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.

Showing 1-6 of 6 results.

A322636 Numbers that are sums of consecutive heptagonal numbers (A000566).

Original entry on oeis.org

0, 1, 7, 8, 18, 25, 26, 34, 52, 55, 59, 60, 81, 89, 107, 112, 114, 115, 136, 148, 170, 188, 189, 193, 195, 196, 235, 248, 260, 282, 286, 300, 307, 308, 337, 341, 342, 396, 403, 424, 430, 448, 449, 455, 456, 469, 521, 530, 540, 572, 585, 616, 619, 628, 637, 644, 645, 684, 697
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A000566, A002413, A034705, A034706, A319184, A319185, A322637.

Programs

  • Maple
    N:= 1000: # for terms up to N
Hepta:= [seq(n*(5*n-3)/2,n=0..floor((3+sqrt(9+40*N))/10))]:
PS:= ListTools:-PartialSums(Hepta):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025
  • Mathematica
    terms = 59;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(5n-3)/2, {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

Original entry on oeis.org

0, 1, 8, 9, 21, 29, 30, 40, 61, 65, 69, 70, 96, 105, 126, 133, 134, 135, 161, 176, 201, 222, 225, 229, 230, 231, 280, 294, 309, 334, 341, 355, 363, 364, 401, 405, 408, 470, 481, 505, 510, 531, 534, 539, 540, 560, 621, 630, 645, 681, 695, 735, 736, 749, 756, 764, 765, 814, 833, 846
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A000567, A002414, A034705, A034706, A319184, A319185, A322636.

Programs

  • Maple
    N:= 1000: # for terms up to N
Octa:= [seq(n*(3*n-2),n=0..floor((1+sqrt(1+3*N))/3))]:
PS:= ListTools:-PartialSums(Octa):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025
  • Mathematica
    terms = 60;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(3n-2), {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A322638 Numbers that are sums of consecutive centered pentagonal numbers (A005891).

Original entry on oeis.org

0, 1, 6, 7, 16, 22, 23, 31, 47, 51, 53, 54, 76, 82, 98, 104, 105, 106, 127, 141, 158, 174, 180, 181, 182, 226, 233, 247, 264, 276, 280, 286, 287, 322, 323, 331, 374, 391, 405, 407, 421, 427, 428, 456, 502, 504, 526, 548, 555, 586, 601, 602, 607, 608, 609, 654, 681, 683, 722
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A004068, A005891, A152043, A319184, A322610, A322611, A322640.

Programs

  • Maple
    L:= [seq((5*n^2+5*n+2)/2,n=0..30)]: N:= L[-1]:
S:=[0,op(ListTools:-PartialSums(L))]:
R:=select(`<=`,{0,seq(seq(S[n]-S[m],m=1..n-1),n=1..nops(S))},N):
sort(convert(R,list)); # Robert Israel, Mar 19 2023
  • Mathematica
    terms = 59;
nmax = 16; kmax = 9; (* empirical *)
T = Table[(5n^2 + 5n + 2)/2, {n, 0, nmax}];
Union[{0}, T, Table[k MovingAverage[T, k], {k, 2, kmax}] // Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 287, 472320, 89051435880
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			Let S(k, m) denote the sum of m pentagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(14, 1) = S(2, 7);
a(3) = S(103, 24) = S(19, 80) = S(67, 41);
a(4) = S(10833, 484) = S(4542, 1936) = S(9153, 660) = S(2817, 3036);

Links

Crossrefs

Cf. A000326, A054859, A068314, A186337, A298467, A319184, A334007, A334010, A334011, A334012.

Extensions

a(4) from Giovanni Resta, Apr 13 2020

A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

Original entry on oeis.org

0, 1, 6, 7, 15, 21, 22, 28, 43, 45, 49, 50, 66, 73, 88, 91, 94, 95, 111, 120, 139, 153, 154, 157, 160, 161, 190, 202, 211, 230, 231, 245, 251, 252, 273, 276, 277, 322, 325, 343, 350, 364, 365, 371, 372, 378, 421, 430, 435, 463, 475, 496, 503, 507, 518, 524, 525, 554, 561
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A000384, A002412, A034705, A034706, A319184, A322636, A322637.

Programs

  • Mathematica
    anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k*(2*k-1), {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A321450 Numbers that are sums of consecutive pentagonal pyramidal numbers (A002411).

Original entry on oeis.org

0, 1, 6, 7, 18, 24, 25, 40, 58, 64, 65, 75, 115, 126, 133, 139, 140, 196, 201, 241, 259, 265, 266, 288, 322, 397, 405, 437, 455, 461, 462, 484, 550, 610, 685, 693, 725, 726, 743, 749, 750, 889, 936, 955, 1015, 1090, 1130, 1148, 1154, 1155, 1183, 1243, 1276, 1439
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A001296, A002411, A319184, A322468, A322479, A322638, A322651, A322652, A322653.
Showing 1-6 of 6 results.

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