A023660 - OEIS

%I #7 Jul 18 2022 09:59:53

%S 1,3,5,8,12,16,20,24,28,33,39,45,51,57,63,69,75,81,87,94,102,110,118,

%T 126,134,142,150,158,166,174,182,190,198,206,215,225,235,245,255,265,

%U 275,285,295,305,315,325,335,345,355,365,375,385,395,405

%N Convolution of odd numbers and A023533.

%H G. C. Greubel, <a href="/A023660/b023660.txt">Table of n, a(n) for n = 1..5000</a>

%F From _G. C. Greubel_, Jul 17 2022: (Start)

%F a(n) = Sum_{j=0..n-1} (2*j+1)*A023533(n-j).

%F a(n) = 2*A023543(n-1) + A056556(n).

%F T(n, k) = (2*k+1)*n + 6*binomial(n+2, 4), for 0 <= k <= n*(n+3)/2 and n >= 1 (as an irregular triangle). (End)

%t Table[(2*k+1)*n + 6*Binomial[n+2,4], {n, 7}, {k,0,n*(n+3)/2}]//Flatten (* _G. C. Greubel_, Jul 17 2022 *)

%o (Magma)

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

%o [(&+[(2*k+1)*A023533(n-k): k in [0..n-1]]): n in [1..80]]; // _G. C. Greubel_, Jul 17 2022

%o (SageMath)

%o def A023660(n, k): return (2*k+1)*n + 6*binomial(n+2, 4)

%o flatten([[A023660(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # _G. C. Greubel_, Jul 17 2022

%Y Cf. A005408, A023533, A023543, A056556.

%K nonn

%O 1,2

%A _Clark Kimberling_