A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.
0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 3, 5, 5, 0, 1, 2, 3, 8, 8, 6, 0, 1, 2, 3, 4, 9, 9, 7, 0, 1, 2, 3, 4, 16, 10, 10, 8, 0, 1, 2, 3, 4, 5, 17, 16, 13, 9, 0, 1, 2, 3, 4, 5, 32, 18, 17, 16, 10, 0, 1, 2, 3, 4, 5, 6, 33, 19, 24, 17, 11, 0, 1, 2, 3, 4, 5, 6, 64, 34, 32, 27, 18, 12
Offset: 1
Examples
Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, ... 4, 5, 8, 4, 4, 4, 4, 4, 4, 4, 4, ... 5, 8, 9, 16, 5, 5, 5, 5, 5, 5, 5, ... 6, 9, 10, 17, 32, 6, 6, 6, 6, 6, 6, ... 7, 10, 16, 18, 33, 64, 7, 7, 7, 7, 7, ... 8, 13, 17, 19, 34, 65, 128, 8, 8, 8, 8, ... 9, 16, 24, 32, 35, 66, 129, 256, 9, 9, 9, ... 10, 17, 27, 33, 36, 67, 130, 257, 512, 10, 10, ...
Links
- Alois P. Heinz, Antidiagonals n = 1..141, flattened
Crossrefs
Programs
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Maple
A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end, proc(n, k) option remember; local b; b:= proc(x, y) option remember; `if`(x<0 or y<1, {}, {0, b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]}) end; while nops(w(k)) < n do forget(b); l(k):= [l(k)[], (nops(l(k))+1)^k]; w(k):= sort([select(h-> h -
Mathematica
b[n_, k_, i_, t_] := b[n, k, i, t] = n == 0 || i > 0 && t > 0 && (b[n, k, i - 1, t] || i^k <= n && b[n - i^k, k, i, t - 1]); A[n_, k_] := A[n, k] = Module[{m}, For[m = 1 + If[n == 1, -1, A[n - 1, k]], !b[m, k, m^(1/k) // Floor, k], m++]; m]; Table[A[n, 1+d-n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, Dec 03 2020, using Alois P. Heinz's code for columns *)
Formula
A(n,k) = n-1 for n <= k+1.