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.

A236770 a(n) = n*(n + 1)*(3*n^2 + 3*n - 2)/8.

Original entry on oeis.org

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962
Offset: 0

Views

Author

Bruno Berselli, Jan 31 2014

Keywords

Comments

After 0, first trisection of A011779 and right border of A177708.

Links

Crossrefs

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

Programs

  • Magma
    [n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];
  • Mathematica
    Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)
  • PARI
    for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

Formula

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A264853 a(n) = n*(n + 1)*(5*n^2 + 5*n - 4)/12.

Original entry on oeis.org

0, 1, 13, 56, 160, 365, 721, 1288, 2136, 3345, 5005, 7216, 10088, 13741, 18305, 23920, 30736, 38913, 48621, 60040, 73360, 88781, 106513, 126776, 149800, 175825, 205101, 237888, 274456, 315085, 360065, 409696, 464288, 524161, 589645, 661080, 738816, 823213, 914641
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2015

Keywords

Comments

Partial sums of centered 10-gonal (or decagonal) pyramidal numbers.
Subsequence of A204221. In fact, a(n) is of the form (k^2-1)/15 for k = 5*n*(n+1)/2-1. - Bruno Berselli, Nov 27 2015

Links

Crossrefs

Cf. A004466 (first differences), A201106 (partial sums), A204221.
Cf. similar sequences listed in A264854.

Programs

  • Magma
    [n*(n+1)*(5*n^2+5*n-4)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
  • Mathematica
    Table[n (n + 1) (5 n^2 + 5 n - 4)/12, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,56,160},40] (* Harvey P. Dale, Aug 14 2017 *)
  • PARI
    a(n)=n*(n+1)*(5*n^2+5*n-4)/12 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 8*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004466(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

A264888 a(n) = n*(n + 1)*(13*n^2 + 13*n - 14)/24.

Original entry on oeis.org

0, 1, 16, 71, 205, 470, 931, 1666, 2766, 4335, 6490, 9361, 13091, 17836, 23765, 31060, 39916, 50541, 63156, 77995, 95305, 115346, 138391, 164726, 194650, 228475, 266526, 309141, 356671, 409480, 467945, 532456, 603416, 681241, 766360, 859215, 960261, 1069966
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2015

Keywords

Comments

Partial sum of centered 13-gonal (or tridecagonal) pyramidal numbers.
This is the case k=13 of the formula n*(n + 1)*(k*n^2 + k*n - 2*k + 12)/24.

Links

Crossrefs

Cf. A062025.
Cf. similar sequences listed in A264854.

Programs

  • Mathematica
    Table[n (n + 1) (13 n^2 + 13 n - 14)/24, {n, 0, 50}]
  • PARI
    a(n)=n*(n+1)*(13*n^2+13*n-14)/24 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 11*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A062025(k).
Showing 1-3 of 3 results.

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