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A099016
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a(n) = 3*L(2*n)/5 - (-1)^n/5, where L = A000032.
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4
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1, 2, 4, 11, 28, 74, 193, 506, 1324, 3467, 9076, 23762, 62209, 162866, 426388, 1116299, 2922508, 7651226, 20031169, 52442282, 137295676, 359444747, 941038564, 2463670946, 6449974273, 16886251874, 44208781348, 115740092171
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OFFSET
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0,2
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COMMENTS
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Let M = an infinite triangle with (1,2,2,3,3,4,4,...) as the left border and all other columns = (0,1,2,3,4,5,...). Then lim_{n->infinity} M^n = A099016, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 26 2010
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LINKS
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FORMULA
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G.f.: (1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = 2*F(n)^2 + F(n)*F(n-1) + F(n-1)^2, where F = A000045.
a(n) = 3*((3/2 - sqrt(5)/2)^n + (3/2 + sqrt(5)/2)^n)/5 - (-1)^n/5.
a(n) = 3*A000032(2*n)/5 - (-1)^n/5.
a(n) = 3*F(n)^2 + (-1)^n.
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MAPLE
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with(combinat):seq(3*fibonacci(n)^2+(-1)^n, n= 0..27)
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MATHEMATICA
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CoefficientList[Series[(1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)
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PROG
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(Magma) F:=Fibonacci; [F(n+1)^2+F(n)*F(n-2): n in [0..30]]; // Bruno Berselli, Feb 15 2017
(PARI) x='x+O('x^30); Vec((1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2))) \\ G. C. Greubel, Dec 31 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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