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 1-4 of 4 results.

A224823 Number of solutions to n = x + y + 3*z where x, y, z are triangular numbers.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 4, 0, 6, 6, 3, 5, 6, 2, 6, 8, 0, 7, 8, 4, 9, 6, 1, 11, 10, 0, 8, 6, 5, 9, 12, 3, 7, 14, 0, 11, 8, 5, 13, 10, 4, 8, 8, 0, 14, 16, 5, 11, 12, 1, 16, 10, 0, 14, 14, 7, 9, 12, 5, 14, 14, 0, 7, 16, 7, 18, 14, 4, 19, 10, 0, 12, 16, 9, 13, 20, 0
Offset: 0

Views

Author

Michael Somos, Jul 20 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(A224829(n)) = 0. - Reinhard Zumkeller, Jul 21 2013

Examples

			G.f. = 1 + 2*x + x^2 + 3*x^3 + 4*x^4 + x^5 + 5*x^6 + 4*x^7 + 6*x^9 + 6*x^10 + ...
G.f. = q^5 + 2*q^13 + q^21 + 3*q^29 + 4*q^37 + q^45 + 5*q^53 + 4*q^61 + 6*q^77 + ...
a(3) = 3 since 3 = 0 + 0 + 3*1 = 0 + 3 + 3*0 = 3 + 0 + 3*0 are the 3 solutions of 3 = x + y + 3*z in triangular numbers.
a(4) = 4 since 4 = 1 + 0 + 3*1 = 0 + 1 + 3*1 = 3 + 1 + 3*0 = 1 + 3 + 3*0 are the 4 solutions of 4 = x + y + 3*z in triangular numbers.

Links

Crossrefs

Cf. A227595.
Cf. A000217.

Programs

  • Haskell
    a224823 n = length [() | let ts = takeWhile (<= n) a000217_list,
            x <- ts, y <- ts, z <- takeWhile (<= div (n - x - y) 3) ts,
            x + y + 3 * z == n]
-- Reinhard Zumkeller, Jul 21 2013
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, 0, x^(3/2)] / (8 x^(5/8)), {x, 0, n}];
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A)), n))};

Formula

Expansion of psi(x)^2 * psi(x^3) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(-5/8) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 3, -2, 2, -3, ...].
G.f.: (Sum_{k>0} x^((k^2-k)/2))^2 * (Sum_{k>0} x^(3 * (k^2-k)/2)).
-2 * a(n) = A227595(3*n + 1).

A185220 Expansion of phi(x^3) * psi(x)^2 / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 4, 5, 5, 5, 7, 7, 9, 7, 6, 11, 8, 10, 8, 9, 14, 10, 15, 7, 7, 14, 14, 16, 8, 13, 13, 12, 18, 14, 13, 15, 15, 16, 9, 11, 22, 16, 19, 16, 11, 17, 16, 23, 19, 9, 22, 18, 16, 15, 18, 27, 12, 23, 11, 15, 24, 24, 27, 9, 23, 23, 20, 21, 19, 15, 22, 24, 22, 17
Offset: 0

Views

Author

Michael Somos, Aug 29 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + 3*x + 4*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 7*x^9 + ...
q^7 + 3*q^31 + 4*q^55 + 5*q^79 + 5*q^103 + 5*q^127 + 7*q^151 + 7*q^175 + ...

Links

Crossrefs

Cf. A224825, A227595.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 / (eta(x + A)^3 * eta(x^6 + A)), n))}

Formula

Expansion of q^(-7/24) * eta(q^2)^5 * eta(q^3)^2 / (eta(q)^3 * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 3, -2, 1, -2, 3, -3, ...].
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)).
a(n) = A224825(3*n) = A227595(3*n).

A224833 Expansion of phi(-x)^2 * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 8, -4, 4, -13, 12, -4, 5, -16, 24, -8, 4, -20, 12, -8, 9, -20, 32, -4, 12, -29, 12, -8, 8, -36, 40, -8, 8, -20, 24, -16, 8, -25, 40, -12, 12, -32, 24, -12, 13, -48, 40, -8, 8, -40, 36, -8, 16, -20, 56, -16, 12, -52, 12, -20, 13, -36, 56, -16, 20, -40, 24
Offset: 0

Views

Author

Michael Somos, Jul 21 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 - 5*x + 8*x^2 - 4*x^3 + 4*x^4 - 13*x^5 + 12*x^6 - 4*x^7 + 5*x^8 - 16*x^9 + ...
q - 5*q^4 + 8*q^7 - 4*q^10 + 4*q^13 - 13*q^16 + 12*q^19 - 4*q^22 + 5*q^25 + ...

Links

Crossrefs

Cf. A224822, A227595.

Programs

  • Mathematica
    eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]^5*eta[q^6]^2/(eta[q^2]^3*eta[q^3]), {q, 0, n}];  Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 19 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A)), n))}

Formula

Expansion of q^(-1/3) * eta(q)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -5, -2, -4, -2, -5, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 73728^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227595.
-2 * a(n) = A224822(3*n + 1).

A259659 Expansion of phi(x^6) * f(-x)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 0, 6, -3, 0, 1, -9, 0, 12, -3, 0, 6, -12, 0, 6, -3, 0, 7, -15, 0, 18, -6, 0, 0, -15, 0, 24, -6, 0, 6, -15, 0, 6, -9, 0, 7, -21, 0, 30, -3, 0, 6, -21, 0, 24, -6, 0, 12, -27, 0, 0, -9, 0, 12, -21, 0, 36, -6, 0, 1, -18, 0, 36, -12, 0, 6, -33, 0, 18, -9, 0
Offset: 0

Views

Author

Michael Somos, Jul 02 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 3*x + 6*x^3 - 3*x^4 + x^6 - 9*x^7 + 12*x^9 - 3*x^10 + 6*x^12 + ...
G.f. = q^3 - 3*q^7 + 6*q^15 - 3*q^19 + q^27 - 9*q^31 + 12*q^39 - 3*q^43 + ...

Links

Crossrefs

Cf. A227595, A259655.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] QPochhammer[ x]^3 / QPochhammer[ x^3], {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-3/4)* eta[q]^3*eta[q^12]^2/(eta[q^3]*eta[q^6]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 17 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^6 + A)), n))};

Formula

Expansion of phi(x^6) * b(x) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
Expansion of q^(-3/4) * eta(q)^3 * eta(q^12)^2 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -3, -2, -3, -3, -1, -3, -3, -2, -3, -3, -3, ...].
a(2*n + 1) = -3 * A227595(n). a(3*n + 1) = -3 * A259655(n). a(3*n + 2) = 0.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.