A280112 -id:A280112 - 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)).

A280113 Triangular numbers (A000217) that are also centered 10-gonal numbers (A062786).

Original entry on oeis.org

1, 1711, 2467531, 3558178261, 5130890585101, 7398740665537651, 10668978908814707911, 15384660187770143270281, 22184669321785637781037561, 31990277777354701910112892951, 46129958370276158368745010598051, 66519367979660443013028395169496861

Offset: 1

Colin Barker, Dec 26 2016

			1711 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..300
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Cf. A000217, A062786, A280111, A280112.

RecurrenceTable[{a[n] == 1443 a[n - 1] - 1443 a[n - 2] + a[n - 3], a[1] == 1, a[2] == 1711, a[3] == 2467531}, a, {n, 12}] (* or *)
Rest@ CoefficientList[Series[x (1 + 268 x + x^2)/((1 - x) (1 - 1442 x + x^2)), {x, 0, 12}], x] (* Michael De Vlieger, Dec 26 2016 *)
LinearRecurrence[{1443,-1443,1},{1,1711,2467531},20] (* Harvey P. Dale, Dec 29 2017 *)

Vec(x*(1 + 268*x + x^2) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^15))

a(n) = 1443*a(n-1) - 1443*a(n-2) + a(n-3) for n>3.

G.f.: x*(1 + 268*x + x^2) / ((1 - x)*(1 - 1442*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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