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A125764
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Array of partial sums of rows of array in A086271, read by antidiagonals.
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2
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1, 3, 2, 6, 7, 3, 10, 15, 12, 4, 15, 26, 27, 18, 5, 21, 40, 48, 42, 25, 6, 28, 57, 75, 76, 60, 33, 7, 36, 77, 108, 120, 110, 81, 42, 8, 45, 100, 147, 174, 175, 150, 105, 52, 9, 55, 126, 192, 238, 255, 240, 196, 132, 63, 10, 66, 155, 243, 312, 350, 351, 315, 248, 162, 75, 11
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OFFSET
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1,2
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COMMENTS
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Row 3 is = 3rd triangular number + 3rd square + 3rd pentagonal number + 3rd hexagonal number + ... + 3rd k-gonal number. First column is triangular numbers. A086271 Rectangular array T(n,k) of polygonal numbers, by diagonals.
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LINKS
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Eric Weisstein's World of Mathematics, Polygonal Number. See equation (4), our partial sums are on this as array element values.
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FORMULA
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a(k,n) = (k*(k-1)/2)n^2 + (k*(k+3)/4)n. a(k,n) = row k of array of partial sums = k-th triangular number + k-th square + k-th pentagonal number + k-th hexagonal number + ... = A000217(k) + A000290(k) + A000326(k) + A000384(k) + ... a(1,n) = n. a(2,n) = (n(n+1)/2)-3 = A000217(n) - 3. a(3,n) = 3*n(n+3)/2 = A000096 with offset 3.
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EXAMPLE
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Partial row sum array begins:
1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... n.
2 | 3, 7, 12, 18, 25, 33, 42, 52, (n(n+1)/2)-3.
3 | 6, 15, 27, 42, 60, 81, 105, ... (3/2)n^2 + (9/2) n.
4 | 10, 26, 48, 76, 110, 150, ... 3n^2 + 7n.
5 | 15, 40, 75, ... 5n^2 + 10n.
6 | 21, 57, 108, ... (15/2)n^2 + (27/2)n.
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MAPLE
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A086271 := proc(n, k) k*binomial(n, 2)+n ; end: A125764 := proc(n, k) add(A086271(n, i), i=1..k) ; end: for d from 1 to 15 do for k from 1 to d do printf("%d, ", A125764(d-k+1, k)) ; od: od: # R. J. Mathar, Nov 02 2007
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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