F(n)^3 + (-1)^n*7*5^2*F(n)^5 + 5^3*F(n)^7)/13, n >= 0. See the general D. Jennings formula given in comment on triangle A111125, where also the reference is given. Here the fourth row (k=3) applies. - Wolfdieter Lang, Sep 01 2012
G.f.: G(0)*x/(2-29*x), where G(k)= 1 + 1/(1 - (x*(845*k-841))/((x*(845*k+4)) - 58/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
O.g.f.: x*exp(Sum_{n >= 1} Lucas(7*n)*x^n/n) = x + 29*x^2 + 842*x^3 + .... - Peter Bala, Oct 11 2019

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

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

A138473 a(n) = Fibonacci(8*n).

Original entry on oeis.org

0, 21, 987, 46368, 2178309, 102334155, 4807526976, 225851433717, 10610209857723, 498454011879264, 23416728348467685, 1100087778366101931, 51680708854858323072, 2427893228399975082453, 114059301025943970552219, 5358359254990966640871840

Offset: 0

Zerinvary Lajos, May 09 2008

Colin Barker, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (47,-1).

Cf. A000032, A000045, A049668, A134498.

[Fibonacci(8*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Fibonacci[8Range[0,20]] (* Harvey P. Dale, Jun 22 2013 *)

MuPAD
```
numlib::fibonacci(8*n) $ n = 0..25;
```

concat(0, Vec(21*x / (1 - 47*x + x^2) + O(x^30))) \\ Colin Barker, Apr 06 2017

[fibonacci(8*n) for n in range(0, 15)] # Zerinvary Lajos, May 15 2009

a(n) = Fibonacci(4*n)*Lucas(4*n) = 21*A049668(n).
G.f.: 21*x / ( 1-47*x+x^2 ). - R. J. Mathar, Sep 30 2013
From Colin Barker, Apr 06 2017: (Start)
a(n) = (47 + 21*sqrt(5))^(1-n)*(-2^n+(2207 + 987*sqrt(5))^n) / (105 + 47*sqrt(5)).
a(n) = 47*a(n-1) - a(n-2) for n > 1.
(End)

A167398 a(n) = Fibonacci(11*n).

Original entry on oeis.org

0, 89, 17711, 3524578, 701408733, 139583862445, 27777890035288, 5527939700884757, 1100087778366101931, 218922995834555169026, 43566776258854844738105, 8670007398507948658051921

Offset: 0

Zak Seidov, Nov 02 2009

nonn

Index entries for linear recurrences with constant coefficients, signature (199, 1).

Cf. A134498 Fibonacci(7n).

[Fibonacci(11*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

(*1*)Table[Fibonacci[11k],{k,0,20}]
(*2*){a,b}={0,89};Do[Print[c={a,b}.{1,199}];a=b;b=c,{20}]

a(0)=0, a(1)=89; a(n) = 199*a(n-1) + a(n-2) for n > 1.

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…

cp's OEIS Frontend

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

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

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A049667 a(n) = Fibonacci(7*n)/13.

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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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