A253622 -id:A253622 - cp's OEIS frontend

A133272 Indices of centered heptagonal numbers (A069099) which are also heptagonal numbers (A000566).

1, 7, 78, 924, 11005, 131131, 1562562, 18619608, 221872729, 2643853135, 31504364886, 375408525492, 4473397941013, 53305366766659, 635191003258890, 7568986672340016, 90192649064821297, 1074742802105515543

Offset: 1

Richard Choulet, Oct 16 2007

Numbers X such that 140*X^2-140*X+49 is a square.
Also positive integers x in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of y being A253621. - Colin Barker, Jan 06 2015
Also indices of centered pentagonal numbers (A005891) which are also centered heptagonal numbers (A069099). - Colin Barker, Jan 06 2015

Colin Barker, Table of n, a(n) for n = 1..930
Index entries for linear recurrences with constant coefficients, signature (13,-13,1).

Cf. A000566, A069099, A128919, A253621, A253622.

LinearRecurrence[{13,-13,1},{1,7,78},25] (* Paolo Xausa, Jan 07 2024 *)

Vec(x*(6*x-1)/((x-1)*(x^2-12*x+1)) + O(x^100)) \\ Colin Barker, Jan 06 2015

a(n+2) = 12*a(n+1) - a(n) - 5.
a(n+1) = 6*a(n) - 5/2 + (1/2)*sqrt(140*a(n)^2 - 140*a(n) + 49).
G.f.: x*(-1+6*x)/((-1+x)*(1-12*x+x^2)). - R. J. Mathar, Nov 14 2007
a(n) = 13*a(n-1) - 13*a(n-2) + a(n-3). - Colin Barker, Jan 06 2015

More terms from Paolo P. Lava, Jul 14 2008

A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 6, 66, 781, 9301, 110826, 1320606, 15736441, 187516681, 2234463726, 26626048026, 317278112581, 3780711302941, 45051257522706, 536834378969526, 6396961290111601, 76226701102369681, 908323451938324566, 10823654722157525106, 128975533213951976701

Offset: 1

Colin Barker, Jan 06 2015

Also positive integers y in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of x being A133272.

			6 is in the sequence because the 6th centered heptagonal number is 106, which is also the 7th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..930
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 (13,-13,1).

Cf. A005891, A069099, A133272, A253622.

I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2)-5: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

RecurrenceTable[{a[1] == 1, a[2] == 6, a[n] == 12 a[n-1] - a[n-2] - 5}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-12*x+1)) + O(x^100))

a(n) = 13*a(n-1)-13*a(n-2)+a(n-3).
G.f.: -x*(x^2-7*x+1) / ((x-1)*(x^2-12*x+1)).
a(n) = (14-(-7+sqrt(35))*(6+sqrt(35))^n+(6-sqrt(35))^n*(7+sqrt(35)))/28. - Colin Barker, Mar 05 2016
a(n) = 12*a(n-1) - a(n-2) - 5. - Vincenzo Librandi, Mar 05 2016
a(n) = (5*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016

cp's OEIS Frontend

A133272 Indices of centered heptagonal numbers (A069099) which are also heptagonal numbers (A000566).

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