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A280112
Indices of centered 10-gonal numbers (A062786) that are also triangular numbers (A000217).
3
1, 19, 703, 26677, 1013005, 38467495, 1460751787, 55470100393, 2106403063129, 79987846298491, 3037431756279511, 115342418892322909, 4379974486151991013, 166323688054883335567, 6315920171599414760515, 239838642832722877563985, 9107552507471869932670897
OFFSET
1,2
COMMENTS
Also positive integers y in the solutions to x^2 - 10*y^2 + x + 10*y - 2 = 0, the corresponding values of x being A280111.
FORMULA
a(n) = 1/2 + (19 + 6*sqrt(10))^(-n)*(10+3*sqrt(10) + (10-3*sqrt(10))*(19+6*sqrt(10))^(2*n)) / 40.
a(n) = 39*a(n-1) - 39*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 - 20*x + x^2) / ((1 - x)*(1 - 38*x + x^2)).
EXAMPLE
19 is in the sequence because the 19th centered 10-gonal number is 1711, which is also the 58th triangular number.
MATHEMATICA
Table[Simplify[1/2 + (19 + 6 #)^(-n) (10 + 3 # + (10 - 3 #) (19 + 6*#)^(2 n))/40] &@ Sqrt@ 10, {n, 17}] (* or *)
Rest@ CoefficientList[Series[x (1 - 20 x + x^2)/((1 - x) (1 - 38 x + x^2)), {x, 0, 17}], x] (* Michael De Vlieger, Dec 26 2016 *)
PROG
(PARI) Vec(x*(1 - 20*x + x^2) / ((1 - x)*(1 - 38*x + x^2)) + O(x^20))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Dec 26 2016
STATUS
approved