A213759 Principal diagonal of the convolution array A213783.

1, 4, 11, 22, 39, 62, 93, 132, 181, 240, 311, 394, 491, 602, 729, 872, 1033, 1212, 1411, 1630, 1871, 2134, 2421, 2732, 3069, 3432, 3823, 4242, 4691, 5170, 5681, 6224, 6801, 7412, 8059, 8742, 9463, 10222, 11021, 11860, 12741, 13664, 14631

Offset: 1

Author

Clark Kimberling, Jun 22 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A213783, A213500.

Partial sums of A047838. - Guenther Schrack, May 24 2018

Programs

Mathematica

b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
LinearRecurrence[{3,-2,-2,3,-1},{1,4,11,22,39},50] (* Harvey P. Dale, Jul 22 2014 *)

Formula

a(n) = (3 - 3*(-1)^n - 4*n + 18*n^2 + 4*n^3)/24.

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).

G.f.: x*(1 + x + x^2 - x^3)/((1 - x)^4 *(1 + x)).

a(n+1) = a(n) + A047838(n+2) for n > 0. - Guenther Schrack, May 24 2018

a(n) = A212964(n+2) - n for n > 0. - Guenther Schrack, May 30 2018