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Search: a134501 -id:a134501
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A103134
a(n) = Fibonacci(6n+4).
+10
12
3, 55, 987, 17711, 317811, 5702887, 102334155, 1836311903, 32951280099, 591286729879, 10610209857723, 190392490709135, 3416454622906707, 61305790721611591, 1100087778366101931, 19740274219868223167, 354224848179261915075, 6356306993006846248183
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
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.
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [Fibonacci(6*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(6*n+4) \\ Charles R Greathouse IV, Feb 05 2013
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Jan 24 2005
EXTENSIONS
Edited by N. J. A. Sloane, Aug 10 2010
STATUS
approved
A134498
a(n) = Fibonacci(7n).
+10
10
0, 13, 377, 10946, 317811, 9227465, 267914296, 7778742049, 225851433717, 6557470319842, 190392490709135, 5527939700884757, 160500643816367088, 4660046610375530309, 135301852344706746049, 3928413764606871165730
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (29,1)
FORMULA
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
MATHEMATICA
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 *)
PROG
(MuPAD) numlib::fibonacci(7*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008
(Sage) [fibonacci(7*n) for n in range(0, 16)] # Zerinvary Lajos, May 15 2009
(Magma) [Fibonacci(7*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(7*n) \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
Cf. A000045, A134499, A134500, A134501, A134502, A134503, A134504, A049667.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011
STATUS
approved
A134504
a(n) = Fibonacci(7n + 6).
+10
10
8, 233, 6765, 196418, 5702887, 165580141, 4807526976, 139583862445, 4052739537881, 117669030460994, 3416454622906707, 99194853094755497, 2880067194370816120, 83621143489848422977, 2427893228399975082453
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..14.
Index entries for linear recurrences with constant coefficients, signature (29,1).
FORMULA
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
MATHEMATICA
Table[Fibonacci[7n+6], {n, 0, 30}]
LinearRecurrence[{29, 1}, {8, 233}, 20] (* Harvey P. Dale, Jul 21 2021 *)
PROG
(Magma) [Fibonacci(7*n +6): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(7*n+6) \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011
STATUS
approved
A134500
a(n) = Fibonacci(7n + 2).
+10
6
1, 34, 987, 28657, 832040, 24157817, 701408733, 20365011074, 591286729879, 17167680177565, 498454011879264, 14472334024676221, 420196140727489673, 12200160415121876738, 354224848179261915075, 10284720757613717413913
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (29,1)
FORMULA
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
MATHEMATICA
Table[Fibonacci[7n+2], {n, 0, 30}]
PROG
(Magma) [Fibonacci(7*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(7*n+2) \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
Cf. A000045, A134498, A134499, A134500, A134501, A134502, A134503, A134504.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011
STATUS
approved
A134499
a(n) = Fibonacci(7*n+1).
+10
5
1, 21, 610, 17711, 514229, 14930352, 433494437, 12586269025, 365435296162, 10610209857723, 308061521170129, 8944394323791464, 259695496911122585, 7540113804746346429, 218922995834555169026, 6356306993006846248183
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (29,1).
FORMULA
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
MATHEMATICA
Table[Fibonacci[7 n + 1], {n, 0, 30}]
PROG
(Magma) [Fibonacci(7*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011
(PARI) a(n)=fibonacci(7*n+1) \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
Cf. A000032, A000045, A134498, A134500, A134501, A134502, A134503, A134504.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset corrected by Vincenzo Librandi, Apr 16 2011
STATUS
approved
A134503
a(n) = Fibonacci(7n + 5).
+10
5
5, 144, 4181, 121393, 3524578, 102334155, 2971215073, 86267571272, 2504730781961, 72723460248141, 2111485077978050, 61305790721611591, 1779979416004714189, 51680708854858323072, 1500520536206896083277
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..14.
Index entries for linear recurrences with constant coefficients, signature (29,1)
FORMULA
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)
MATHEMATICA
Table[Fibonacci[7n+5], {n, 0, 30}]
LinearRecurrence[{29, 1}, {5, 144}, 20] (* Harvey P. Dale, Apr 24 2017 *)
PROG
(Magma) [Fibonacci(7*n+5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(7*n+5) \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
Cf. A000045, A134498, A134499, A134500, A134501, A134502, A134504.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011
STATUS
approved
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