A299921 Squares that differ from a triangular number by 1.

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

Author

N. J. A. Sloane, Mar 17 2018

Keywords

• nonn

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 1..1959
• Index entries for linear recurrences with constant coefficients, signature (0,0,35,0,0,-35,0,0,1).

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