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A081422
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Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.
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9
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1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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LINKS
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T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number
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FORMULA
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Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.
T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015
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EXAMPLE
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The array starts
1 1 3 10 ...
1 2 6 16 ...
1 3 9 22 ...
1 4 12 28 ...
The triangle starts
1;
1, 1;
1, 2, 3;
1, 3, 6, 10;
1, 4, 9, 16, 25;
...
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MATHEMATICA
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Table[PolygonalNumber[n, i], {n, 0, 10}, {i, n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)
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PROG
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(PARI) tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", "); ); print(); ); } \\ Michel Marcus, Jun 22 2015
(Magma) [[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018
(Sage) [[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019
(GAP) Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019
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CROSSREFS
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Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.
Sequence in context: A208516 A111808 A247046 * A213742 A213743 A213744
Adjacent sequences: A081419 A081420 A081421 * A081423 A081424 A081425
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KEYWORD
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easy,nonn,tabl,look
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AUTHOR
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Paul Barry, Mar 21 2003
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STATUS
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approved
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