A253546 -id:A253546 - cp's OEIS frontend

A253457 Indices of centered hexagonal numbers (A003215) which are also centered heptagonal numbers (A069099).

1, 14, 351, 9100, 236237, 6133050, 159223051, 4133666264, 107316099801, 2786084928550, 72330892042487, 1877817108176100, 48750913920536101, 1265645944825762514, 32858043651549289251, 853043488995455758000, 22146272670230300418737, 574950045936992355129150

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers x in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of y being A253458.

			14 is in the sequence because the 14th centered hexagonal number is 547, which is also the 13th centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..708
Index entries for linear recurrences with constant coefficients, signature (27,-27,1).

Cf. A003215, A069099, A253458, A253546.

Vec(x*(13*x-1)/((x-1)*(x^2-26*x+1)) + O(x^100))

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).
G.f.: x*(13*x-1) / ((x-1)*(x^2-26*x+1)).
a(n) = sqrt((-2-(13-2*sqrt(42))^n-(13+2*sqrt(42))^n)*(2-(13-2*sqrt(42))^(1+n)-(13+2*sqrt(42))^(1+n)))/(4*sqrt(6)). - Gerry Martens, Jun 04 2015

A253458 Indices of centered heptagonal numbers (A069099) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 13, 325, 8425, 218713, 5678101, 147411901, 3827031313, 99355402225, 2579413426525, 66965393687413, 1738520822446201, 45134575989913801, 1171760454915312613, 30420637251808214125, 789764808092098254625, 20503464373142746406113, 532300308893619308304301

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers y in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of x being A253457.

			13 is in the sequence because the 13th centered heptagonal number is 547, which is also the 14th centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..708
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (27,-27,1).

Cf. A003215, A069099, A253457, A253546.

LinearRecurrence[{27,-27,1},{1,13,325},20] (* Harvey P. Dale, Oct 13 2022 *)

Vec(-x*(x^2-14*x+1)/((x-1)*(x^2-26*x+1)) + O(x^100))

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).
G.f.: -x*(x^2-14*x+1) / ((x-1)*(x^2-26*x+1)).
a(n) = 1/2+(13+2*sqrt(42))^(-n)*(7+sqrt(42)-(-7+sqrt(42))*(13+2*sqrt(42))^(2*n))/28. - Colin Barker, Mar 03 2016

cp's OEIS Frontend

A253457 Indices of centered hexagonal numbers (A003215) which are also centered heptagonal numbers (A069099).

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