A100506 Bisection of A001523.

1, 4, 15, 47, 130, 330, 784, 1765, 3804, 7898, 15880, 31048, 59220, 110484, 202070, 362974, 641368, 1116325, 1916184, 3247088, 5436972, 9002752, 14752316, 23938188, 38487496, 61344055, 96974176, 152110204, 236837795, 366177506, 562373990, 858193804, 1301654610

Offset: 0

Author

N. J. A. Sloane, Nov 24 2004

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A001523.

Programs

Magma

m:=200;
R:=PowerSeriesRing(Integers(), m);
b:= Coefficients(R!( 1 + (&+[ x^n*(1-x^n)/(&*[(1-x^j)^2: j in [1..n]]): n in [1..m+2]]) ));
A100506:= func< n | b[2*n+2] >;
[A100506(n): n in [0..80]]; // G. C. Greubel, Apr 03 2023

Maple

seq(coeff(convert(series(add(-(-1)^k*x^(k*(k+1)/2),k=1..100)/(mul(1-x^k,k=1..100))^2,x,100),polynom),x,2*n+1),n=0..45); # (C. Ronaldo)
# second Maple program:
b:= proc(n, i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(2*n+1, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

Mathematica

b[n_, i_]:= b[n, i]= If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i + 1]*(j+1), {j, 0, n/i}]];
a[n_]:= If[n==0, 1, b[2*n+1, 1]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 18 2018, after Alois P. Heinz *)

SageMath

@CachedFunction
def b(n, k): # Indranil Ghosh's code of A001523
    if k>n: return 0
    if n%k==0: x=1
    else: x=0
    return x + sum(b(n-k*j, k+1)*(j+1) for j in range(n//k + 1))
def A100506(n): return 1 if n==0 else b(2*n+1, 1)
[A100506(n) for n in range(41)] # G. C. Greubel, Apr 03 2023

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005