A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

0, 0, 1, 1, 3, 4, 7, 8, 12, 14, 19, 21, 27, 30, 37, 40, 48, 52, 61, 65, 75, 80, 91, 96, 108, 114, 127, 133, 147, 154, 169, 176, 192, 200, 217, 225, 243, 252, 271, 280, 300, 310, 331, 341, 363, 374, 397, 408, 432, 444, 469, 481, 507, 520, 547, 560, 588, 602, 631

Offset: 0

Author

Paul Curtz, May 20 2007

Keywords

• nonn

Comments

Interleaving of A077043 and A006578.

First differences are in A124072.

If the values of the second, fourth, sixth, ... column are replaced by the corresponding negative values, the antidiagonal sums of the resulting triangular array are 0, 0, 1, 1, -1, -2, -1, -2, -6, -8, -7, -9, ... .

Row sums of triangle A168316 = (1, 1, 3, 4, 7, 8, 12, ...). - Gary W. Adamson, Nov 22 2009

Examples

			First seven rows of T are
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3, 2;
  0, 1, 4, 2, 3;
  0, 1, 5, 2, 4, 3;
  0, 1, 6, 2, 5, 3, 4;.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006578, A077043, A124072, A168316.

Programs

Magma

m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jul 16 2007

Magma

A129819:= func< n | Floor(((n-1)*(3*n+1) +(2*n+5)*((n+1) mod 2))/16) >;
[A129819(n): n in [0..70]]; // G. C. Greubel, Sep 19 2024

Mathematica

CoefficientList[Series[x^2*(1+x^2+x^3)/((1-x)*(1-x^2)*(1-x^4)), {x, 0, 70}], x] (* G. C. Greubel, Sep 19 2024 *)

PARI

{vector(59, n, (n-2+n%2)*(n+n%2)/8+floor((n-2-n%2)^2/16))} \\ Klaus Brockhaus, Jul 16 2007

SageMath

def A129819(n): return ((n-1)*(3*n+1) + (2*n+5)*((n+1)%2))//16
[A129819(n) for n in range(71)] # G. C. Greubel, Sep 19 2024

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n > 6, with a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 4, a(6) = 7.

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

a(n) = (3/16)*(n+2)*(n+1) - (5/8)*(n+1) + 7/32 + (3/32)*(-1)^n + (1/16)*(n+1)*(-1)^n - (1/8)*cos(n*Pi/2) + (1/8)*sin(n*Pi/2). - Richard Choulet, Nov 27 2008

Extensions

Edited and extended by Klaus Brockhaus, Jul 16 2007