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A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals. 40
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; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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.

The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Index to sequences related to polygonal numbers

FORMULA

T(n,k) = n*(k-1)*k/2+k.

T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016

EXAMPLE

The square array of nonnegatives together with polygonal numbers begins:

=========================================================

....................... A   A   .   .   A    A    A    A

....................... 0   0   .   .   0    0    1    1

....................... 0   0   .   .   1    1    3    3

....................... 0   0   .   .   6    7    9    9

....................... 0   0   .   .   9    3    6    6

....................... 0   1   .   .   5    2    0    0

....................... 4   2   .   .   7    9    6    7

=========================================================

Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...

Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...

Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...

Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...

Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...

Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...

Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...

9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...

10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...

11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...

12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...

...

=========================================================

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.

MATHEMATICA

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

CROSSREFS

Cf. A139617, A139618, A139620.

Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011

Sequence in context: A179329 A089112 A155584 * A198321 A166278 A242379

Adjacent sequences:  A139597 A139598 A139599 * A139601 A139602 A139603

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol, Apr 27 2008

EXTENSIONS

Edited by Omar E. Pol, Jan 05 2009

STATUS

approved

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Last modified August 24 06:49 EDT 2017. Contains 291052 sequences.