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A024458 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = [ (n+1)/2 ], s = (Fibonacci numbers). 2
1, 1, 3, 5, 12, 19, 40, 65, 130, 210, 404, 654, 1227, 1985, 3653, 5911, 10720, 17345, 31090, 50305, 89316, 144516, 254568, 411900, 720757, 1166209, 2029095, 3283145, 5684340, 9197455, 15855964, 25655489, 44061862, 71293590, 122032508 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

From Wolfdieter Lang, Jan 02 2012: (Start)

chat(n):=a(n+1), n>=0, is the half-convolution of the sequence A000045(n+1), n>=0, with itself. For the definition of half-convolution see a comment on A201204, where also the rule to find the o.g.f. is given. Here the o.g.f. is obtained from (U(x)^2 + U2(x^2))/2 with U(x)=1/(1-x-x^2),the o.g.f. of A000045(n+1), n>=0, and U2(x):=(1-x)/((1+x)*(1-3*x+x^2) the o.g.f. of A007598(n+1), n>=0. This coincides with the o.g.f. given below in the formula section after x has been divided.

For the bisection of this half-convolution see A027991(n+1) and A001870(n), n>=0.

(End)

LINKS

Table of n, a(n) for n=1..35.

FORMULA

G.f.: x(x^3-x^2+1)/[(1+x^2)(1+x-x^2)(1-x-x^2)^2].

a(n) = ((13 - 5*(-1)^n + 10*n)*A000045(n) + (1 - (-1)^n + 2*n)*A000032(n) + 8*sin(Pi*n/2))/40. - Vladimir Reshetnikov, Oct 03 2016

MATHEMATICA

Table[((13 - 5 (-1)^n + 10 n) Fibonacci[n] + (1 - (-1)^n + 2 n) LucasL[n] + 8 Sin[Pi n/2])/40, {n, 1, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *)

CROSSREFS

Sequence in context: A323866 A082740 A010067 * A143643 A321679 A266819

Adjacent sequences:  A024455 A024456 A024457 * A024459 A024460 A024461

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from James A. Sellers, May 03 2000

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)