A048910 Indices of 9-gonal numbers that are also square.

1, 2, 18, 49, 529, 1458, 15842, 43681, 474721, 1308962, 14225778, 39225169, 426298609, 1175446098, 12774732482, 35224157761, 382815675841, 1055549286722, 11471695542738, 31631254443889, 343768050606289, 947882084029938, 10301569822645922, 28404831266454241

Offset

1,2

Comments

From Ant King, Nov 18 2011: (Start)

lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)) = 1/25 * (9 + 2 * sqrt(14))^2.

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

(14 * a(n) - 5)^2 - 56 * A048911(n) ^ 2 = 25.

(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 (1,30,-30,-1,1).

Formula

From Ant King, Nov 18 2011: (Start)

a(n) = 30 * a(n - 2) - a(n-4) - 10.

a(n) = a(n - 1) + 30 * a(n - 2) - 30 * a(n - 3) - a(n - 4) + a(n - 5).

Let p = 9 + 4 * sqrt(2) + sqrt(7) + 2 * sqrt(14) and q = 9 - 4 * sqrt(2) - sqrt(7) + 2 * sqrt(14). Then

a(n) = 1/56 * ( ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) + ( p + q * (-1)^n) * ( 2 * sqrt(2) - sqrt(7))^n + 20 ).

a(n) = ceiling (1/56 * ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) ).

G.f.: x * (1 + x - 14 * x^2 + x^3 + x^4) / ((1 - x) * (1 - 30 * x^2 + x^4)).

(End)

Mathematica

LinearRecurrence[ {1, 30, - 30, -1, 1 }, {1, 2, 18, 49, 529}, 21 ] (* Ant King, Nov 18 2011 *)

Prog

(PARI) Vec(-x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^4-30*x^2+1)) + O(x^50)) \\ Colin Barker, Jun 22 2015

Crossrefs

Cf. A048911, A036411.

Sequence in context: A052681 A208652 A223469 * A356712 A077591 A050808

Adjacent sequences: A048907 A048908 A048909 * A048911 A048912 A048913

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved