A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Author

Robert G. Wilson v, Apr 27 2020

Keywords

• easy
• sign
• tabl

Comments

\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

r\

_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998

_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956

_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477

_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217

_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290

_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326

_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384

_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566

_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567

_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A001106

10 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A001107

11 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A051682

12 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A051624

13 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A051865

14 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A051866

15 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867

...

Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

Links

• E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
• Eric Weisstein's World of Mathematics, Polygonal Number.
• Wikipedia, Polygonal number.
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A317302 (the same array) but read by ascending antidiagonals.

Sub-arrays: A089000, A139600, A206735;

Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;

Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;

Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,

Number of times k>1 appears: A129654, First occurrence of k: A063778.

Programs

Mathematica

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

Formula

P(r, c) = (r - 2)(c(c-1)/2) + c.