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.

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A131751 Numbers that are both centered triangular and centered pentagonal.

Original entry on oeis.org

1, 31, 1891, 117181, 7263301, 450207451, 27905598631, 1729696907641, 107213302675081, 6645495068947351, 411913480972060651, 25531990325198812981, 1582571486681354344141, 98093900183918770523731, 6080239239916282418127151, 376876738974625591153359601
Offset: 1

Views

Author

Richard Choulet, Sep 20 2007

Keywords

Comments

We solve 0.5*(3*p^2+3*p+2)=0.5*(5*r^2+5*r+2), i.e., 3*(2*p+1)^2=5*(2*r+1)^2-2.
The Diophantine equation 3*X^2=5*Y^2-2is such that : X is given by A057080 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2+10)^0.5, Y is given by A070997 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2-6)^0.5 while r is given by the sequence 0,3,27,216,1704,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*(60*a(n)^2+60*a(n)+9)^0.5, p is given by the sequence 0,4,35,279,2200,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*sqrt(60*a(n)^2+60*a(n)+25).

Links

Crossrefs

Cf. A050807 A070997.

Programs

  • Maple
    A131751 := proc(n) coeftayl(x*(1-32*x+x^2)/(1-x)/(1-62*x+x^2),x=0,n) ; end: seq(A131751(n),n=1..20) ; # R. J. Mathar, Oct 24 2007
  • Mathematica
    LinearRecurrence[{63,-63,1},{1,31,1891},20] (* Harvey P. Dale, Oct 01 2017 *)

Formula

a(n+2) = 62*a(n+1) - a(n) - 30, a(n+1) = 31*a(n) - 15 + sqrt(960*a(n)^2 - 960*a(n)+225).
G.f.: f(z) = a(1)*z+a(2)*z^2+... = z*(1-32*z+z^2)/((1-z)*(1-62*z+z^2)).
A005891 INTERSECT A005448. - R. J. Mathar, Oct 24 2007

Extensions

Corrected and extended by R. J. Mathar, Oct 24 2007

A145607 Numbers k such that (3*(2*k + 1)^2 + 2)/5 is a square.

Original entry on oeis.org

0, 4, 35, 279, 2200, 17324, 136395, 1073839, 8454320, 66560724, 524031475, 4125691079, 32481497160, 255726286204, 2013328792475, 15850904053599, 124793903636320, 982500325036964, 7735208696659395, 60899169248238199
Offset: 1

Views

Author

Richard Choulet, Oct 14 2008

Keywords

Comments

Square roots of (3*(2*k+1)^2+2)/5 are listed in A070997, therefore (3*(2*a(n) + 1)^2 + 2)/5 = A070997(n-1)^2.

Links

Crossrefs

Cf. A070997, A131751.
Cf. A001091 (first differences).

Formula

a(n+2) = 8*a(n+1) - a(n) + 3.
From R. J. Mathar, Oct 24 2008: (Start)
G.f.: x^2*(4 - x)/((1 - x)*(1 - 8*x + x^2)).
a(n) = (A057080(n-1)-1)/2. (End)

Extensions

a(4) corrected, extended, definition corrected by R. J. Mathar, Oct 24 2008
Offset changed by Bruno Berselli, Apr 06 2018
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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