A280111 -id:A280111 - cp's OEIS frontend

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

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)).

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

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

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