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 1-4 of 4 results.

A329753 Doubly square pyramidal numbers.

Original entry on oeis.org

0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035, 19096385, 43312841, 91753025, 183453270, 349074740, 636310340, 1117143236, 1897397285, 3129084635, 5026125195, 7884086595, 12104671656, 18225763270, 26957923950, 39228339150, 56233289775, 79500340101, 110961532605
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000290, A000330, A000583, A140236, A329754, A329755, A329756, A329757.

Programs

  • Mathematica
    A000330[n_] := n (n + 1) (2 n + 1)/6; a[n_] := A000330[A000330[n]]; Table[a[n], {n, 0, 27}]
Table[Sum[k^2, {k, 0, n (n + 1) (2 n + 1)/6}], {n, 0, 27}]
nmax = 27; CoefficientList[Series[x (1 + 45 x + 510 x^2 + 1660 x^3 + 1715 x^4 + 519 x^5 + 30 x^6)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035}, 28]

Formula

G.f.: x*(1 + 45*x + 510*x^2 + 1660*x^3 + 1715*x^4 + 519*x^5 + 30*x^6)/(1 - x)^10.
a(n) = A000330(A000330(n)).
a(n) = Sum_{k=0..A000330(n)} A000290(k).
a(n) = n *(2*n+1) *(n+2) *(n+1) *(2*n^2-n+3) *(2*n^3+3*n^2+n+3) /648. - R. J. Mathar, Nov 28 2019

A329754 Doubly pentagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075, 83338750, 191592126, 410450976, 828497488, 1589341950, 2917620000, 5154021376, 8801526501, 14585352318, 23529456550, 37052820000, 57089119626, 86233820926, 127923156648, 186649920000, 268221484375, 380065968126
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000326, A002411, A140236, A232713, A329753, A329755, A329756, A329757.

Programs

  • Mathematica
    A002411[n_] := n^2 (n + 1)/2; a[n_] := A002411[A002411[n]]; Table[a[n], {n, 0, 26}]
Table[Sum[k (3 k - 1)/2, {k, 0, n^2 (n + 1)/2}], {n, 0, 26}]
nmax = 26; CoefficientList[Series[x (1 + 116 x + 1863 x^2 + 7570 x^3 + 9350 x^4 + 3474 x^5 + 304 x^6 + 2 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075}, 27]

Formula

G.f.: x*(1 + 116*x + 1863*x^2 + 7570*x^3 + 9350*x^4 + 3474*x^5 + 304*x^6 + 2*x^7)/(1 - x)^10.
a(n) = A002411(A002411(n)).
a(n) = Sum_{k=0..A002411(n)} A000326(k).
a(n) = n^4 *(n^3+n^2+2) *(n+1)^2 /16. - R. J. Mathar, Nov 28 2019

A329755 Doubly hexagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475, 243939410, 564840991, 1217275137, 2469392562, 4757404575, 8765621740, 15534503236, 26603512517, 44196596312, 71459197125, 112756874195, 174046844356, 263335062397, 391232840362, 571628456750, 822490729775
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000384, A002412, A063249, A140236, A329753, A329754, A329756, A329757.

Programs

  • Mathematica
    A002412[n_] := n (n + 1) (4 n - 1)/6; a[n_] := A002412[A002412[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (2 k - 1), {k, 0, n (n + 1) (4 n - 1)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 242 x + 4862 x^2 + 22425 x^3 + 30465 x^4 + 12424 x^5 + 1248 x^6 + 13 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475}, 26]

Formula

G.f.: x*(1 + 242*x + 4862*x^2 + 22425*x^3 + 30465*x^4 + 12424*x^5 + 1248*x^6 + 13*x^7)/(1 - x)^10.
a(n) = A002412(A002412(n)).
a(n) = Sum_{k=0..A002412(n)} A000384(k).
a(n) = n *(4*n-1) *(n+1) *(4*n^3+3*n^2-n+6) *(8*n^3+6*n^2-2*n-3) / 648 . - R. J. Mathar, Nov 28 2019

A329756 Doubly heptagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235, 568280240, 1321714636, 2858876956, 5817509516, 11237224740, 20751835560, 36849296016, 63215722181, 105182448536, 170297734920, 269047574180, 415753060646, 629674964556, 936359517556
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000566, A002413, A140236, A264891, A329753, A329754, A329755, A329757.

Programs

  • Mathematica
    A002413[n_] := n (n + 1) (5 n - 2)/6; a[n_] := A002413[A002413[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (5 k - 3)/2, {k, 0, n (n + 1) (5 n - 2)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 446 x + 10461 x^2 + 52420 x^3 + 75580 x^4 + 32544 x^5 + 3504 x^6 + 44 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235}, 26]

Formula

G.f.: x*(1 + 446*x + 10461*x^2 + 52420*x^3 + 75580*x^4 + 32544*x^5 + 3504*x^6 + 44*x^7)/(1 - x)^10.
a(n) = A002413(A002413(n)).
a(n) = Sum_{k=0..A002413(n)} A000566(k).
a(n) = n *(5*n-2) *(n+1) *(5*n^3+3*n^2-2*n+6) *(25*n^3+15*n^2-10*n-12)/1296. - R. J. Mathar, Nov 28 2019
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.