A059989 Numbers k such that 3*k+1 and 4*k+1 are both squares.

0, 56, 10920, 2118480, 410974256, 79726887240, 15466605150360, 3000441672282656, 582070217817684960, 112918621814958599640, 21905630561884150645256, 4249579410383710266580080, 824396499983877907565890320, 159928671417461930357516142056, 31025337858487630611450565668600

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..400

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Formula

a(n) = (A001570(n)^2 - 1)/3.

G.f.: 56*x^2 / (1-195*x+195*x^2-x^3).

From Colin Barker, Mar 03 2016: (Start)

a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n > 3.

a(n) = (-1)*((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(7+4*sqrt(3)+(-7+4*sqrt(3))*(97+56*sqrt(3))^n))/48. (End)

Sum_{n>=2} 1/a(n) = (7 - 4*sqrt(3))/4. - Amiram Eldar, Jan 27 2026

Example

3*56+1 = 13^2 and 4*56+1 = 15^2.

Maple

f:= proc(n) local u;
  u:= <<7, 8>|<6, 7>>^n . <1, -1>;
  (u[1]^2-1)/3
end proc:
map(f, [$1..30]); # Robert Israel, Mar 03 2016

Mathematica

CoefficientList[Series[56 x/(1 - 195 x + 195 x^2 - x^3), {x, 0, 13}], x] (* Michael De Vlieger, Mar 03 2016 *)

Prog

(PARI) isok(n) = issquare(3*n+1) && issquare(4*n+1) \\ Michel Marcus, Jun 08 2013
(PARI) concat(0, Vec(56*x^2/((1-x)*(1-194*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 03 2016

Crossrefs

Cf. A001570, A245031.

Sequence in context: A034204 A275921 A091546 * A352602 A184125 A213865

Adjacent sequences: A059986 A059987 A059988 * A059990 A059991 A059992

Keyword

nonn,easy

Author

David Radcliffe, Mar 07 2001

Extensions

Offset changed to 1 by Joerg Arndt, Mar 03 2016

Status

approved