A116410 Expansion of (1-x-2x^2+sqrt(1-2x-3x^2))/(2*(1-2x-3x^2)).
1, 1, 3, 8, 23, 66, 192, 561, 1647, 4850, 14318, 42351, 125468, 372191, 1105275, 3285288, 9772767, 29090826, 86646486, 258208671, 769820418, 2296067565, 6850744365, 20447143866, 61045757604, 182303186391, 544550917797
Offset: 0
Programs
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Maple
a:= proc(n) option remember; `if`(n<4, [1$2, 3, 8][n+1], ((5*n-4)*a(n-1)-(3*n-6)*a(n-2)-(9*n-18)*a(n-3))/n) end: seq(a(n), n=0..30); # Alois P. Heinz, May 30 2016 -
Mathematica
Table[ (4 3^k + 3 (-1)^k DifferenceRoot[Function[{y, n}, {(-3 + 3 n) y[n] + (2 - 5 n) y[1 + n] + (-1 + n) y[2 + n] + (2 + n) y[3 + n] == 0, y[-1] == 0, y[0] == 0, y[1] == 1}]][1 + k] + 3^(2 + k)DifferenceRoot[Function[{y, n}, {(-1 + n) y[n] + (2 + n) y[1 + n] + (-7 - 5 n) y[2 + n] + (6 + 3 n) y[3 + n] == 0, y[-1] == 0, y[0] == 0, y[1] == 1}]][1 + k])/24, {k, 1, 20}] (* Benedict W. J. Irwin, May 30 2016 *)
Formula
a(n) = ((3^n+2*0^n)/3 + Sum_{k=0..floor(n/2)} C(n,2k)C(2k,k))/2.
From Benedict W. J. Irwin, May 30 2016: (Start)
Let y1(-1)=0, y1(0)=0, y1(1)=1,
Let (3n-3)*y1(n)+(2-5n)*y1(n+1)+(n-1)*y1(n+2)+(n+2)*y1(n+3)=0,
Let y2(-1)=0, y2(0)=0, y2(1)=1,
Let (n-1)*y2(n)+(2+n)*y2(n+1)-(5n+7)*y2(n+2)+(3n+6)*y2(n+3)=0,
a(n) = (4*3^n+3*(-1)^n*y1(n+1)+3^(n+2)*y2(n+1))/24, n>0.
(End)
a(n) ~ 3^(n-1)/2 * (1 + 3*sqrt(3/(Pi*n))/2). - Vaclav Kotesovec, May 30 2016
D-finite with recurrence: a(n) = ((5*n-4)*a(n-1) -(3*n-6)*a(n-2) -(9*n-18)*a(n-3))/n for n>3. - Alois P. Heinz, May 30 2016
Comments