A332690
Sum of all numbers in bijective base-9 numeration with digit sum n.
Original entry on oeis.org
0, 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999862, 12499999623, 124999998144, 1249999984364, 12499999840480, 124999998308464, 1249999981991936, 12499999808733888, 124999997974967808, 1249999978624935680, 12499999774999871588
Offset: 0
a(2) = 12 = 2 + 10 = 2_bij9 + 11_bij9.
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Wikipedia, Bijective numeration
- Index entries for linear recurrences with constant coefficients, signature (10,1,-8,-17,-26,-35,-44,-53,-62,-81,-72,-63,-54,-45,-36,-27,-18,-9).
-
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p->
[p[1], p[2]*9+p[1]*d])(b(n-d)), d=1..min(n, 9)))
end:
a:= n-> b(n)[2]:
seq(a(n), n=0..23);
A332691
Bijective base-9 representation of the sum of all numbers in bijective base-9 numeration with digit sum n.
Original entry on oeis.org
1, 13, 147, 1636, 18124, 199399, 2314581, 25461653, 281178597, 3192976395, 35233852789, 387573484456, 4374418444135, 48228613881184, 541525753635894, 5956784387951128, 66635738355523786, 743994232656361639, 8285146556418623572, 92246623188575957748
Offset: 1
a(2) = 13_bij9 = 12 = 2 + 10 = 2_bij9 + 11_bij9.
-
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p->
[p[1], p[2]*9+p[1]*d])(b(n-d)), d=1..min(n, 9)))
end:
g:= proc(n) local d, l, m; m, l:= n, "";
while m>0 do d:= irem(m, 9, 'm');
if d=0 then d:=9; m:= m-1 fi; l:= d, l
od; parse(cat(l))
end:
a:= n-> g(b(n)[2]):
seq(a(n), n=1..23);
A332711
Sum of all numbers in bijective base-n numeration with digit sum n.
Original entry on oeis.org
0, 1, 5, 28, 203, 1936, 23517, 349504, 6149495, 124999936, 2881935953, 74300836864, 2118007738035, 66142897770496, 2245609694259557, 82351536043343872, 3244079458377786863, 136619472483668525056, 6125138252818308310041, 291271111111111111081984
Offset: 0
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.
-
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);
-
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 *)
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