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 A024458 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (Fibonacci numbers). 4
 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 G. C. Greubel, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (1,3,-2,0,-2,-3,1,1). FORMULA G.f.: x*(1-x^2+x^3)/((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 From G. C. Greubel, Apr 06 2022: (Start) a(2*n) = (1/5)*(n*Lucas(2*n+1) + Fibonacci(2*n)), n >= 1. a(2*n+1) = (1/5)*((-1)^n + (n+1)*Lucas(2*n+2) + Fibonacci(2*n+1)), n >= 0. a(n) = Sum_{j=0..floor((n-1)/2)} fibonacci(j+1)*Fibonacci(n-j). (End) MATHEMATICA Table[((13-5(-1)^n +10n)Fibonacci[n] + (1-(-1)^n +2n)LucasL[n] +8Sin[Pi*n/2])/40, {n, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *) LinearRecurrence[{1, 3, -2, 0, -2, -3, 1, 1}, {1, 1, 3, 5, 12, 19, 40, 65}, 40] (* Harvey P. Dale, Mar 02 2023 *) PROG (Magma) [(&+[Fibonacci(j+1)*Fibonacci(n-j): j in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Apr 06 2022 (SageMath) def A024458(n): return sum(fibonacci(j+1)*fibonacci(n-j) for j in (0..((n-1)//2)) ) [A024458(n) for n in (1..50)] # G. C. Greubel, Apr 06 2022 CROSSREFS Cf. A000032, A000045, A058071. Sequence in context: A082740 A010067 A341710 * A143643 A321679 A266819 Adjacent sequences: A024455 A024456 A024457 * A024459 A024460 A024461 KEYWORD nonn,easy AUTHOR Clark Kimberling EXTENSIONS More terms from James A. Sellers, May 03 2000 STATUS approved

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Last modified October 4 10:09 EDT 2023. Contains 365874 sequences. (Running on oeis4.)