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 1-3 of 3 results.

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3*n*(4*n-3).

Original entry on oeis.org

0, 3, 30, 81, 156, 255, 378, 525, 696, 891, 1110, 1353, 1620, 1911, 2226, 2565, 2928, 3315, 3726, 4161, 4620, 5103, 5610, 6141, 6696, 7275, 7878, 8505, 9156, 9831, 10530, 11253, 12000, 12771, 13566, 14385, 15228, 16095, 16986, 17901, 18840, 19803, 20790, 21801
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2008

Keywords

Comments

3*A172078(n) = n*a(n) - Sum_{k=0..n-1} a(k). - Bruno Berselli, Dec 12 2010

Examples

			For n=8, a(8) = (1*3 + 5*7 + 9*11 +..+ 29*31) - (2*4 + 6*8 + 10*12 +..+ 26*28) = 696 (see Problem 1052 in References). - _Bruno Berselli_, Dec 12 2010

References

  • "Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Jan. 1910 p. 47 (Problem 1052).

Links

Crossrefs

Cf. A001107, A001539, A002939, A139271, A153794, A172078.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=24: see Comments lines of A226492.

Programs

Formula

a(n) = 12*n^2 - 9*n = 3*A001107(n).
a(n) = a(n-1) + 24*n - 21, n > 0. - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k=0..n-1} A001539(k) - Sum_{k=0..n-1} 4*A002939(k) if n > 0 (see References, Problem 1052). - Bruno Berselli, Dec 08 2010 - Jan 21 2011
G.f.: -3*x*(1+7*x)/(x-1)^3.
a(0)=0, a(1)=3, a(2)=30, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 26 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 4*x).
a(n) = A153794(n) - n. (End)

A153808 8 times octagonal numbers: 8*n*(3*n-2).

Original entry on oeis.org

0, 8, 64, 168, 320, 520, 768, 1064, 1408, 1800, 2240, 2728, 3264, 3848, 4480, 5160, 5888, 6664, 7488, 8360, 9280, 10248, 11264, 12328, 13440, 14600, 15808, 17064, 18368, 19720, 21120, 22568, 24064, 25608, 27200, 28840, 30528, 32264
Offset: 0

Views

Author

Omar E. Pol, Jan 19 2009

Keywords

Links

Crossrefs

Cf. A000567 (octagonal numbers), A064201 (9 times octagonal numbers), A139267 (twice octagonal numbers), A152751 (3 times octagonal numbers), A153794 (4 times octagonal numbers).

Programs

  • Magma
    [ 8*n*(3*n-2): n in [0..40] ];
  • Mathematica
    Table[8*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,8,64}, 25] (* G. C. Greubel, Aug 29 2016 *)
8*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, Nov 22 2023 *)
  • PARI
    a(n)=24*n^2-16*n \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 24*n^2 - 16*n = 8*A000567(n) = 4*A139267(n) = 2*A153794(n).
a(n) = a(n-1) + 48*n - 40 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 8*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 8*x*(1 + 3*x)*exp(x). (End)

A153795 5 times octagonal numbers: a(n) = 5*n*(3*n-2).

Original entry on oeis.org

0, 5, 40, 105, 200, 325, 480, 665, 880, 1125, 1400, 1705, 2040, 2405, 2800, 3225, 3680, 4165, 4680, 5225, 5800, 6405, 7040, 7705, 8400, 9125, 9880, 10665, 11480, 12325, 13200, 14105, 15040, 16005, 17000, 18025, 19080, 20165, 21280
Offset: 0

Views

Author

Omar E. Pol, Jan 20 2009

Keywords

Links

Crossrefs

Cf. A000567, A153794, A153796.

Programs

  • Mathematica
    Table[5 * n * (3 * n - 2) , {n, 0, 25}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 40}, 25] (* G. C. Greubel, Aug 28 2016 *)
  • PARI
    a(n)=5*n*(3*n-2) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 15*n^2 - 10*n = A000567(n)*5.
a(n) = 30*n + a(n-1) - 25 for n > 0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 5*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 5*x*(1 + 3*x)*exp(x). (End)
Showing 1-3 of 3 results.

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