A121171 -id:A121171 - cp's OEIS frontend

A001946 a(n) = 11*a(n-1) + a(n-2).

2, 11, 123, 1364, 15127, 167761, 1860498, 20633239, 228826127, 2537720636, 28143753123, 312119004989, 3461452808002, 38388099893011, 425730551631123, 4721424167835364, 52361396397820127, 580696784543856761, 6440026026380244498, 71420983074726546239

Offset: 0

N. J. A. Sloane, Simon Plouffe

For odd n there is the Aurifeuillian factorization a(n) = Lucas[5n] = Lucas[n]*A[n]*B[n] = A000032[n]*A124296[n]*A124297[n], where A[n] = A124296[n] = 5*F(n)^2 - 5*F(n) + 1 and B[n] = A124297[n] = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci[n]. The largest prime divisors of a(n) for n>0 are listed in A121171[n] = {11, 41, 31, 2161, 151, 2521, 911, ...}. - Alexander Adamchuk, Oct 25 2006

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 139.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (11, 1).

Cf. A000032, A000045, A121171, A124296, A124297.

[ Lucas(5*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

A001946:=(-2+11*z)/(-1+11*z+z**2); # Conjectured by Simon Plouffe in his 1992 dissertation

Table[Fibonacci[5n-1]+Fibonacci[5n+1],{n,0,30}] (* Alexander Adamchuk, Oct 25 2006 *)
LinearRecurrence[{11,1},{2,11},20] (* Harvey P. Dale, Jan 25 2024 *)

a(n) = Lucas(5n) = Fibonacci(5n-1) + Fibonacci(5n+1). - Alexander Adamchuk, Oct 25 2006
a(n) = ((11 + 5*sqrt(5))/2)^n + ((11 - 5*sqrt(5))/2)^n. - Tanya Khovanova, Feb 06 2007
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 11*A097842(n), a(2n) = A065705(n).
a(3n+1) = A041226(5n), a(3n+2) = A041226(5n+3), a(3n+3) = 2* A041226(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A001946(n) + A049666(n)*sqrt(125))/2.
Limit(A001946(n)/A049666(n), n=infinity) = sqrt(125). (End)
From Peter Bala, Mar 22 2015: (Start)
a(n) = Fibonacci(10*n)/Fibonacci(5*n) for n >= 1.
a(n) = ( Fibonacci(5*n + 2*k) - F(5*n - 2*k) )/Fibonacci(2*k) for nonzero integer k.
a(n) = ( Fibonacci(5*n + 2*k + 1) + F(5*n - 2*k - 1) )/Fibonacci(2*k + 1) for arbitrary integer k.
a(n) = Sum_{k = 0..2*n} binomial(2*n,k)*Lucas(n + k). (End)
a(n) = [x^n] ( (1 + 11*x + sqrt(1 + 22*x + 125*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 26 2015
E.g.f.: 2*cosh(5*sqrt(5)*x/2)*(cosh(11*x/2) + sinh(11*x/2)). - Stefano Spezia, Jan 18 2025

A124296 a(n) = 5F(n)^2 - 5F(n) + 1, where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 11, 31, 101, 281, 781, 2101, 5611, 14851, 39161, 102961, 270281, 708761, 1857451, 4865911, 12744061, 33372361, 87382901, 228792301, 599019851, 1568309051, 4105974961, 10749725281, 28143378001, 73680695281, 192899171531

Offset: 0

Alexander Adamchuk, Oct 25 2006

11 = Lucas(5) divides a(3+10k), a(7+10k), and a(8+10k). The last digit of a(n) is 1, so a(n) mod 10 = 1. For odd n there exists the so-called Aurifeuillian factorization A001946(n) = Lucas(5n) = Lucas(n)*A(n)*B(n) = A000032(n)*A124296(n)*A124297(n), where A(n) = A124296(n) = 5*F(n)^2 - 5*F(n) + 1 and B(n) = A124297(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

John Cerkan, Table of n, a(n) for n = 0..2375
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization
Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Cf. A000032, A000045, A121171, A001946, A124297.

Bisections: A001604, A156094.

Table[5*Fibonacci[n]^2-5*Fibonacci[n]+1,{n,0,50}]
5#^2-5#+1&/@Fibonacci[Range[0,30]] (* Harvey P. Dale, Nov 29 2011 *)

a(n)=subst(5*t*(t-1)+1, t, fibonacci(n)) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = 5*Fibonacci(n)^2 - 5*Fibonacci(n) + 1.

G.f.: -(x^5+9*x^4-15*x^3+x^2+3*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Jan 03 2013]

A124297 a(n) = 5F(n)^2 + 5F(n) + 1, where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 11, 11, 31, 61, 151, 361, 911, 2311, 5951, 15401, 40051, 104401, 272611, 712531, 1863551, 4875781, 12760031, 33398201, 87424711, 228859951, 599129311, 1568486161, 4106261531, 10750188961, 28144128251, 73681909211, 192901135711

Offset: 0

Alexander Adamchuk, Oct 25 2006

11 = Lucas(5) divides a(1+10k), a(2+10k), and a(9+10k). Last digit of a(n) is 1, or a(n) mod 10 = 1. For odd n there exists the so-called Aurifeuillian factorization A001946(n) = Lucas(5n) = Lucas(n)*A(n)*B(n) = A000032(n)*A124296(n)*A124297(n), where A(n) = A124296(n) = 5*F(n)^2 - 5*F(n) + 1 and B(n) = A124297(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

John Cerkan, Table of n, a(n) for n = 0..2373
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization
Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Cf. A000032, A000045, A121171, A001946, A124296.

Bisections: A001604, A156095.

Table[5*Fibonacci[n]^2+5*Fibonacci[n]+1,{n,0,50}]
LinearRecurrence[{4,-2,-6,4,2,-1},{1,11,11,31,61,151},30] (* Harvey P. Dale, Feb 23 2023 *)

a(n)=subst(5*t*(t+1)+1,t,fibonacci(n)) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = 5*Fibonacci(n)^2 + 5*Fibonacci(n) + 1.

G.f.: -(11*x^5-21*x^4-15*x^3+31*x^2-7*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Jan 03 2013]

A133320 Numbers k such that both A124296(k) = 5F(k)^2 - 5F(k) + 1 and A124297(k) = 5F(k)^2 + 5F(k) + 1 are prime, where F(k) = Fibonacci(k).

Original entry on oeis.org

3, 4, 5, 10, 40

Offset: 1

Alexander Adamchuk, Oct 18 2007

Cf. A124297 (5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n)).
Cf. A124296 (5*F(n)^2 - 5*F(n) + 1, where F(n) = Fibonacci(n)).
Cf. A000032, A000045, A121171, A001946.

Do[ F=Fibonacci[n]; f=5*F^2-5*F+1; g=5*F^2+5*F+1; If[ PrimeQ[f], If[ PrimeQ[g], Print[ {n,f,g} ] ] ], {n,1,1000} ]

cp's OEIS Frontend

A001946 a(n) = 11*a(n-1) + a(n-2).

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A124296 a(n) = 5*F(n)^2 - 5*F(n) + 1, where F(n) = Fibonacci(n).

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A124297 a(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

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A133320 Numbers k such that both A124296(k) = 5*F(k)^2 - 5*F(k) + 1 and A124297(k) = 5*F(k)^2 + 5*F(k) + 1 are prime, where F(k) = Fibonacci(k).

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Mathematica

A124296 a(n) = 5F(n)^2 - 5F(n) + 1, where F(n) = Fibonacci(n).

A124297 a(n) = 5F(n)^2 + 5F(n) + 1, where F(n) = Fibonacci(n).

A133320 Numbers k such that both A124296(k) = 5F(k)^2 - 5F(k) + 1 and A124297(k) = 5F(k)^2 + 5F(k) + 1 are prime, where F(k) = Fibonacci(k).