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%I #67 Oct 23 2024 01:26:02
%S 10,7,17,24,41,65,106,171,277,448,725,1173,1898,3071,4969,8040,13009,
%T 21049,34058,55107,89165,144272,233437,377709,611146,988855,1600001,
%U 2588856,4188857,6777713,10966570,17744283,28710853,46455136,75165989,121621125
%N Fibonacci sequence beginning 10, 7.
%C 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
%C From _Greg Dresden_ and Yiming Wu, Sep 10 2023: (Start)
%C For n >= 3, a(n) is the number of ways to tile this shape of length n+2 with squares and dominos:
%C _ _
%C _|_|___________________|_|_
%C |_|_|_|_|_|_|_|_|_|_|_|_|_|_|
%C |_| |_|. (End)
%C 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
%H Vincenzo Librandi, <a href="/A190996/b190996.txt">Table of n, a(n) for n = 0..1000</a>
%H Jia Huang, <a href="https://arxiv.org/abs/1405.1636">Hecke algebras with independent parameters</a>, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KayakPaddleGraph.html">Kayak Paddle Graph</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F 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
%F a(n) = 7*Fibonacci(n) + 10*Fibonacci(n-1). - _Charles R Greathouse IV_, Jun 08 2011
%F G.f.: (10-3*x)/(1-x-x^2). - _Colin Barker_, Jan 11 2012
%F a(n) = 4*Fibonacci(n+1) + 3*LucasL(n). - _G. C. Greubel_, Oct 26 2022
%F a(n) = A000285(n)+3*A000285(n-1). - _Feryal Alayont_, Sep 28 2024
%p seq(coeff(series((10-3*x)/(1-x-x^2),x,n+1), x, n), n = 0 .. 40); # _Muniru A Asiru_, Jan 22 2019
%t LinearRecurrence[{1, 1}, {10, 7}, 100]
%o (PARI) a(n)=7*fibonacci(n)+10*fibonacci(n-1) \\ _Charles R Greathouse IV_, Jun 08 2011
%o (Magma) [n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Feb 15 2012
%o (SageMath) [7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # _G. C. Greubel_, Oct 26 2022
%Y Cf. A000032, A000045, A190994, A190995.
%K nonn,easy
%O 0,1
%A _Vladimir Joseph Stephan Orlovsky_, Jun 07 2011