A280111 Indices of triangular numbers (A000217) that are also centered 10-gonal numbers (A062786).

1, 58, 2221, 84358, 3203401, 121644898, 4619302741, 175411859278, 6661031349841, 252943779434698, 9605202587168701, 364744754532975958, 13850695469665917721, 525961683092771897458, 19972693262055666185701, 758436382275022543159198, 28800609833188800973863841

Offset

1,2

Comments

Also positive integers x in the solutions to x^2 - 10*y^2 + x + 10*y - 2 = 0, the corresponding values of y being A280112.

Links

Colin Barker, Table of n, a(n) for n = 1..600

Index entries for linear recurrences with constant coefficients, signature (39,-39,1).

Formula

a(n) = (-2 - (3+sqrt(10))*(19+6*sqrt(10))^(-n) + (-3+sqrt(10))*(19+6*sqrt(10))^n) / 4.

a(n) = 39*a(n-1) - 39*a(n-2) + a(n-3) for n>3.

G.f.: x*(1 + 19*x - 2*x^2) / ((1 - x)*(1 - 38*x + x^2)).

Example

58 is in the sequence because the 58th triangular number is 1711, which is also the 19th centered 10-gonal number.

Mathematica

Table[Simplify[(-2 - (3 + #) (19 + 6 #)^(-n) + (-3 + #) (19 + 6 #)^n)/4] &@ Sqrt@ 10, {n, 17}] (* or *)
Rest@ CoefficientList[Series[x (1 + 19 x - 2 x^2)/((1 - x) (1 - 38 x + x^2)), {x, 0, 17}], x] (* Michael De Vlieger, Dec 26 2016 *)

Prog

(PARI) Vec(x*(1 + 19*x - 2*x^2) / ((1 - x)*(1 - 38*x + x^2)) + O(x^20))

Crossrefs

Cf. A000217, A062786, A280112, A280113.

Sequence in context: A017774 A035724 A017721 * A281031 A225352 A278097

Adjacent sequences: A280108 A280109 A280110 * A280112 A280113 A280114

Keyword

nonn,easy

Author

Colin Barker, Dec 26 2016

Status

approved