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.

A259157 Positive triangular numbers (A000217) that are hexagonal numbers (A000384) divided by 2.

Original entry on oeis.org

3, 3570, 4119885, 4754343828, 5486508657735, 6331426236682470, 7306460390622912753, 8431648959352604634600, 9730115592632515125415755, 11228544962248963102125146778, 12957731156319710787337293966165, 14953210525847983999624135111807740
Offset: 1

Views

Author

Colin Barker, Jun 19 2015

Keywords

Comments

Intersection of A000217 and A033991 (even hexagonal numbers divided by 2). - Michel Marcus, Jun 20 2015

Examples

			3 is in the sequence because 3 is the 2nd triangular number, and 2*3 is the 2nd hexagonal number.

Links

Crossrefs

Cf. A000217, A000384, A033991, A259156, A259158-A259167.

Programs

  • Mathematica
    LinearRecurrence[{1155, -1155, 1}, {3, 3570, 4119885}, 20] (* Vincenzo Librandi, Jun 20 2015 *)
  • PARI
    Vec(-3*x*(35*x+1)/((x-1)*(x^2-1154*x+1)) + O(x^20))

Formula

G.f.: -3*x*(35*x+1) / ((x-1)*(x^2-1154*x+1)).
a(n) = 1155*a(n-1) - 1155*a(n-2) + a(n-3). - Wesley Ivan Hurt, Aug 04 2025

A259159 Positive squares (A000290) that are heptagonal numbers (A000566) divided by 2.

Original entry on oeis.org

9, 938961, 97353360225, 10093791093915321, 1046544448101974957529, 108507821458015176452612289, 11250307943363385076857772396401, 1166454428075294670080752381151042025, 120940328000452394039949183305644566845481
Offset: 1

Views

Author

Colin Barker, Jun 19 2015

Keywords

Examples

			9 is in the sequence because 9 is the 3rd square, and 2*9 is the 3rd heptagonal number.

Links

Crossrefs

Cf. A000290, A000566, A259156-A259158, A259160-A259167.

Programs

  • Mathematica
    LinearRecurrence[{103683, -103683, 1},{9, 938961, 97353360225}, 20] (* Vincenzo Librandi, Jun 20 2015 *)
  • PARI
    Vec(-9*x*(x^2+646*x+1)/((x-1)*(x^2-103682*x+1)) + O(x^20))

Formula

G.f.: -9*x*(x^2+646*x+1) / ((x-1)*(x^2-103682*x+1)).

A309437 Number of prime parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 11, 17, 30, 45, 72, 95, 138, 183, 253, 326, 433, 545, 706, 873, 1100, 1345, 1666, 2009, 2451, 2928, 3524, 4169, 4961, 5818, 6859, 7982, 9324, 10778, 12496, 14350, 16519, 18866, 21585, 24521, 27893, 31533, 35688, 40165
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A259158, A309436.

Programs

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} (c(i) + c(j) + c(k) + c(l) + c(m) + c(o) + c(p) + c(n-i-j-k-l-m-o-p)), where c = A010051.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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