A190996 Fibonacci sequence beginning 10, 7.

10, 7, 17, 24, 41, 65, 106, 171, 277, 448, 725, 1173, 1898, 3071, 4969, 8040, 13009, 21049, 34058, 55107, 89165, 144272, 233437, 377709, 611146, 988855, 1600001, 2588856, 4188857, 6777713, 10966570, 17744283, 28710853, 46455136, 75165989, 121621125

Offset

0,1

Comments

For n >= 5, the number a(n-3) is the dimension of a commutative Hecke algebra of affine type D_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019

From Greg Dresden and Yiming Wu, Sep 10 2023: (Start)

For n >= 3, a(n) is the number of ways to tile this shape of length n+2 with squares and dominos:

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|_| |_|. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551.

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

Formula

a(n) = (5 + 2*sqrt(5)/5)*((1 + sqrt(5))/2)^n + (5 - 2*sqrt(5)/5)*((1 - sqrt(5))/2)^n. - Antonio Alberto Olivares, Jun 07 2011

a(n) = 7*Fibonacci(n) + 10*Fibonacci(n-1). - Charles R Greathouse IV, Jun 08 2011

G.f.: (10-3*x)/(1-x-x^2). - Colin Barker, Jan 11 2012

a(n) = 4*Fibonacci(n+1) + 3*LucasL(n). - G. C. Greubel, Oct 26 2022

Maple

seq(coeff(series((10-3*x)/(1-x-x^2), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Jan 22 2019

Mathematica

LinearRecurrence[{1, 1}, {10, 7}, 100]

Prog

(PARI) a(n)=7*fibonacci(n)+10*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011
(Magma) [n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; \\ Vincenzo Librandi, Feb 15 2012
(SageMath) [7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

Crossrefs

Cf. A000032, A000045, A190994, A190995.

Sequence in context: A324087 A068444 A210285 * A210283 A227473 A038309

Adjacent sequences: A190993 A190994 A190995 * A190997 A190998 A190999

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

Status

approved