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.

User: Nikolaos Pantelidis

Nikolaos Pantelidis's wiki page.

Nikolaos Pantelidis has authored 3 sequences.

A360488 31-gonal numbers: a(n) = n*(29*n-27)/2.

Original entry on oeis.org

0, 1, 31, 90, 178, 295, 441, 616, 820, 1053, 1315, 1606, 1926, 2275, 2653, 3060, 3496, 3961, 4455, 4978, 5530, 6111, 6721, 7360, 8028, 8725, 9451, 10206, 10990, 11803, 12645, 13516, 14416, 15345, 16303, 17290, 18306, 19351, 20425, 21528, 22660, 23821, 25011, 26230
Offset: 0

Views

Author

Nikolaos Pantelidis, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A360436, A051867, A051869.
Cf. A126385, A126418, A126499.

Programs

  • Mathematica
    Table[n (29 n - 27)/2, {n, 0, 30}]

Formula

G.f.: x*(1 + 28*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 29*x^2/2).

A360436 32-gonal numbers: a(n) = n*(15*n-14).

Original entry on oeis.org

0, 1, 32, 93, 184, 305, 456, 637, 848, 1089, 1360, 1661, 1992, 2353, 2744, 3165, 3616, 4097, 4608, 5149, 5720, 6321, 6952, 7613, 8304, 9025, 9776, 10557, 11368, 12209, 13080, 13981, 14912, 15873, 16864, 17885, 18936, 20017, 21128, 22269, 23440, 24641, 25872
Offset: 0

Views

Author

Nikolaos Pantelidis, Feb 07 2023

Keywords

Links

Crossrefs

Cf. A051868, A254474, A161935, A282853.

Programs

Formula

G.f.: x*(1 + 29*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 15*x^2).

A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

Original entry on oeis.org

1, 11, 253, 8855, 416185, 24554915, 1743398965, 144702114095, 13746700839025, 1470896989775675, 175036741783305325, 22929813173612997575, 3278963283826658653225, 508239308993132091249875, 84875964601853059238729125, 15192797663731697603732513375
Offset: 0

Views

Author

Nikolaos Pantelidis, Aug 06 2021

Keywords

Links

Crossrefs

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), this sequence (m=12).

Programs

  • Magma
    m:=12; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 16 2022
  • Mathematica
    CoefficientList[Series[(1-12*x)^(-11/12),{x,0,20}], x] * Range[0, 20]!
FullSimplify[Table[12^n Gamma[n+11/12]/Gamma[11/12],{n,0,15}]] (* Stefano Spezia, Aug 07 2021 *)
  • Sage
    m=12; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 16 2022

Formula

G.f.: 1/(1-11*x/(1-12*x/(1-23*x/(1-24*x/(1-35*x/(1-36*x/(1-47*x/(1-48*x/(1-59*x/(1-60*x/(1-...))))))))))) (Stieltjes continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(12*k+11)/(1 - x*(12*k+12)/Q(k+1) ) (continued fraction).
G.f.: 1/(1-11*x-132*x^2/(1-35*x-552*x^2/(1-59*x-1260*x^2/(1-83*x-2256*x^2/(1-107*x-3540*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/G(0) where G(k) = 1 - x*(24*k+11) - 12*(k+1)*(12*k+11)*x^2/G(k+1) (continued fraction).
a(n) = 12^n*Gamma(n+11/12)/Gamma(11/12). - Stefano Spezia, Aug 07 2021
Sum_{n>=0} 1/a(n) = 1 + (e/12)^(1/12)*(Gamma(11/12) - Gamma(11/12, 1/12)). - Amiram Eldar, Dec 22 2022

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