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A195152
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Square array read by antidiagonals with T(n,k) = n*((k+2)*n-k)/2, n=0, +- 1, +- 2,..., k>=0.
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19
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0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 4, 5, 3, 1, 0, 9, 7, 6, 4, 1, 0, 9, 12, 10, 7, 5, 1, 0, 16, 15, 15, 13, 8, 6, 1, 0, 16, 22, 21, 18, 16, 9, 7, 1, 0, 25, 26, 28, 27, 21, 19, 10, 8, 1, 0, 25, 35, 36, 34, 33, 24, 22, 11, 9, 1, 0, 36, 40, 45, 46, 40, 39, 27, 25, 12, 10, 1, 0
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OFFSET
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0,7
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COMMENTS
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Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).
Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).
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LINKS
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FORMULA
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T(n,k) = (k+2)*n*(n+1)/8+(k-2)*((2*n+1)*(-1)^n-1)/16, n >= 0 and k >= 0. - Omar E. Pol, Oct 01 2011
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EXAMPLE
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Array begins:
. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...
. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,...
. 4, 7, 10, 13, 16, 19, 22, 25, 28, 31,...
. 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
. 9, 15, 21, 27, 33, 39, 45, 51, 57, 63,...
. 16, 22, 28, 34, 40, 46, 52, 58, 64, 70,...
. 16, 26, 36, 46, 56, 66, 76, 86, 96, 106,...
. 25, 35, 45, 55, 65, 75, 85, 95, 105, 115,...
. 25, 40, 55, 70, 85, 100, 115, 130, 145, 160,...
...
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CROSSREFS
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Column 0 gives A008794, except its first term.
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KEYWORD
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AUTHOR
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STATUS
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approved
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