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

Author

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

Keywords

• nonn
• easy

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)

For n >= 3, a(n) is the number of edge covers of the kayak paddle graphs KP(3,3,n-3), where we interpret KP(3,3,0) as two C_3's with one common vertex. - Feryal Alayont, Sep 28 2024

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.
• Eric Weisstein's World of Mathematics, Kayak Paddle Graph.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A000045, A190994, A190995.

Programs

Magma

[n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

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]

PARI

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

SageMath

[7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

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

a(n) = A000285(n)+3*A000285(n-1). - Feryal Alayont, Sep 28 2024

a(2*n) = A000285(n)^2 + A000285(n-1)^2. - Greg Dresden, Feb 28 2025