Roger L. Bagula - cp's OEIS frontend

A362455 Squarefree positive integers d such that the dimension of the space of cuspidal harmonic automorphic forms for SL(2, O_{-d}) is zero, where O_{-d} is the ring of integers in Q(sqrt(-d)).

1, 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, 39, 47, 71

Offset: 1

Gary W. Adamson and N. J. A. Sloane, May 06 2023, following a suggestion from Roger L. Bagula

Reid, Alan W., and Colin MacLachlan. "The Arithmetic of hyperbolic 3-manifolds." Graduate Texts in Mathematics 219, Springer (2003).

Alan W. Reid, Arithmetic hyperbolic manifolds, slides of a talk, Cornell University, June 2014.
Karen Vogtmann, Rational homology of Bianchi groups, Mathematische Annalen 272.3 (1985): 399-419.

A321044 Irregular table related to f[(ax+b)/(cx+d)]=(cx+d)^(2n)f[x], and f[x]=1/(x+1), f[x]=(ax+b)/(c*x+d).

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 1, 9, 16, 14, 6, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 1, 25, 144, 506, 1210, 2079, 2640, 2508, 1782, 935, 352, 90, 14, 1, 1, 29, 196, 819, 2366, 5005, 8008, 9867, 9438, 7007

Offset: 0

Roger L. Bagula, Oct 26 2018

The author writes: This derivation is an interesting Fibonacci modular form. The modular form function is: f[(a*x+b)/(c*x+d)] = (c*x+d)^(2*n)*f[x] and I have used f[x]=1/(x+1) and f[x]=(a*x+b)/(c*x+d) for Möbius function matrix: {{0,1},{1,1}}. The polynomial is solved as a zero based form.

The graph of the root structures of the polynomial are mostly on a circle with center at -1, see Mathematica.

			{{1},
{1, 1},
{1, 5, 4, 1},
{1, 9, 16, 14, 6, 1},
{1, 13, 36, 55, 50, 27, 8,  1},
{1, 17, 64, 140, 196, 182, 112, 44, 10, 1},
{1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1},
{1, 25, 144, 506, 1210, 2079, 2640, 2508, 1782,  935, 352, 90, 14, 1},
{1, 29, 196, 819, 2366, 5005, 8008, 9867, 9438, 7007, 4004, 1729, 546, 119, 16, 1},
{1, 33, 256, 1240, 4200, 10556, 20384, 30888,37180, 35750, 27456, 16744, 8008, 2940, 800, 152, 18, 1},
{1, 37, 324, 1785,6936, 20196, 45696, 82212, 119340, 140998, 136136, 107406, 68952, 35700, 14688, 4692, 1122, 189, 20, 1}}

g[x_, n_] = If[n == 0, 1, (2 + x)*(1 + x)^(-1 + 2 n)/(1 + x) - 1]
Show[Table[ListPlot[{Re[x], Im[x]} /. NSolve[g[x, n] == 0, x], PlotStyle -> Red], {n, 1, 10}]]
a = Table[CoefficientList[g[x, n], x], {n, 0, 10}] (* Roger L. Bagula, Oct 26 2018 *)
row[n_] = If[n > 0, CoefficientList[(x+2)*(x+1)^(2n-2)-1, x], {1}]; (* Charles R Greathouse IV, Oct 30 2018 *)

row(n)=if(n, Vec((x+2)*(x+1)^(2*n-2)-1), [1]) \\ Charles R Greathouse IV, Oct 30 2018

For n > 0, the n-th row sum is 3*2^(2n-2) - 1. - Charles R Greathouse IV, Oct 30 2018

A297189 Expansion of (x + 3x^2 - 2x^3 - 3x^4)/(1 - 8x^2 + 9*x^4).

Original entry on oeis.org

0, 1, 3, 6, 21, 39, 141, 258, 939, 1713, 6243, 11382, 41493, 75639, 275757, 502674, 1832619, 3340641, 12179139, 22201062, 80939541, 147542727, 537904077, 980532258, 3574776747, 6516373521, 23757077283, 43306197846, 157883627541, 287802221079

Offset: 0

N. J. A. Sloane, Jan 04 2018, following a suggestion from Roger L. Bagula

Related to a tiling of the plane by heptagons.

Colin Barker, Table of n, a(n) for n = 0..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (0,8,0,-9).

concat(0, Vec((x + 3*x^2 - 2*x^3 - 3*x^4)/(1 - 8*x^2 + 9*x^4) + O(x^40))) \\ Colin Barker, Jan 05 2018

a(n) = 8*a(n-2) - 9*a(n-4). - Colin Barker, Jan 05 2018

a(2*n)/a(2*n-1) ~ 2*a(2*n+1)/a(2*n) ~ 1 + sqrt(7). - Kyle MacLean Smith, Oct 11 2019

A291311 Expansion of (1-x^2)/((1-x-x^2)*(1-x-x^4)).

Original entry on oeis.org

1, 2, 3, 5, 9, 16, 27, 45, 75, 125, 207, 341, 560, 918, 1502, 2453, 4000, 6515, 10601, 17235, 28000, 45461, 73773, 119665, 194033, 314519, 509685, 825768, 1337612, 2166360, 3508085, 5680122, 9196043, 14886981, 24097953, 39005540, 63131935, 102176733, 165362855, 267614381

Offset: 0

Roger L. Bagula, Aug 21 2017

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,-1,-1).

Cf. A131298.

LinearRecurrence[{2,0,-1,1,-1,-1},{1,2,3,5,9,16},40] (* Harvey P. Dale, Apr 18 2018 *)

my(x='x+O('x^50)); Vec((1 - x^2)/((1-x-x^2)*(1-x-x^4))) \\ Michel Marcus, Jun 25 2023

G.f.: (1-x^2)/((1-x-x^2)*(1-x-x^4)).

A260994 Numbers n such that the modular curve X_0(n) has genus >= 2 and contains infinitely many points of degree 2 over some number field L.

Original entry on oeis.org

22, 23, 26, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 65, 69, 71, 72, 75, 79, 81, 83, 89, 92, 94, 95, 101, 119, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991, A260992, A260993.

A260993 Numbers n such that the modular curve X_0(n) contains infinitely many rational points of degree 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 43, 46, 47, 48, 50, 53, 59, 61, 65, 71, 79, 83, 89, 101, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991, A260992, A260994.

A260992 Numbers n such that the modular curves X(n) and X_1(n) are not bielliptic.

Original entry on oeis.org

52, 57, 58, 66, 67, 68, 70, 73, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 132, 133, 134

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

nonn

The list includes all n >= 132.

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991.

A260991 Numbers n such that the modular curve X_0(n) has a bielliptic involution of Atkin-Lehner type.

Original entry on oeis.org

22, 26, 28, 30, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 48, 50, 51, 53, 54, 55, 56, 60, 61, 62, 63, 64, 65, 69, 75, 79, 81, 83, 89, 92, 94, 95, 101, 119, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Same as A260990 except that now 72 is excluded.

Cf. A260992, A260993, A260994.

A260990 Numbers n such that the modular curve X_0(n) is bielliptic.

Original entry on oeis.org

22, 26, 28, 30, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 48, 50, 51, 53, 54, 55, 56, 60, 61, 62, 63, 64, 65, 69, 72, 75, 79, 81, 83, 89, 92, 94, 95, 101, 119, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

J. S. Balakrishnan, B. Mazur, and N. Dogra, Ogg's torsion conjecture: fifty years later, Bull. Amer. Math. Soc., 62:2 (2025), 235-268.

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260991, A260992, A260993, A260994.

A245560 Row sums of triangle in A144480.

Original entry on oeis.org

1, 2, 6, 14, 36, 82, 196, 436, 1000, 2186, 4884, 10540, 23128, 49428, 107048, 227048, 486864, 1026394, 2183860, 4581244, 9686776, 20237372, 42571896, 88632664, 185653936, 385380932, 804316296, 1665340856, 3464899440, 7158117736, 14853106384

Offset: 0

Roger L. Bagula, Oct 11 2008

Cf. A144480.

f:=n->if (n mod 2) = 0 then (n+2)*2^(n-1)-(n/2)*binomial(n,n/2)
else (n+2)*2^(n-1)-((n+1)/4)*binomial(n+1,(n+1)/2); fi;
[seq(f(n),n=0..40)];

From N. J. A. Sloane, Aug 07 2014: if n is even, a(n) = (n+2)*2^(n-1)-(n/2)*binomial(n,n/2) otherwise a(n) = (n+2)*2^(n-1)-((n+1)/4)*binomial(n+1,(n+1)/2). This follows easily from the definition.

Edited with more terms by N. J. A. Sloane, Aug 07 2014

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User: Roger L. Bagula

A362455 Squarefree positive integers d such that the dimension of the space of cuspidal harmonic automorphic forms for SL(2, O_{-d}) is zero, where O_{-d} is the ring of integers in Q(sqrt(-d)).

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A321044 Irregular table related to f[(a*x+b)/(c*x+d)]=(c*x+d)^(2*n)*f[x], and f[x]=1/(x+1), f[x]=(a*x+b)/(c*x+d).

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A297189 Expansion of (x + 3*x^2 - 2*x^3 - 3*x^4)/(1 - 8*x^2 + 9*x^4).

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A291311 Expansion of (1-x^2)/((1-x-x^2)*(1-x-x^4)).

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Mathematica

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A260994 Numbers n such that the modular curve X_0(n) has genus >= 2 and contains infinitely many points of degree 2 over some number field L.

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A260993 Numbers n such that the modular curve X_0(n) contains infinitely many rational points of degree 2.

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A260992 Numbers n such that the modular curves X(n) and X_1(n) are not bielliptic.

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A260991 Numbers n such that the modular curve X_0(n) has a bielliptic involution of Atkin-Lehner type.

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A260990 Numbers n such that the modular curve X_0(n) is bielliptic.

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A245560 Row sums of triangle in A144480.

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A321044 Irregular table related to f[(ax+b)/(cx+d)]=(cx+d)^(2n)f[x], and f[x]=1/(x+1), f[x]=(ax+b)/(c*x+d).

A297189 Expansion of (x + 3x^2 - 2x^3 - 3x^4)/(1 - 8x^2 + 9*x^4).