A280113 -id:A280113 - cp's OEIS frontend

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

Colin Barker, Dec 26 2016

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.

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

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (39,-39,1).

Cf. A000217, A062786, A280112, A280113.

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 *)

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

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)).

A280112 Indices of centered 10-gonal numbers (A062786) that are also triangular numbers (A000217).

Original entry on oeis.org

1, 19, 703, 26677, 1013005, 38467495, 1460751787, 55470100393, 2106403063129, 79987846298491, 3037431756279511, 115342418892322909, 4379974486151991013, 166323688054883335567, 6315920171599414760515, 239838642832722877563985, 9107552507471869932670897

Offset: 1

Colin Barker, Dec 26 2016

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.

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

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (39,-39,1).

Cf. A000217, A062786, A280111, A280113.

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 *)

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

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)).

cp's OEIS Frontend

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

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