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A254782
Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).
3
1, 11, 231, 5061, 111101, 2439151, 53550211, 1175665481, 25811090361, 566668322451, 12440892003551, 273132955755661, 5996484134620981, 131649518005905911, 2890292911995309051, 63454794545890893201, 1393115187097604341361, 30585079321601404616731
OFFSET
1,2
COMMENTS
Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 5*x + 6*y = 0, the corresponding values of x being A133285.
The numbers (as opposed to the indices) are A133141.
LINKS
FORMULA
a(n) = 23*a(n-1)-23*a(n-2)+a(n-3).
G.f.: -x*(x^2-12*x+1) / ((x-1)*(x^2-22*x+1)).
a(n) = 1/2+1/24*(11+2*sqrt(30))^(-n)*(6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n)). - Colin Barker, Mar 03 2016
EXAMPLE
11 is in the sequence because the 11th centered hexagonal number is 331, which is also the 12th centered pentagonal number.
MATHEMATICA
LinearRecurrence[{23, -23, 1}, {1, 11, 231}, 20] (* Harvey P. Dale, Mar 01 2022 *)
PROG
(PARI) Vec(-x*(x^2-12*x+1)/((x-1)*(x^2-22*x+1)) + O(x^100))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Feb 07 2015
STATUS
approved