cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A112206 Coefficients of replicable function number "72b".

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 4, 4, 7, 7, 6, 10, 11, 11, 14, 16, 17, 21, 22, 24, 32, 34, 34, 44, 49, 50, 60, 66, 72, 84, 90, 98, 117, 125, 132, 156, 171, 181, 206, 226, 245, 277, 298, 322, 369, 397, 422, 480, 522, 557, 620, 674, 728, 807, 868, 936, 1043, 1121, 1198
Offset: 0

Views

Author

Michael Somos, Aug 28 2005

Keywords

Comments

From Michael Somos, Oct 28 2019: (Start)
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution squared is A112173.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
Given G.f. A(x), then B(q) = q^(-1) * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + (u^2 - v)*v*w^2 + (u^2 + v)*v^2.
(End)

Examples

			G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
G.f. = q^-1 + q^5 + 2*q^17 + 2*q^23 + q^29 + 2*q^35 + 2*q^41 + ...

Links

Crossrefs

Cf. A112173, A112175, A328789, A328790.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^k)*(1 + x^(3*k)) / ((1 + x^(2*k))*(1 + x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; h:= q^(1/6)*((eta[q^2]*eta[q^6])^2/(eta[q]*eta[q^3]*eta[q^4]*eta[q^12])); a:= CoefficientList[Series [h, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jun 01 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3, x^6], {x, 0 ,n}]; (* Michael Somos, Oct 28 2019 *)
  • PARI
    q='q+O('q^50); h=((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))); Vec(h) \\ G. C. Greubel, Jun 01 2018
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; /* Michael Somos, Oct 28 2019 */

Formula

a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
Expansion of q^(1/6)*((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))) in powers of q. - G. C. Greubel, Jun 01 2018
From Michael Somos, Oct 28 2019: (Start)
Expansion of chi(x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.
Euler transform of period 12 sequence [1, -1, 2, 0, 1, -2, 1, 0, 2, -1, 1, 0, ...].
G.f.: Product_{k>=0} (1 + x^(2*k + 1)) * (1 + x^(6*k + 3)).
a(n) = (-1)^n * A112175(n). a(2*n) = A328789(n). a(2*n + 1) = A328790(n).
(End)

A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 7, 11, 17, 22, 27, 31, 37, 41, 47, 51, 57, 62, 67, 71, 77, 82, 87, 91, 97, 102, 107, 111, 117, 121, 127, 131, 137, 142, 147, 151, 157, 161, 167, 171, 177, 182, 187, 191, 197, 201, 207, 211, 217, 222, 227, 231, 237, 242, 247, 251, 257, 262, 267, 271, 277
Offset: 0

Views

Author

Clark Kimberling, May 04 2018

Keywords

Comments

Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
aa(n) = least new;
bb(n) = aa(n) + cc(n-1);
cc(n) = least new;
where "least new k" means the least positive integer not yet placed.
***
The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >= 0,
(1) 5*n - 1 - 2*aa(n) is in {0,1,2},
(2) (aa(n) mod 5) is in {1,2,4},
(3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
(4) (bb(n) mod 5) is in {1,2}.
From N. J. A. Sloane, Nov 05 2019: (Start)
Conjecture: For t >= 0, bb(2t) = 10t + 1 (+1 if binary expansion of t ends in an odd number of 0's), bb(2t+1) = 10t + 7.
The first part may also be written as bb(2t) = 10t + 1 + A328789(t-1).
(End)

Examples

			n:  0 1 2 3 4 5 6 7 8 9 10
aa: 1 4 6 9 12 14 16 19 21 24 26
bb: 2 7 11 17 22 27 31 37 41 47 51
cc: 3 5 8 10 13 15 18 20 23 25 28

Links

Crossrefs

Cf. A299634, A298468 (aa), A047218 (cc), A328789.

Programs

  • Mathematica
    z = 500;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[b, Last[a] + Last[c]];
  AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
Take[a, 100] (* A298468 *)
Take[b, 100] (* A297469 *)
Take[c, 100] (* A047218 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Extensions

Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - N. J. A. Sloane, Nov 03 2019

A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 6, 9, 12, 14, 16, 19, 21, 24, 26, 29, 32, 34, 36, 39, 42, 44, 46, 49, 52, 54, 56, 59, 61, 64, 66, 69, 72, 74, 76, 79, 81, 84, 86, 89, 92, 94, 96, 99, 101, 104, 106, 109, 112, 114, 116, 119, 122, 124, 126, 129, 132, 134, 136, 139, 141, 144, 146, 149
Offset: 0

Views

Author

Clark Kimberling, May 04 2018

Keywords

Comments

Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
aa(n) = least new;
bb(n) = aa(n) + cc(n-1);
cc(n) = least new;
where "least new k" means the least positive integer not yet placed.
***
The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >=0,
(1) 5*n - 1 - 2*aa(n) is in {0,1,2},
(2) (aa(n) mod 5) is in {1,2,4},
(3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
(4) (bb(n) mod 5) is in {1,2}.
From N. J. A. Sloane, Nov 05 2019: (Start)
Conjecture: For t >= 1, aa(2t) = 5t+1(+1 if binary expansion of t ends in an odd number of 0's), and for t >= 0, aa(2t+1) = 5t+4.
The first part may also be written as aa(2t) = 5t+1+A328789(t-1).
(End)

Examples

			n:  0 1 2 3 4 5 6 7 8 9 10
aa: 1 4 6 9 12 14 16 19 21 24 26
bb: 2 7 11 17 22 27 31 37 41 47 51
cc: 3 5 8 10 13 15 18 20 23 25 28

Links

Crossrefs

Cf. A299634, A297469 (bb), A047218 (cc), A328789.

Programs

  • Mathematica
    z = 500;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[b, Last[a] + Last[c]];
  AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
Take[a, 100] (* A298468 *)
Take[b, 100] (* A297469 *)
Take[c, 100] (* A047218 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Extensions

Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - N. J. A. Sloane, Nov 03 2019

A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 6, 2, 0, 3, 6, 4, 0, 4, 8, 4, 0, 7, 14, 7, 0, 6, 16, 10, 0, 11, 20, 11, 0, 14, 32, 16, 0, 17, 38, 21, 0, 22, 46, 24, 0, 32, 66, 34, 0, 34, 78, 44, 0, 49, 96, 50, 0, 60, 130, 66, 0, 72, 154, 84, 0, 90, 186, 98, 0, 117, 244
Offset: 0

Views

Author

Michael Somos, Oct 28 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328802.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.

Examples

			G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...

Links

Crossrefs

Cf. A112206, A328789, A328790, A328797, A328798, A328802.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6])^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A))^2 / (eta(x+ A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Formula

Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 - x^(6*k-3))^2.
a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

A328797 Expansion of (chi(-x) * chi(x^3))^2 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 1, 0, 0, -2, 2, 0, 2, -2, 1, 0, 2, -6, 2, 0, 3, -6, 4, 0, 4, -8, 4, 0, 7, -14, 7, 0, 6, -16, 10, 0, 11, -20, 11, 0, 14, -32, 16, 0, 17, -38, 21, 0, 22, -46, 24, 0, 32, -66, 34, 0, 34, -78, 44, 0, 49, -96, 50, 0, 60, -130, 66, 0, 72, -154, 84, 0, 90, -186
Offset: 0

Views

Author

Michael Somos, Oct 27 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328800.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328790.

Examples

			G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q^-1 - 2*q^2 + q^5 - 2*q^14 + 2*q^17 + 2*q^23 - 2*q^26 + ..

Links

Crossrefs

Cf. A112206, A328789, A328790, A328795, A328798, A328800.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of q^(1/3) * (eta(q) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [-2, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, ...].
G.f.: Product_{k>=1} (1 - x^(2*k-1))^2 * (1 + x^(6*k-3))^2.
a(n) = (-1)^n * A328795(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = -2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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