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 1-2 of 2 results.

A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.

Original entry on oeis.org

0, 3, 18, 105, 612, 3567, 20790, 121173, 706248, 4116315, 23991642, 139833537, 815009580, 4750223943, 27686334078, 161367780525, 940520349072, 5481754313907, 31950005534370, 186218278892313, 1085359667819508, 6325939728024735, 36870278700328902, 214895732473948677
Offset: 1

Views

Author

Pierre CAMI, Apr 29 2005

Keywords

Comments

The ratio k(n) /(2*j(n)) tends to sqrt(2) as n increases.
The squares of the numbers in this sequence are one less than a triangular number: a(n)^2 = A164080(n). For example, 18^2 is 324, and 325 is a triangular number. a(n)^2 + 1 = A164055(n). a(n)^2 = A072221(n)(A072221(n)+1)/2 - 1. - Tanya Khovanova & Alexey Radul, Aug 09 2009
For n > 0, a(n+1) is the n-th almost balancing number of first type (see Tekcan and Erdem). - Stefano Spezia, Nov 25 2022

Links

Crossrefs

Cf. A001109, A005319, A072221, A103328, A164080, A164055.

Programs

  • Haskell
    a106328 n = a106328_list !! (n-1)
a106328_list = 0 : 3 : zipWith (-) (map (* 6) (tail a106328_list)) a106328_list
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    s=0;lst={};Do[s+=n;If[Sqrt[s-1]==Floor[Sqrt[s-1]],AppendTo[lst,Sqrt[s-1]]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Rest@ CoefficientList[Series[3 x^2/(1 - 6 x + x^2), {x, 0, 24}], x] (* Michael De Vlieger, Nov 02 2020 *)
  • PARI
    concat(0, Vec(3*x^2/(1-6*x+x^2) + O(x^40))) \\ Michel Marcus, Sep 07 2016
  • PARI
    a(n)=([0,1;-1,6]^n*[-3;0])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

Formula

a(1)=0, a(2)=3 then a(n) = 6*a(n-1) - a(n-2).
a(n) = ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1))*3/4/sqrt(2). - Max Alekseyev, Jan 11 2007
a(n) = 3*A001109(n). - M. F. Hasler, R. J. Mathar, Jun 03 2009
a(n) = (3/4)*A005319(n-1).
G.f.: 3*x^2/(1 - 6*x + x^2). - Philippe Deléham, Nov 17 2008
E.g.f.: 3 - 3*exp(3*x)*(4*cosh(2*sqrt(2)*x) - 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, Nov 25 2022

Extensions

More terms from Max Alekseyev, Jan 11 2007

A299921 Squares that differ from a triangular number by 1.

Original entry on oeis.org

0, 1, 4, 9, 16, 121, 324, 529, 4096, 11025, 17956, 139129, 374544, 609961, 4726276, 12723489, 20720704, 160554241, 432224100, 703893961, 5454117904, 14682895929, 23911673956, 185279454481, 498786237504, 812293020529, 6294047334436, 16944049179225, 27594051024016
Offset: 1

Views

Author

N. J. A. Sloane, Mar 17 2018

Keywords

Comments

Squares k such that 8*k-7 or 8*k+9 is a square. - Robert Israel, Mar 18 2018

Links

Crossrefs

Cf. A001110, A164080, A182334, A229131.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n+9) = 35*a(n+6) - 35*a(n+3) + a(n), seq(a(i)=[0, 1, 4, 9, 16, 121, 324, 529, 4096][i],i=1..9)}, a(n), remember):
map(f, [$1..50]); # Robert Israel, Mar 18 2018
  • Mathematica
    LinearRecurrence[{0, 0, 35, 0, 0, -35, 0, 0, 1}, {0, 1, 4, 9, 16, 121, 324, 529, 4096}, 50] (* Jean-François Alcover, Sep 17 2022 *)
  • PARI
    isok(n) = issquare(n) && (ispolygonal(n+1, 3) || ispolygonal(n-1, 3)); \\ Michel Marcus, Mar 17 2018

Formula

From Robert Israel, Mar 18 2018: (Start)
G.f.: x^2*(1+4*x+9*x^2-19*x^3-19*x^4+9*x^5+4*x^6+x^7)/(1-35*x^3+35*x^6-x^9).
a(n) = 35*a(n-3) - 35*a(n-6) + a(n-9). (End)

Extensions

More terms from Altug Alkan, Mar 17 2018
Showing 1-2 of 2 results.

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

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