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%I #46 Sep 08 2022 08:46:18
%S 1,46,47,93,140,233,373,606,979,1585,2564,4149,6713,10862,17575,28437,
%T 46012,74449,120461,194910,315371,510281,825652,1335933,2161585,
%U 3497518,5659103,9156621,14815724,23972345,38788069,62760414,101548483,164308897,265857380,430166277
%N a(n) = 11*Fibonacci(n+3) + Fibonacci(n-8) with n>=0.
%C Similar sequences with the formula h*Fibonacci(n+k) + Fibonacci(n+k-h):
%C h= 1, k=-1: A000045;
%C h= 2, k= 1: A013655;
%C h= 3, k=-2: A118658 = 2*A212804;
%C h= 4, k= 2: A022379 = 3*A000204;
%C h= 5, k= 1: A022113;
%C h= 6, k= 2: A022125;
%C h= 7, k= 3: A097657;
%C h= 8, k= 2: A022355 = 21*A000045;
%C h= 9, k= 3: 10, 32, 42, 74, 116, 190, 306, 496, 802, ... = 2*A022140;
%C h=10, k= 3: 33, 22, 55, 77, 132, 209, 341, 550, 891, ... = 11*A013655;
%C h=11, k= 3: this sequence.
%H Indranil Ghosh, <a href="/A282465/b282465.txt">Table of n, a(n) for n = 0..4769</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F G.f.: (1 + 45*x)/(1 - x - x^2).
%F a(n) = a(n-1) + a(n-2).
%F a(n) = a(i)*Fibonacci(n-i+1) + a(i-1)*Fibonacci(n-i). Examples:
%F for i= 3, a(3)=93, a(2)= 47: a(n) = 93*Fibonacci(n-2) + 47*Fibonacci(n-3);
%F for i=-1, a(-1)=45, a(-2)=-44: a(n) = 45*Fibonacci(n+2) - 44*Fibonacci(n+1).
%F Other formulae:
%F a(n) = 44*Fibonacci(n) + Fibonacci(n+2),
%F a(n) = 45*Fibonacci(n) + Fibonacci(n+1),
%F a(n) = 46*Fibonacci(n) + Fibonacci(n-1),
%F a(n) = 47*Fibonacci(n) - Fibonacci(n-2).
%F a(n) = ((91 + sqrt(5))*((1 + sqrt(5))/2)^n - (91 - sqrt(5))*((1 - sqrt(5))/2)^n)/sqrt(20).
%t Table[11 Fibonacci[n + 3] + Fibonacci[n - 8], {n, 0, 40}]
%t LinearRecurrence[{1,1},{1,46},36] (* or *) CoefficientList[Series[(1 + 45*x)/(1 - x - x^2) , {x,0,35}],x] (* _Indranil Ghosh_, Feb 22 2017 *)
%o (Magma) [11*Fibonacci(n+3)+Fibonacci(n-8): n in [0..40]];
%o (PARI) a(n) = 11*fibonacci(n+3) + fibonacci(n-8) \\ _Indranil Ghosh_, Feb 23 2017
%Y Cf. A000045, A013655, A022113, A022125, A022355, A022379, A022379, A097657, A118658.
%Y Cf. sequences with g.f. (1 + r*x)/(1 - x - x^2) for r = 2..31, respectively: A000204, A000285, A022095 - A022110, A022391 - A022402.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Feb 20 2017
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