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

1,3

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).

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)

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 *)

Prog

(PARI) isok(n) = issquare(n) && (ispolygonal(n+1, 3) || ispolygonal(n-1, 3)); \\ Michel Marcus, Mar 17 2018

Crossrefs

Cf. A001110, A164080, A182334, A229131.

Sequence in context: A352919 A263094 A226354 * A089149 A082920 A204434

Adjacent sequences: A299918 A299919 A299920 * A299922 A299923 A299924

Keyword

nonn

Author

N. J. A. Sloane, Mar 17 2018

Extensions

More terms from Altug Alkan, Mar 17 2018

Status

approved