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Search: id:A129819
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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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+0
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; listen)
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OFFSET
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0,5
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COMMENT
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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, ... .
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FORMULA
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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)).
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EXAMPLE
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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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PROGRAM
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(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
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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
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KEYWORD
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nonn
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AUTHOR
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Paul Curtz (bpcrtz(AT)free.fr), May 20 2007
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EXTENSIONS
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Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 16 2007
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