A030221 -id:A030221 - cp's OEIS frontend

A221365 The simple continued fraction expansion of F(x) := Product_{n >= 0} (1 - x^(4n+3))/(1 - x^(4n+1)) when x = 1/2*(5 - sqrt(21)).

1, 3, 1, 21, 1, 108, 1, 525, 1, 2523, 1, 12096, 1, 57963, 1, 277725, 1, 1330668, 1, 6375621, 1, 30547443, 1, 146361600, 1, 701260563, 1, 3359941221, 1, 16098445548, 1, 77132286525, 1, 369562987083, 1, 1770682648896, 1

Offset: 0

Peter Bala, Jan 15 2013

The function F(x) := Product_{n >= 0} (1 - x^(4*n+3))/(1 - x^(4*n+1)) is analytic for |x| < 1. When x is a quadratic irrational of the form x = 1/2*(N - sqrt(N^2 - 4)), N an integer greater than 2, the real number F(x) has a predictable simple continued fraction expansion. The first examples of these expansions, for N = 2, 4, 6 and 8, are due to Hanna. See A174500 through A175503. The present sequence is the case N = 5. See also A221364 (N = 3), A221366 (N = 7) and A221367 (N = 9).

If we denote the present sequence by [1, c(1), 1, c(2), 1, c(3), ...] then for k = 1, 2, ..., the simple continued fraction expansion of F((1/2*(5 - sqrt(21)))^k) is given by the sequence [1; c(k), 1, c(2*k), 1, c(3*k), 1, ...].

			F(1/2*(5 - sqrt(21))) = 1.25274 83510 08359 27965 ... = 1 + 1/(3 + 1/(1 + 1/(21 + 1/(1 + 1/(108 + 1/(1 + 1/(525 + ...))))))).
F((1/2*(5 - sqrt(21)))^2) = 1.04545 84663 16495 30047 ... = 1 + 1/(21 + 1/(1 + 1/(525 + 1/(1 + 1/(12096 + 1/(1 + 1/(277725 + ...))))))).
F((1/2*(5 - sqrt(21)))^3) = 1.00917 43188 83793 73068 ... = 1 + 1/(108 + 1/(1 + 1/(12096 + 1/(1 + 1/(1330668 + 1/(1 + 1/(146361600 + ...))))))).

Peter Bala, Some simple continued fraction expansions for an infinite product, Part 1
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-6,0,1).

Cf. A004254, A030221, A054493, A174500 (N = 4), A221364 (N = 3), A221366 (N = 7), A221369 (N = 9).

LinearRecurrence[{0,6,0,-6,0,1},{1,3,1,21,1,108},40] (* Harvey P. Dale, Jun 06 2023 *)

a(2*n-1) = (1/2*(5 + sqrt(21)))^n + (1/2*(5 - sqrt(21)))^n - 2 = 3*A054493(n); a(2*n) = 1.
a(4*n+1) = 3*(A030221(n))^2; a(4*n-1) = 21*(A004254(n))^2.
a(n) = 6*a(n-2)-6*a(n-4)+a(n-6). G.f.: -(x^4+3*x^3-5*x^2+3*x+1) / ((x-1)*(x+1)*(x^4-5*x^2+1)). - Colin Barker, Jan 20 2013

A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 24, 552, 12673, 290928, 6678672, 153318529, 3519647496, 80798573880, 1854847551745, 42580695116256, 977501140122144, 22439945527693057, 515141245996818168, 11825808712399124808, 271478459139183052417, 6232178751488811080784, 143068632825103471805616

Offset: 0

Francesca Arici, Sep 11 2020

Michael De Vlieger, Table of n, a(n) for n = 0..735
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A097778 (first differences).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

CoefficientList[Series[x/((1 - x) (x^2 - 23 x + 1)), {x, 0, 18}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = A004254(n)*A004254(n+1)/5 = A160695(n+1)/5.
G.f.: x/((1-x)*(x^2-23*x+1)). - Alois P. Heinz, Sep 11 2020
From Klaus Purath, Jun 18 2025: (Start)
a(n) = (A004253(n+1)^2 - 1) / 15.
a(n) = (A030221(n)^2 - 1) / 35.
a(n) + a(n+1) = A004253(n+1)^2. (End)

a(13)-a(14) corrected and more terms added by Alois P. Heinz, Sep 11 2020

A340475 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4sin(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 29, 29, 1, 1, 139, 500, 139, 1, 1, 666, 8329, 8329, 666, 1, 1, 3191, 138301, 463736, 138301, 3191, 1, 1, 15289, 2295701, 25543057, 25543057, 2295701, 15289, 1, 1, 73254, 38105729, 1404312491, 4614756624, 1404312491, 38105729, 73254, 1

Offset: 0

Seiichi Manyama, Jan 09 2021

			Square array begins:
  1,   1,      1,        1,          1, ...
  1,   6,     29,      139,        666, ...
  1,  29,    500,     8329,     138301, ...
  1, 139,   8329,   463736,   25543057, ...
  1, 666, 138301, 25543057, 4614756624, ...

Rows and columns 0..1 give A000012, A030221.
Main diagonal gives A127605.
Cf. A116469, A187617, A340476.

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin(a*Pi/(2*n+1))^2+4*sin(b*Pi/(2*k+1))^2)))}

T(n,k) = T(k,n).

A387017 Expansion of (Product_{k>=1} (1 - x^k)^2/(1 - 5x^k + x^(2k)) - 1)/3.

Original entry on oeis.org

1, 6, 28, 139, 660, 3192, 15260, 73254, 350848, 1681650, 8056608, 38604748, 184963130, 886226880, 4246152960, 20344613659, 97476826932, 467039887908, 2237722185188, 10721572793580, 51370139753240, 246129134364792, 1179275522335680, 5650248517615128

Offset: 1

Christian Kassel, Aug 13 2025

nonn

a(n) is the value at q = (5 + sqrt(21))/2 of C_n(q)/(q^{n-1}(q - 1)^2), where C_n(q) is the number of codimension n ideals of the algebra of two-variable Laurent polynomials over a finite field of order q. The number C_n(q) is a palindromic polynomial of degree 2n with integer coefficients in the variable q and it is divisible by (q-1)^2.

Christian Kassel and Christophe Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, arXiv:1505.07229 [math.AG], 2015-2016; Michigan Math. J. 67 (2018), 715-741.
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016; The Ramanujan Journal 46 (2018), 633-655.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025.

Cf. A030221, A386706, A329156.

nmax = 25; Rest[CoefficientList[Series[(Product[(1 - x^k)^2/(1 - 5*x^k + x^(2*k)), {k, 1, nmax}] - 1)/3, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 14 2025 *)

G.f.: (Product_{k>=1} (1 - x^k)^2/(1 - 5*x^k + x^(2*k)) - 1)/3
a(2^k) = A030221(2^k-1). (Follows from Cor. 4.5 of Kassel and Reutenauer (2025).)
a(n) ~ (3 + sqrt(21))^(2*n-1) / (2^(2*n-1) * 3^n). - Vaclav Kotesovec, Aug 14 2025

A107389 Expansion of x(1-6x+7x^2)/( (1-x)(1+x)(1-5x+x^2)).

Original entry on oeis.org

0, 1, -1, 2, 5, 31, 144, 697, 3335, 15986, 76589, 366967, 1758240, 8424241, 40362959, 193390562, 926589845, 4439558671, 21271203504, 101916458857, 488311090775, 2339638995026, 11209883884349, 53709780426727, 257339018249280, 1232985310819681, 5907587535849119

Offset: 0

Roger L. Bagula, May 24 2005, corrected Sep 04 2008

Index entries for linear recurrences with constant coefficients, signature (5,0,-5,1).

LinearRecurrence[{5,0,-5,1},{0,1,-1,2},30] (* Harvey P. Dale, Sep 17 2020 *)

concat(0,Vec((1-6*x+7*x^2)/(1-x)/(1+x)/(1-5*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Jan 25 2012

a(n)-a(n-2) = A030221(n-3), n>2. - R. J. Mathar, Dec 17 2017

Irregular sign at a(2) switched by R. J. Mathar, Jan 24 2012

cp's OEIS Frontend

A221365 The simple continued fraction expansion of F(x) := Product_{n >= 0} (1 - x^(4*n+3))/(1 - x^(4*n+1)) when x = 1/2*(5 - sqrt(21)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A340475 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*sin(b*Pi/(2*k+1))^2).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A387017 Expansion of (Product_{k>=1} (1 - x^k)^2/(1 - 5*x^k + x^(2*k)) - 1)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A107389 Expansion of x*(1-6*x+7*x^2)/( (1-x)*(1+x)*(1-5*x+x^2)).

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

Extensions

A221365 The simple continued fraction expansion of F(x) := Product_{n >= 0} (1 - x^(4n+3))/(1 - x^(4n+1)) when x = 1/2*(5 - sqrt(21)).

A340475 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4sin(bPi/(2k+1))^2).

A387017 Expansion of (Product_{k>=1} (1 - x^k)^2/(1 - 5x^k + x^(2k)) - 1)/3.

A107389 Expansion of x(1-6x+7x^2)/( (1-x)(1+x)(1-5x+x^2)).