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A129819
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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.
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7
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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; refs; listen; history; internal format)
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OFFSET
| 0,5
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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,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009]
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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)).
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) [From Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 27 2008]
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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 ].
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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 */
(PARI) {vector(59, n, (n-2+n%2)*(n+n%2)/8+floor((n-2-n%2)^2/16))} /* Klaus Brockhaus, Jul 16 2007 */
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CROSSREFS
| Cf. A077043, A006578, A124072.
A168316 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009]
Sequence in context: A051201 A026449 A165157 * A025032 A003141 A157419
Adjacent sequences: A129816 A129817 A129818 * A129820 A129821 A129822
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KEYWORD
| nonn
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), May 20 2007
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EXTENSIONS
| Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 16 2007
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