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

0,5

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

Links

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

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

Example

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;.

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 *)

Prog

(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
(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

Crossrefs

Cf. A006578, A077043, A124072, A168316.

Sequence in context: A282166 A165157 A182079 * A025032 A207524 A003141

Adjacent sequences: A129816 A129817 A129818 * A129820 A129821 A129822

Keyword

nonn

Author

Paul Curtz, May 20 2007

Extensions

Edited and extended by Klaus Brockhaus, Jul 16 2007

Status

approved