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

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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

A264854 a(n) = n*(n + 1)*(11*n^2 + 11*n - 10)/24.

Original entry on oeis.org

0, 1, 14, 61, 175, 400, 791, 1414, 2346, 3675, 5500, 7931, 11089, 15106, 20125, 26300, 33796, 42789, 53466, 66025, 80675, 97636, 117139, 139426, 164750, 193375, 225576, 261639, 301861, 346550, 396025, 450616, 510664, 576521, 648550, 727125, 812631, 905464, 1006031
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2015

Keywords

Comments

Partial sums of centered 11-gonal (or hendecagonal) pyramidal numbers.

Links

Crossrefs

Cf. A004467.
Cf. similar sequences provided by the partial sums of centered k-gonal pyramidal numbers: A006522 (k=1), A006007 (k=2), A002817 (k=3), A006325 (k=4), A006322 (k=5), A000537 (k=6), A006323 (k=7), A006324 (k=8), A236770 (k=9), A264853 (k=10), this sequence (k=11), A062392 (k=12), A264888 (k=13).

Programs

  • Magma
    [n*(n+1)*(11*n^2+11*n-10)/24: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
  • Mathematica
    Table[n (n + 1) (11 n^2 + 11 n - 10)/24, {n, 0, 50}]
  • PARI
    a(n)=n*(n+1)*(11*n^2+11*n-10)/24 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 9*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004467(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

A270694 Alternating sum of centered heptagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -23, 51, -94, 157, -242, 354, -495, 670, -881, 1133, -1428, 1771, -2164, 2612, -3117, 3684, -4315, 5015, -5786, 6633, -7558, 8566, -9659, 10842, -12117, 13489, -14960, 16535, -18216, 20008, -21913, 23936, -26079, 28347, -30742, 33269
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2016

Keywords

Comments

More generally, the ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers is -x*(1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^4).

Links

Crossrefs

Cf. A004126 (centered heptagonal pyramidal numbers).
Cf. A000330, A006323 (partial sums of centered heptagonal pyramidal numbers), A019298, A232599.

Programs

  • Magma
    [((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016
  • Maple
    A270694:= n-> ((-1)^n*(2*n+1)*(14*n^2+14*n-9) + 9)/48; seq(A270694(n), n=0..40); # G. C. Greubel, Apr 02 2021
  • Mathematica
    LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -23, 51}, 39]
Table[((-1)^n (2 n + 1) (14 n^2 + 14 n - 9) + 9)/48, {n, 0, 38}]
  • PARI
    my(x='x+O('x^50)); concat(0, Vec(-x*(1-5*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016
  • Sage
    [((-1)^n*(2*n+1)*(14*n^2+14*n-9) +9)/48 for n in (0..40)] # G. C. Greubel, Apr 02 2021

Formula

G.f.: -x*(1 - 5*x + x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48.
E.g.f.: (1/48)*(9*exp(x) - (9 + 66*x - 126*x^2 + 28*x^3)*exp(-x)). - G. C. Greubel, Mar 28 2016
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