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.

A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

Original entry on oeis.org

0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720
Offset: 0

Views

Author

Bruno Berselli, Feb 10 2014

Keywords

Comments

Also 17-gonal (or heptadecagonal) pyramidal numbers.
This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

Examples

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  16;
   3,  32,  31;
   4,  48,  62,  46;
   5,  64,  93,  92,  61;
   6,  80, 124, 138, 122,  76;
   7,  96, 155, 184, 183, 152,  91;
   8, 112, 186, 230, 244, 228, 182, 106;
   9, 128, 217, 276, 305, 304, 273, 212, 121;
  10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 15*r-14 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.

References

  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).

Links

Crossrefs

Cf. A051869, A104728.
Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).

Programs

  • GAP
    List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019
  • Magma
    [n*(n+1)*(5*n-4)/2: n in [0..40]];
  • Magma
    I:=[0,1,18,66]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 12 2014
  • Maple
    seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019
  • Mathematica
    Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* Harvey P. Dale, Jan 11 2015 *)
  • PARI
    a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015
  • Sage
    [n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

G.f.: x*(1 + 14*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
a(n) = A104728(A001844(n-1)) for n>0.
Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - Vaclav Kotesovec, Dec 07 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

A172085 a(n) = n*(27*n^3 + 22*n^2 - 21*n - 16)/12.

Original entry on oeis.org

0, 1, 41, 212, 660, 1585, 3241, 5936, 10032, 15945, 24145, 35156, 49556, 67977, 91105, 119680, 154496, 196401, 246297, 305140, 373940, 453761, 545721, 650992, 770800, 906425, 1059201, 1230516, 1421812, 1634585, 1870385, 2130816, 2417536
Offset: 0

Views

Author

Vincenzo Librandi, Jan 25 2010

Keywords

Comments

The sequence is related to A172082 by a(n) = n*A172082(n)-sum(A172082(i), i=0..n-1).
This is the case d=9 in the identity n^2*(n+1)*(2*d*n -2*d +3)/6 - Sum_{k=0..n-1} k*(k+1)*(2*d*k -2*d +3)/6 = n*(n+1)*(3*d*n^2 - d*n + 4*n - 2*d + 2)/12. - Bruno Berselli, May 07 2010, Jan 28 2011

Links

Crossrefs

Cf. A172082.

Programs

  • GAP
    List([0..40], n-> n*(n+1)*(27*n^2 -5*n -16)/12); # G. C. Greubel, Aug 30 2019
  • Magma
    [n*(n+1)*(27*n^2-5*n-16)/12: n in [0..40]]; // Vincenzo Librandi, Jan 02 2014
  • Maple
    seq(n*(n+1)*(27*n^2 -5*n -16)/12, n=0..40); # G. C. Greubel, Aug 30 2019
  • Mathematica
    CoefficientList[Series[x(1 +36x +17x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 02 2014 *)
Table[n*(n+1)*(27*n^2-5*n-16)/12, {n,0,40}] (* G. C. Greubel, Aug 30 2019 *)
  • PARI
    vector(40, n, m=n-1; n*m*(27*m^2 -5*m -16)/12) \\ G. C. Greubel, Aug 30 2019
  • Sage
    [n*(n+1)*(27*n^2 -5*n -16)/12 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

a(n) = n*(n+1)*(27*n^2 -5*n -16)/12.
From Bruno Berselli, Jan 28 2011: (Start)
G.f.: x*(1 +36*x +17*x^2)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
E.g.f.: x*(12 + 234*x + 184*x^2 + 27*x^3)*exp(x)/12. - G. C. Greubel, Aug 30 2019

Extensions

Librandi's contribution restored and rewritten from Bruno Berselli, Feb 29 2012

A264852 a(n) = n*(n + 1)*(n + 2)*(9*n - 7)/12.

Original entry on oeis.org

0, 1, 22, 100, 290, 665, 1316, 2352, 3900, 6105, 9130, 13156, 18382, 25025, 33320, 43520, 55896, 70737, 88350, 109060, 133210, 161161, 193292, 230000, 271700, 318825, 371826, 431172, 497350, 570865, 652240, 742016, 840752, 949025, 1067430, 1196580, 1337106
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2015

Keywords

Comments

Partial sums of 20-gonal (or icosagonal) pyramidal numbers. Therefore, this is the case k=9 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

Links

Crossrefs

Cf. A172082.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

Programs

  • Magma
    [n*(n+1)*(n+2)*(9*n-7)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
  • Mathematica
    Table[n (n + 1) (n + 2) (9 n - 7)/12, {n, 0, 50}]
  • PARI
    a(n)=n*(n+1)*(n+2)*(9*n-7)/12 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 17*x)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A172082(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015
Showing 1-3 of 3 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.