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A048910 Indices of 9-gonal numbers that are also square. 2
1, 2, 18, 49, 529, 1458, 15842, 43681, 474721, 1308962, 14225778, 39225169, 426298609, 1175446098, 12774732482, 35224157761, 382815675841, 1055549286722, 11471695542738, 31631254443889, 343768050606289, 947882084029938, 10301569822645922, 28404831266454241 (list; graph; refs; listen; history; text; internal format)
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 * A077591 A050808 A058653

Adjacent sequences:  A048907 A048908 A048909 * A048911 A048912 A048913

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified April 21 10:03 EDT 2019. Contains 322328 sequences. (Running on oeis4.)