A360096 To get a(n), replace 0's in the binary expansion of n with (-1) and interpret the result in base n.

0, 1, 1, 4, 11, 21, 41, 57, 439, 640, 909, 1222, 1859, 2354, 2953, 3616, 61167, 78303, 98837, 123121, 152379, 185641, 224113, 268227, 344999, 405601, 473901, 550423, 637363, 732483, 837929, 954305, 32472031, 37912414, 44058661, 50977186, 58741163, 67420476

Offset: 0

Author

Alois P. Heinz, Jan 25 2023

Keywords

• nonn
• base

Comments

The empty bit string is used as binary expansion of 0, so a(0) = 0.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16383

Crossrefs

Main diagonal of A360099.

Cf. A030300, A057427.

Programs

Maple

b:= proc(n, k) option remember; local m;
      `if`(n=0, 0, k*b(iquo(n, 2, 'm'), k)+2*m-1)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);
# second Maple program:
a:= n-> (l-> add((2*l[i]-1)*n^(i-1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..44);

Formula

a(n) = [x^n] g_n(x) where g_k(x) satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).

a(n) = A(n,n) where A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.

a(n) = A360099(n,n).

a(n) mod 2 = A057427(n) if n is even; a(n) mod 2 = A030300(n) if n is odd.