A048911 Indices of square numbers which are also 9-gonal.

1, 3, 33, 91, 989, 2727, 29637, 81719, 888121, 2448843, 26613993, 73383571, 797531669, 2199058287, 23899336077, 65898365039, 716182550641, 1974751892883, 21461577183153, 59176658421451, 643131132943949, 1773325000750647, 19272472411135317, 53140573364097959

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

From Ant King, Nov 18 2011: (Start)

lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)).

lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).

(End)

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Nonagonal Square Number.
• Index entries for linear recurrences with constant coefficients, signature (0,30,0,-1)

Crossrefs

Cf. A048910, A036411.

Programs

Mathematica

LinearRecurrence[ {0, 30, 0, - 1 }, { 1, 3, 33, 91 } , 21 ] (* Ant King, Nov 18 2011 *)

PARI

Vec(x*(x+1)^3/(x^4-30*x^2+1) + O(x^50)) \\ Colin Barker, Jun 22 2015

Formula

From Ant King, Nov 18 2011: (Start)

a(n) = 30 * a(n-2) - a(n-4).

G.f.: x * (1 + x) ^ 3 / (1 - 30 * x ^ 2 + x ^ 4).

Let p = 8 * sqrt(7) + 9 * sqrt(14) - 7 * sqrt(2) - 28 and q = 7 * sqrt(2) + 9 * sqrt(14) - 8 * sqrt(7) - 28. Then

a(n) = 1/112 * ( ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n - ( p - q * (-1) ^ n) * ( 2 * sqrt(2) - sqrt(7)) ^ ( n - 1) ).

a(n) = floor ( 1/112 * ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n ).

(End)