OFFSET
0,3
COMMENTS
The number of numbers in bijective base-n numeration with digit sum n equals the number of compositions of n: A000079(n).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..387
Wikipedia, Bijective numeration
Wikipedia, Digit sum
FORMULA
a(n) = ((n+1)^n - 2^n) / (n - 1) for n >= 2. - Peter Bala, Sep 28 2023
EXAMPLE
a(0) = 0.
a(1) = 1 = 1_bij1.
a(2) = 5 = 3 + 2 = 11_bij2 + 2_bij2.
a(3) = 28 = 13 + 7 + 5 + 3 = 111_bij3 + 21_bij3 + 12_bij3 + 3_bij3.
MAPLE
b:= proc(n, k) option remember; `if`(n=0, [1, 0], add((p->
[p[1], p[2]*k+p[1]*d])(b(n-d, k)), d=1..min(n, k)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..23);
MATHEMATICA
b[n_, k_] := b[n, k] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*k + p[[1]]*d}][b[n - d, k]], {d, 1, Min[n, k]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 23] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 20 2020
STATUS
approved