A133141 -id:A133141 - cp's OEIS frontend

A133285 Indices of the centered 12-gonal numbers which are also 12-gonal number, or numbers X such that 120X^2-120X+36 is a square.

1, 12, 253, 5544, 121705, 2671956, 58661317, 1287877008, 28274632849, 620754045660, 13628314371661, 299202162130872, 6568819252507513, 144214821393034404, 3166157251394249365, 69511244709280451616

Offset: 1

Richard Choulet, Oct 16 2007

Partial sums of A077422. - R. J. Mathar, Nov 27 2011

Indices of centered pentagonal numbers (A005891) which are also centered hexagonal numbers (A003215). - Colin Barker, Feb 07 2015

Colin Barker, Table of n, a(n) for n = 1..746
Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

Cf. A003215, A005891, A077422, A133141, A254782.

Table[Cos[n*ArcCos[11]] // Round, {n, 0, 15}] // Accumulate  (* Jean-François Alcover, Dec 19 2013, after R. J. Mathar *)
LinearRecurrence[{23,-23,1},{1,12,253},20] (* Harvey P. Dale, Jul 04 2018 *)

Vec(x*(11*x-1)/((x-1)*(x^2-22*x+1)) + O(x^100)) \\ Colin Barker, Feb 07 2015

a(n+2) = 22*a(n+1)-a(n)-10 ; a(n+1)=11*a(n)-5+(120*a(n)^2-120*a(n)+36)^0.5

G.f. x*(-1+11*x) / ( (x-1)*(x^2-22*x+1) ). - R. J. Mathar, Nov 27 2011

More terms from Paolo P. Lava, Aug 06 2008

A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 11, 231, 5061, 111101, 2439151, 53550211, 1175665481, 25811090361, 566668322451, 12440892003551, 273132955755661, 5996484134620981, 131649518005905911, 2890292911995309051, 63454794545890893201, 1393115187097604341361, 30585079321601404616731

Offset: 1

Colin Barker, Feb 07 2015

Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 5*x + 6*y = 0, the corresponding values of x being A133285.

The numbers (as opposed to the indices) are A133141.

			11 is in the sequence because the 11th centered hexagonal number is 331, which is also the 12th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..746
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 (23,-23,1).

Cf. A003215, A005891, A133141, A133285.

LinearRecurrence[{23,-23,1},{1,11,231},20] (* Harvey P. Dale, Mar 01 2022 *)

Vec(-x*(x^2-12*x+1)/((x-1)*(x^2-22*x+1)) + O(x^100))

a(n) = 23*a(n-1)-23*a(n-2)+a(n-3).
G.f.: -x*(x^2-12*x+1) / ((x-1)*(x^2-22*x+1)).
a(n) = 1/2+1/24*(11+2*sqrt(30))^(-n)*(6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n)). - Colin Barker, Mar 03 2016

cp's OEIS Frontend

A133285 Indices of the centered 12-gonal numbers which are also 12-gonal number, or numbers X such that 120X^2-120X+36 is a square.

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A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

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cp's OEIS Frontend

A133285 Indices of the centered 12-gonal numbers which are also 12-gonal number, or numbers X such that 120*X^2-120*X+36 is a square.

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A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

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Crossrefs

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Mathematica

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A133285 Indices of the centered 12-gonal numbers which are also 12-gonal number, or numbers X such that 120X^2-120X+36 is a square.