cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

A282850 38-gonal numbers: a(n) = n*(18*n-17).

Original entry on oeis.org

0, 1, 38, 111, 220, 365, 546, 763, 1016, 1305, 1630, 1991, 2388, 2821, 3290, 3795, 4336, 4913, 5526, 6175, 6860, 7581, 8338, 9131, 9960, 10825, 11726, 12663, 13636, 14645, 15690, 16771, 17888, 19041, 20230, 21455, 22716, 24013, 25346, 26715, 28120, 29561
Offset: 0

Views

Author

Haney Moon, Feb 23 2017

Keywords

Links

Crossrefs

Cf. A255187, A254474, A261191.

Programs

  • Mathematica
    Table[n(36n-34)/2, {n,50}]
LinearRecurrence[{3, -3, 1}, {0, 1, 38}, 42] (* or *) CoefficientList[Series[-x * (35 * x + 1) / (x - 1) ^ 3, {x, 0, 41}], x] (* Indranil Ghosh, Feb 27 2017 *)
  • PARI
    for(n=0,20,print1(n*(18*n-17),", ")) \\ Derek Orr, Feb 26 2017
  • PARI
    a(n)=n*(18*n-17) \\ Charles R Greathouse IV, Feb 26 2017

Formula

G.f.: -x*(35*x+1)/(x-1)^3.
E.g.f.: exp(x)*(x + 18*x^2). - Nikolaos Pantelidis, Feb 10 2023

A282851 35-gonal numbers: a(n) = n*(33*n-31)/2.

Original entry on oeis.org

0, 1, 35, 102, 202, 335, 501, 700, 932, 1197, 1495, 1826, 2190, 2587, 3017, 3480, 3976, 4505, 5067, 5662, 6290, 6951, 7645, 8372, 9132, 9925, 10751, 11610, 12502, 13427, 14385, 15376, 16400, 17457, 18547, 19670, 20826, 22015, 23237, 24492
Offset: 0

Views

Author

Kyu Bin Choi, Feb 23 2017

Keywords

Links

Crossrefs

Cf. A255187, A254474, A261191.

Programs

  • Mathematica
    Table[n(33n-31)/2, {n, 50}]
PolygonalNumber[35,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 12 2017 *)
  • PARI
    for(n=0,100,print1(n*(33*n-31)/2,", ")) \\ Derek Orr, Feb 27 2017

Formula

From Nikolaos Pantelidis, Feb 10 2023: (Start)
G.f.: x*(1 + 32*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 33*x^2/2). (End)

A282854 34-gonal numbers: a(n) = n*(32*n-30)/2.

Original entry on oeis.org

0, 1, 34, 99, 196, 325, 486, 679, 904, 1161, 1450, 1771, 2124, 2509, 2926, 3375, 3856, 4369, 4914, 5491, 6100, 6741, 7414, 8119, 8856, 9625, 10426, 11259, 12124, 13021, 13950, 14911, 15904, 16929, 17986, 19075, 20196, 21349, 22534, 23751
Offset: 0

Views

Author

Daniel Mohebiravesh, Feb 23 2017

Keywords

Links

Crossrefs

Cf. A246645, A254474, A261191.

Programs

  • Mathematica
    Table[n(32n-30)/2, {n,50}]
PolygonalNumber[34,Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,1,34},40] (* The first program requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 26 2018 *)
  • PARI
    a(n)=n*(16*n-15) \\ Charles R Greathouse IV, Feb 27 2017

Formula

From Nikolaos Pantelidis, Feb 09 2023 : (Start)
G.f.: x*(1 + 31*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 16*x^2). (End)

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

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Apr 27 2020

Keywords

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

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.
Previous Showing 11-14 of 14 results.

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