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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
7
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 (list; graph; listen)
OFFSET

0,5

COMMENT

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, ... .

FORMULA

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

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

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 ].

PROGRAM

(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 */

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

CROSSREFS

Cf. A077043, A006578, A124072.

Adjacent sequences: A129816 A129817 A129818 this_sequence A129820 A129821 A129822

Sequence in context: A045615 A051201 A026449 this_sequence A025032 A003141 A008368

KEYWORD

nonn

AUTHOR

Paul Curtz (bpcrtz(AT)free.fr), May 20 2007

EXTENSIONS

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 16 2007

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Last modified October 7 14:39 EDT 2008. Contains 144666 sequences.


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