A023543 - OEIS

%I #11 Jul 16 2022 03:53:34

%S 1,2,3,5,7,9,11,13,15,18,21,24,27,30,33,36,39,42,45,49,53,57,61,65,69,

%T 73,77,81,85,89,93,97,101,105,110,115,120,125,130,135,140,145,150,155,

%U 160,165,170,175,180,185,190,195,200,205,210,216,222,228

%N Convolution of natural numbers with A023533.

%H G. C. Greubel, <a href="/A023543/b023543.txt">Table of n, a(n) for n = 1..5000</a>

%F From _G. C. Greubel_, Jul 15 2022: (Start)

%F a(n) = Sum_{j=1..floor((n+1)/2)} (n - j + 1)*A023533(j).

%F a(n) = (m+2)*(n+1) - binomial(n+4, 4), for binomial(n+3, 3) - 2 <= m <= binomial(n+4, 3) - 3, and n >= 1, with a(1) = 1, a(2) = 2. (End)

%t Join[{1,2}, Table[(m+2)*(n+1) -Binomial[n+4,4], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* _G. C. Greubel_, Jul 15 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o [(&+[A023533(k)*(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // _G. C. Greubel_, Jul 15 2022

%o (SageMath)

%o [1,2]+flatten([[(m+2)*(n+1) - binomial(n+4,4) for m in (binomial(n+3,3)-2 .. binomial(n+4,3)-3)] for n in (1..6)]) # _G. C. Greubel_, Jul 15 2022

%Y Cf. A000027, A023533.

%K nonn

%O 1,2

%A _Clark Kimberling_

%E Title updated by _Sean A. Irvine_, Jun 06 2019