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 1-3 of 3 results.

A045831 Number of 4-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 3, 4, 4, 2, 2, 7, 3, 5, 6, 2, 4, 7, 3, 4, 7, 5, 8, 5, 4, 4, 8, 5, 6, 7, 2, 9, 11, 3, 8, 9, 4, 6, 5, 7, 5, 14, 7, 4, 10, 5, 10, 11, 3, 9, 10, 5, 8, 10, 4, 6, 15, 8, 9, 10, 6, 8, 15, 6, 10, 6, 5, 15, 9, 6, 8, 14, 8, 6, 13, 5, 16, 18, 7, 8, 7, 9, 6, 15, 6, 12, 17, 5, 8, 15, 7, 12
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Conjecturally Sum_n a(n)q^(8n+5) equals theta series of sodalite. - Fred Lunnon, Mar 05 2015
Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - Michael Somos, Feb 10 2020

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ... ,
apparently the theta series of the sodalite net, aka edge-skeleton of space honeycomb by truncated octahedra. - _Fred Lunnon_, Mar 05 2015

References

  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. II, p. 23.

Links

Crossrefs

A004024/4, column t=4 of A175595.
Cf. A286953.
Cf. A008443, A045818, A213624.

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Jul 26 2011, updated Nov 29 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 / eta(x + A), n))}; /* Michael Somos, Mar 24 2003 */

Formula

eta(32*z)^4/eta(8*z) = Sum_{x, y, z} q^(x^2+2*y^2+2*z^2), x, y, z >= 1 and odd.
From Michael Somos, Mar 24 2003: (Start)
Euler transform of period 4 sequence [1, 1, 1, -3, ...].
Expansion of q^(-5/8) * eta(q^4)^4/eta(q) in powers of q.
(End)
Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
G.f.: Product_{k>0} (1-q^(4*k))^4/(1-q^k).
Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2008

Extensions

More terms from James Sellers, Feb 11 2000

A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 4, 4, 2, 2, 5, 4, 2, 6, 3, 6, 7, 2, 5, 4, 5, 6, 6, 2, 5, 10, 3, 6, 10, 4, 6, 8, 3, 8, 7, 6, 7, 6, 4, 6, 11, 6, 9, 10, 3, 6, 14, 4, 8, 10, 8, 10, 5, 6, 4, 16, 7, 4, 10, 4, 13, 14, 7, 8, 8, 6, 10, 12, 7, 12, 15, 8, 8, 10, 4, 6, 17, 6, 10, 10
Offset: 0

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Author

Michael Somos, Jun 16 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Dickson writes that Liouville proved several related results about sums of triangular number. In particular, that every nonnegative integer is the sum of t1 + t2 + 4*t3 where t1, t2, t3 are trianglular numbers. - Michael Somos, Feb 10 2020

Examples

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q^3 + 2*q^7 + q^11 + 2*q^15 + 3*q^19 + 2*q^23 + 4*q^27 + 4*q^31 + 2*q^35 + ...

References

  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 23.

Links

Crossrefs

Cf. A008443, A045818,

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};

Formula

Expansion of q^(-3/4) * eta(q^2)^4 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].

A045817 Numbers n written in base 7, where in the list of divisors of n (in base 7), each digit 0-6 appears equally often.

Original entry on oeis.org

3602, 246506, 264533, 266405, 303652, 320556, 324255, 325605, 342560, 345064, 345406, 345604, 346340, 362055, 414056, 430462, 434630, 435065, 436430, 436550, 453605, 500426, 500641, 506022, 524360, 524406, 526433, 530632, 532650, 533402
Offset: 1

Views

Author

Naohiro Nomoto

Keywords

Examples

			E.g., divisors of 342560 (base 7) are (1,2,10,20,15463,34256,154630,342560) (all in base 7); the numbers of digits (0-6) are [0(4),1(4),2(4),3(4),4(4),5(4),6(4)].

Links

Crossrefs

Cf. A038564, A038565, A045818.

Programs

  • Maple
    N:= 7^6:
cv7:= proc(n) local L; L:= convert(n,base,7);
add(L[i]*10^(i-1),i=1..nops(L)) end proc:
V:= Matrix(N,7,datatype=integer[8]):
count:= 0: Res:= NULL:
for i from 1 to N do
  L:= convert(i,base,7);
  M:= Vector[row]([seq(numboccur(d,L),d=0..6)],datatype=integer[8]);
  for r from i to N by i do V[r,..]:= V[r,..] + M od;
  if nops(convert(V[i,..],set))=1 then
    count:= count+1;
    w:= cv7(i);
    Res:= Res,w;
  fi
od:
Res; # Robert Israel, Sep 07 2018

Extensions

Definition clarified by Robert Israel, Sep 07 2018
Showing 1-3 of 3 results.

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

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