A061349 Sum of antidiagonals of A060736.

0, 1, 6, 17, 40, 75, 130, 203, 304, 429, 590, 781, 1016, 1287, 1610, 1975, 2400, 2873, 3414, 4009, 4680, 5411, 6226, 7107, 8080, 9125, 10270, 11493, 12824, 14239, 15770, 17391, 19136, 20977, 22950, 25025, 27240, 29563, 32034, 34619, 37360, 40221

Offset: 0

Author

Henry Bottomley, Jun 07 2001

Keywords

• nonn
• easy

Comments

a(1) = 1, a(2) = 2+4=6, a(3) = 5+3+9=17, a(4) = 10+6+8+16=40.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A001844, A002378, A004526, A005900, A006918, A060736.

Programs

Mathematica

LinearRecurrence[{2,1,-4,1,2,-1},{0,1,6,17,40,75},50] (* Harvey P. Dale, Oct 17 2021 *)
Accumulate[Table[n^2 + (n - 1)^2 - Floor[((n-1)/2)]*Floor[((n+1)/2)],{n,41}]] (* Stefano Spezia, Jun 05 2023 *)

PARI

concat(0, Vec(x*(x^4+4*x^3+4*x^2+4*x+1)/((x-1)^4*(x+1)^2) + O(x^100))) \\ Colin Barker, Sep 13 2014

Formula

a(n) = A005900(n) - A006918(n).

a(n) = a(n-1) + A001844(n-1) - A002378(A004526(n-1)).

a(n) = a(n-1) + n^2 + (n - 1)^2 - floor((n-1)/2)*floor((n+1)/2).

If n is odd then a(n) = (7*n^3 + 5*n)/12;

If n is even then a(n) = (7*n^3 + 8*n)/12.

From Colin Barker, Sep 13 2014: (Start)

a(n) = (n*(13 + 3*(-1)^n + 14*n^2))/24.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

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

E.g.f.: x*((12 + 21*x + 7*x^2)*cosh(x) + (15 + 21*x + 7*x^2)*sinh(x))/12. - Stefano Spezia, Jun 05 2023