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.

A023361 Number of compositions of n into positive triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 25, 40, 61, 94, 147, 227, 351, 546, 846, 1309, 2030, 3147, 4876, 7558, 11715, 18154, 28136, 43609, 67586, 104748, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487153, 5404570, 8376292, 12982016, 20120202, 31183350
Offset: 0

Views

Author

David W. Wilson, Jun 14 1998

Keywords

Comments

Number of compositions [c(1), c(2), c(3), ...] of n such that either c(k) = c(k-1) + 1 or c(k) = 1; see example. Same as fountains of coins (A005169) where each valley is at the lowest level. - Joerg Arndt, Mar 25 2014

Examples

			From _Joerg Arndt_, Mar 25 2014: (Start)
There are a(9) = 25 compositions of 9 such that either c(k) = c(k-1) + 1 or c(k) = 1:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 1 1 1 ]
06:  [ 1 1 1 1 2 1 2 ]
07:  [ 1 1 1 1 2 3 ]
08:  [ 1 1 1 2 1 1 1 1 ]
09:  [ 1 1 1 2 1 1 2 ]
10:  [ 1 1 1 2 1 2 1 ]
11:  [ 1 1 1 2 3 1 ]
12:  [ 1 1 2 1 1 1 1 1 ]
13:  [ 1 1 2 1 1 1 2 ]
14:  [ 1 1 2 1 1 2 1 ]
15:  [ 1 1 2 1 2 1 1 ]
16:  [ 1 1 2 3 1 1 ]
17:  [ 1 2 1 1 1 1 1 1 ]
18:  [ 1 2 1 1 1 1 2 ]
19:  [ 1 2 1 1 1 2 1 ]
20:  [ 1 2 1 1 2 1 1 ]
21:  [ 1 2 1 2 1 1 1 ]
22:  [ 1 2 1 2 1 2 ]
23:  [ 1 2 1 2 3 ]
24:  [ 1 2 3 1 1 1 ]
25:  [ 1 2 3 1 2 ]
The last few, together with the corresponding fountains of coins are:
.  20:  [ 1 2 1 1 2 1 1 ]
.
.     O     O
.    O O O O O O O
.
.
.  21:  [ 1 2 1 2 1 1 1 ]
.
.     O   O
.    O O O O O O O
.
.
.  22:  [ 1 2 1 2 1 2 ]
.
.     O   O   O
.    O O O O O O
.
.
.  23:  [ 1 2 1 2 3 ]
.
.           O
.      O   O O
.     O O O O O
.
.
.  24:  [ 1 2 3 1 1 1 ]
.
.       O
.      O O
.     O O O O O O
.
.
.  25:  [ 1 2 3 1 2 ]
.
.       O
.      O O   O
.     O O O O O
(End)
Applying recursion formula: 40 = a(10) = a(9) + a(7) + a(4) + a(0) = 25 + 11 + 3 + 1. - _Gregory L. Simay_, Jun 14 2016

Links

Crossrefs

Cf. A106332.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(`if`(issqr(8*j+1), a(n-j), 0), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017
  • Mathematica
    (1/(2 - QPochhammer[x^2]/QPochhammer[x, x^2]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 23 2016 *)
a[n_] := a[n] = If[n == 0, 1, Sum[ If[ IntegerQ[ Sqrt[8j+1]], a[n-j], 0], {j, 1, n}]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
  • PARI
    N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,1+sqrtint(2*N), x^binomial(k+1,2) ) ) )
/* Joerg Arndt, Sep 30 2012 */

Formula

G.f. : 1 / (1 - Sum_{k>=1} x^(k*(k+1)/2) ).
a(n) ~ c * d^n, where d = 1/A106332 = 1.5498524695188884304192160776463163555... is the root of the equation d^(1/8) * EllipticTheta(2, 0, 1/sqrt(d)) = 4 and c = 0.492059962414480455851222791075288170662444559041717451009563731799... - Vaclav Kotesovec, May 01 2014, updated Feb 17 2017
a(n) = a(n-1) + a(n-3) + a(n-6) + a(n-10) + ... Gregory L. Simay, Jun 09 2016
G.f.: 1/(2 - (x^2;x^2)inf/(x;x^2)_inf), where (a;q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 23 2016
G.f.: 1/(2 - theta_2(sqrt(q))/(2*q^(1/8))), where theta_2() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A106334 Decimal expansion of the function F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

Constant A106333 divided by this constant equals constant A106335, the radius of convergence of the g.f. of A106336.

Examples

			F(x)=1.9873697211846841452692897833444126183427177298554574703562231
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
at x = 0.6411803884299545796456448886283011... (A106333).

Crossrefs

Cf. A106333, A106335, A106332, A106336.

Programs

  • Mathematica
    digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); f[x0] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
  • PARI
    A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2))

A106333 Decimal expansion of the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

Not equal to exp(-4/9), which agrees with the first 16 decimal places. Related to Jacobi theta constant theta_2 and Dedekind's eta(x^2)^2/eta(x): Sum_{n>=0} x^(n*(n+1)/2) = 1.9873697... (A106334). This constant divided by constant A106334 equals constant A106335, the radius of convergence of the g.f. of A106336.

Examples

			0 = 1 - 2*x^3 - 5*x^6 - 9*x^10 - 14*x^15 - 20*x^21 - 27*x^28 - ...
x=0.641180388429954579645644888628301106553419618910071190877560305051317278

Crossrefs

Cf. A106334, A106335, A106332, A106336.

Programs

  • Mathematica
    digits = 105; g[x_?NumericQ] := NSum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 100]; x /. FindRoot[g[x], {x, 1/2}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 12 2013 *)
  • PARI
    solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2)))

Formula

Sum_{n>=0} (1 - n*(n+1)/2)*x^(n*(n+1)/2) = 0.
Showing 1-3 of 3 results.

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