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Search: id:A139600
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| A139600 |
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Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals. |
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20
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| 0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11
(list; table; graph; listen)
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OFFSET
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0,6
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COMMENT
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A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number.
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FORMULA
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T(n,k) = n*(k-1)*k/2+k, n>=0, k>=0.
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EXAMPLE
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The square array of nonnegatives together with polygonal numbers begins:
===========================================================================
........................ A . A . . . . . A . A .. A .. A .. A .. A .. A ...
........................ 0 . 0 . . . . . 0 . 0 .. 1 .. 1 .. 1 .. 1 .. 1 ...
........................ 0 . 0 . . . . . 1 . 1 .. 3 .. 3 .. 3 .. 3 .. 3 ...
........................ 0 . 0 . . . . . 6 . 7 .. 9 .. 9 .. 9 .. 9 .. 9 ...
........................ 0 . 0 . . . . . 9 . 3 .. 6 .. 6 .. 6 .. 6 .. 6 ...
........................ 0 . 1 . . . . . 5 . 2 .. 0 .. 0 .. 0 .. 0 .. 1 ...
........................ 4 . 2 . . . . . 7 . 9 .. 6 .. 7 .. 8 .. 9 .. 0 ...
===========================================================================
Nonnegatives .. A001477: 0., 1., 2., 3., 4., 5.., 6.., 7.., 8.., 9.., 10.,
Triangulars ... A000217: 0., 1., 3., 6., 10, 15., 21., 28., 36., 45., 55.,
Squares ....... A000290: 0., 1., 4., 9., 16, 25., 36., 49., 64., 81., 100,
Pentagonals ... A000326: 0., 1., 5., 12, 22, 35., 51., 70., 92., 117, 145,
Hexagonals .... A000384: 0., 1., 6., 15, 28, 45., 66., 91., 120, 153, 190,
Heptagonals ... A000566: 0., 1., 7., 18, 34, 55., 81., 112, 148, 189, 235,
Octagonals .... A000567: 0., 1., 8., 21, 40, 65., 96., 133, 176, 225, 280,
9-gonals ...... A001106: 0., 1., 9., 24, 46, 75., 111, 154, 204, 261, 325,
10-gonals ..... A001107: 0., 1., 10, 27, 52, 85., 126, 175, 232, 297, 370,
11-gonals ..... A051682: 0., 1., 11, 30, 58, 95., 141, 196, 260, 333, 415,
12-gonals ..... A051624: 0., 1., 12, 33, 64, 105, 156, 217, 288, 369, 460,
And so on .................................................................
===========================================================================
The column with the numbers 2, 3, 4, 5, 6,..... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15,... is formed by the positive members of A008585.
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MATHEMATICA
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T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2009]
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CROSSREFS
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Cf. A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A000004, A000012, A000027, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A057145, A086271, A139601.
Cf. A139617, A139618, A139620.
Sequence in context: A077884 A089112 A155584 this_sequence A166278 A103438 A167279
Adjacent sequences: A139597 A139598 A139599 this_sequence A139601 A139602 A139603
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Omar E. Pol (info(AT)polprimos.com), Apr 27 2008
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EXTENSIONS
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Edited by Omar E. Pol (info(AT)polprimos.com), Jan 05 2009
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