A046195 Indices of heptagonal numbers (A000566) which are also squares (A000290).

1, 6, 49, 961, 8214, 70225, 1385329, 11844150, 101263969, 1997643025, 17079255654, 146022572641, 2880599856289, 24628274808486, 210564448483921, 4153822995125281, 35513955194580726, 303633788691241009, 5989809878370798481, 51211098762310597974

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

(10 * a(n) - 3)^2 - 40 * (A046196(n))^2 = 9. - Ant King, Nov 12 2011

Also numbers n such that the n-th heptagonal number is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 11 2014

Also indices of heptagonal numbers (A000566) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 19 2015

Also nonnegative integers y in the solutions to 2*x^2-5*y^2+4*x+3*y+2+2 = 0, the corresponding values of x being A251927. - Colin Barker, Dec 11 2014

Links

• Colin Barker, Table of n, a(n) for n = 1..950
• Eric Weisstein's World of Mathematics, Heptagonal Square Number.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1442,-1442,0,-1,1).

Crossrefs

Cf. A000566, A036254, A046196, A251927, A253920.

Programs

Maple

for n from 1 to 10000 do m:=sqrt((5*n^2-3*n)/2):
if (trunc(m)=m) then print(n,m): end if: end do: # Paul Weisenhorn, May 01 2009

Mathematica

LinearRecurrence[{1 ,0, 1442, -1442, 0, -1, 1}, {1, 6, 49, 961, 8214, 70225, 1385329}, 17] (* Ant King, Nov 12 2011 *)

Formula

From Paul Weisenhorn, May 01 2009: (Start)

Pell equations: r^2-10*s^2=1 with solution (19,6)

(10*n-3)^2-10*(2*m)^2=9; basic solutions: (7,-2); (7,+2)((57,18);

with x=10*n-3; y=2*m; A=(19+6*sqrt(10))^2; B=(19-6*sqrt(10))^2 one get

x(3*k)+sqrt(10)*y(3*k)=(7-2*sqrt(10))*A^k;

x(3*k+1)+sqrt(10)*y(3*k+1)=(7+2*sqrt(10))*A^k;

x(3*k+2)+sqrt(10)*y(3*k+2)=(57+18*sqrt(10))*A^k;

with the eigenvalues A=721+228*sqrt(10); B=721-228*sqrt(10)

one get the recurrences with 1442=4*19*19-2

x(k+6)=1442*x(k+3)-x(k); y(k+6)=1442*y(k+3)-y(k);

m(k+6)=1442*m(k+3)-m(k); n(k+6)=1442*n(k+3)-n(k)-432;

and the explicit formulas

x(3*k+1)=(7*(A^k+B^k)+2*sqrt(10)*(A^k-B^k))/2;

x(3*k+2)=(57*(A^k+B^k)+18*sqrt(10)*(A^k-B^k))/2;

x(3*k)=(7*(A^k+B^k)-2*sqrt(10)*(A^k-B^k))/2;

y(3*k+1)=(7*(A^k-B^k)/sqrt(10)+2*(A^k+B^k)/2;

y(3*k+2)=(57*(A^k-B^k)/sqrt(10)+18*(A^k+B^k))/2;

y(3*k)=(7*(A^k-B^k)/sqrt(10)-2*(A^k+B^k))/2;

n(k)=(x(k)+3)/10; m(k)=y(k)/2;

(End)

a(n) = +a(n-1) +1442*a(n-3) -1442*a(n-4) -a(n-6) +a(n-7). G.f.: -x*(1+5*x+43*x^2-530*x^3+43*x^4+5*x^5+x^6) / ( (x-1)*(x^6-1442*x^3+1) ). - R. J. Mathar, Aug 01 2010

a(n) = 1442*a(n-3) - a(n-6) - 432. - Ant King, Nov 12 2011

Extensions

More terms from Colin Barker, Dec 11 2014