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.

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Original entry on oeis.org

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151
Offset: 0

Views

Author

Bruno Berselli, Dec 08 2012

Keywords

Comments

Partial sums of A172073.
Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Links

Crossrefs

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

Programs

  • Magma
    A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];
  • Magma
    I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A264851 a(n) = n*(n + 1)*(n + 2)*(4*n - 3)/6.

Original entry on oeis.org

0, 1, 20, 90, 260, 595, 1176, 2100, 3480, 5445, 8140, 11726, 16380, 22295, 29680, 38760, 49776, 62985, 78660, 97090, 118580, 143451, 172040, 204700, 241800, 283725, 330876, 383670, 442540, 507935, 580320, 660176, 748000, 844305, 949620, 1064490, 1189476
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2015

Keywords

Comments

Partial sums of 18-gonal (or octadecagonal) pyramidal numbers. Therefore, this is the case k=8 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. A172078.
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)*(4*n - 3)/6: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
  • Mathematica
    Table[n (n + 1) (n + 2) (4 n - 3)/6, {n, 0, 50}]
  • PARI
    a(n)=n*(n+1)*(n+2)*(4*n-3)/6 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 15*x)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A172078(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

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.

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