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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
Eric Weisstein's World of Mathematics, Nonagonal Square Number
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
Sequence in context: A052681 A208652 A223469 * A356712 A077591 A050808
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)