A130880 -id:A130880 - cp's OEIS frontend

A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

-1, 2, 7, 11, 16, 20, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 115, 119, 124, 128, 133, 137, 142, 146, 151, 155, 160, 164, 169, 173, 178, 182, 187, 191, 196, 200, 205, 209, 214, 218, 223, 227, 232, 236

Offset: 2

N. J. A. Sloane, Jul 14 2001

It appears that for n > 2 a(n) = floor((9n-22)/2). - Gary Detlefs, Mar 02 2010

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A063232, A063233, A017185 (bisection), A130880, A332438.

Join[{-1}, Table[9*n/2 + (-1)^n/4 - 45/4, {n, 3, 60}]] (* Amiram Eldar, Jan 12 2024 *)

a(n) = 9*n/2 + (-1)^n/4 - 45/4 for n >= 3, with first differences in A010710. - R. J. Mathar, Dec 06 2010
From M. F. Hasler, Mar 05 2012: (Start)
G.f.: x^2*(-1 + 3*x + 6*x^2 + x^3)/(1 - x - x^2 + x^3).
a(n+2) = a(n)+9 (n>2), a(2n+1) = a(2n)+4 (n>1), a(2n) = a(2n-1)+5 (n>1). (End)
Sum_{n>=3} (-1)^(n+1)/a(n) = cot(2*Pi/9)*Pi/9. - Amiram Eldar, Jan 12 2024
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=3} (1 - (-1)^n/a(n)) = 2*sin(Pi/18) + 1 (= A130880 + 1).
Product_{n>=3} (1 + (-1)^n/a(n)) = (1/2) * sec(Pi/9) (= A332438 - 3). (End)

A178959 Decimal expansion of the site percolation threshold for the (3,6,3,6) Kagome Archimedean lattice.

Original entry on oeis.org

6, 5, 2, 7, 0, 3, 6, 4, 4, 6, 6, 6, 1, 3, 9, 3, 0, 2, 2, 9, 6, 5, 6, 6, 7, 4, 6, 4, 6, 1, 3, 7, 0, 4, 0, 7, 9, 9, 9, 2, 4, 8, 6, 4, 5, 6, 3, 1, 8, 6, 1, 2, 2, 5, 5, 2, 7, 5, 1, 7, 2, 4, 3, 7, 3, 5, 8, 6, 8, 3, 5, 5, 7, 2, 1, 9, 7, 0, 5, 2, 9, 1, 5, 6, 9, 6, 6, 7, 7, 3, 6, 8, 5, 2, 0, 0, 8, 5, 1, 9, 7, 6

Offset: 0

Jonathan Vos Post, Dec 22 2012

Consider an infinite graph where vertices are selected with probability p. The site percolation threshold is a unique value p_c such that if p > p_c an infinite connected component of selected vertices will almost surely exist, and if p < p_c an infinite connected component will almost surely not exist. This sequence gives p_c for the (3,6,3,6) Kagome Archimedean lattice.

This is one of the three real roots of x^3 - 3x^2 + 1. The other roots are 1 + A332437 = 2.879385241... and -(A332438 - 3) = - 0.5320888862... . - Wolfdieter Lang, Dec 13 2022

			0.652703644666139302296566746461370407999248645631861225527517243735868355...

M. F. Sykes and J. W. Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, J. Math. Phys. 5, 1117 (1964).
Wikipedia, Percolation threshold

Cf. A130880, A174849, A332437, A332438.

RealDigits[1 - 2 Sin[Pi/18], 10, 105][[1]] (* Alonso del Arte, Dec 22 2012 *)

1-2*sin(Pi/18) \\ Charles R Greathouse IV, Jan 03 2013

Equals 1 - 2*sin(Pi/18) = 1 = 1 - 2*cos(4*Pi/9) = 1 - A130880.

a(98) corrected and more terms from Georg Fischer, Jun 06 2024

A357104 Decimal expansion of the real root of x^3 + 3*x - 1.

Original entry on oeis.org

3, 2, 2, 1, 8, 5, 3, 5, 4, 6, 2, 6, 0, 8, 5, 5, 9, 2, 9, 1, 1, 4, 7, 0, 7, 1, 0, 7, 0, 4, 0, 3, 1, 9, 8, 4, 9, 3, 1, 6, 4, 4, 3, 8, 2, 8, 9, 9, 5, 8, 4, 0, 0, 9, 1, 7, 8, 8, 4, 3, 9, 1, 1, 9, 0, 4, 2, 9, 6, 7, 6, 2, 3, 1, 2, 7, 8, 6

Offset: 0

Wolfdieter Lang, Sep 21 2022

The other two roots are w1*phi^(1/3) - w2*(-1 + phi)^(1/3) = -0.16109267... + 1.75438095...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1; phi = A001622.

The hyperbolic function version is -sinh((1/3)*arcsinh(1/2)) + sqrt(3)*cosh((1/3)*arcsinh(1/2))*i, and its complex conjugate.

			0.32218535462608559291147071070403198493164438289958400917884391190429676...

Cf. A001622, A130880, A332437, A332438 - 2.

h := ((1 + sqrt(5))/2)^(1/3): evalf(h - 1/h, 90); # Peter Luschny, Sep 24 2022

RealDigits[Subtract @@ Surd[GoldenRatio, {3, -3}], 10, 100][[1]] (* Amiram Eldar, Sep 21 2022 *)
RealDigits[Root[x^3+3x-1,1],10,120][[1]] (* Harvey P. Dale, Oct 09 2023 *)

2*sinh((1/3)*asinh(1/2)) \\ Michel Marcus, Sep 23 2022

r = phi^(1/3) - phi^(-1/3), with phi = A001622.
r = phi^(1/3) - (-1 + phi)^(1/3).
r = 2*sinh((1/3)*arcsinh(1/2)).

A245720 Decimal expansion of the mean cluster density for bond percolation on the triangular lattice.

Original entry on oeis.org

1, 1, 1, 8, 4, 4, 2, 7, 5, 2, 8, 4, 5, 4, 9, 6, 9, 5, 6, 7, 5, 3, 4, 4, 3, 3, 6, 0, 5, 1, 6, 1, 1, 1, 8, 0, 3, 8, 2, 7, 5, 1, 1, 9, 4, 4, 1, 3, 2, 1, 2, 2, 0, 1, 4, 5, 8, 2, 8, 0, 0, 2, 3, 3, 5, 4, 6, 7, 2, 7, 3, 4, 0, 2, 4, 6, 6, 3, 2, 4, 6, 3, 0, 1, 5, 2, 4, 6, 6, 0, 9, 7, 2, 3, 1, 1, 5, 9, 5, 2, 2, 8, 4, 7

Offset: 0

Jean-François Alcover, Jul 30 2014

			0.11184427528454969567534433605161118038275119441321220145828...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.18 Percolation Cluster Density Constants, p. 373.

Cf. A130880.

RealDigits[35/4 - 3/(2*Sin[Pi/18]), 10, 104] // First

35/4 - 3/(2*sin(Pi/18)) \\ Stefano Spezia, May 17 2025

35/4 - 3/p_c, where p_c = A130880 = 2*sin(Pi/18) is the critical probability.

cp's OEIS Frontend

A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

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A178959 Decimal expansion of the site percolation threshold for the (3,6,3,6) Kagome Archimedean lattice.

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A357104 Decimal expansion of the real root of x^3 + 3*x - 1.

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A245720 Decimal expansion of the mean cluster density for bond percolation on the triangular lattice.

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