A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n >= 2.

2, 3, 3, 4, 4, 4, 5, 5, 9, 5, 6, 6, 16, 10, 6, 7, 7, 25, 17, 12, 7, 8, 8, 36, 26, 20, 13, 8, 9, 9, 49, 37, 30, 21, 27, 9, 10, 10, 64, 50, 42, 31, 64, 28, 10, 11, 11, 81, 65, 56, 43, 125, 65, 30, 11, 12, 12, 100, 82, 72, 57, 216, 126, 68, 31, 12, 13, 13, 121, 101, 90, 73, 343

Offset: 2

Author

Ralf Stephan, Mar 05 2005

Keywords

• nonn
• tabl

Comments

Sums of distinct powers of a. Numbers having only {0,1} in a-ary representation.

Examples

			Array begins:
  2,  3,  4,  5,  6,  7,   8,   9, ...
  3,  4,  9, 10, 12, 13,  27,  28, ...
  4,  5, 16, 17, 20, 21,  64,  65, ...
  5,  6, 25, 26, 30, 31, 125, 126, ...
  6,  7, 36, 37, 42, 43, 216, 217, ...
  7,  8, 49, 50, 56, 57, 343, 344, ...
  8,  9, 64, 65, 72, 73, 512, 513, ...
  9, 10, 81, 82, 90, 91, 729, 730, ...
  ...
		

Links

• John Tyler Rascoe, Antidiagonals n = 2..140, flattened

Crossrefs

Rows include (essentially) A005836, A000695, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A033050, A033051, A033052.

Columns include A000290, A002522, A002378, A000578, A001093, A034262, A071568, A011379, A098547, A027444.

Main diagonal is A104258.

Programs

Mathematica

T[, 0] = 0; T[2, n] := n; T[a_, 2] := a;
T[a_, n_] := T[a, n] = If[EvenQ[n], a T[a, n/2], a T[a, (n-1)/2]+1];
Table[T[a-n+2, n], {a, 2, 13}, {n, 2, a}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

PARI

T(a, n) = fromdigits(binary(n), a); \\ Michel Marcus, Aug 19 2022

Python

def T(a, n): return n if n < 2 else (max(a, n) if min(a, n) == 2 else a*T(a, n//2) + n%2)
print([T(a-n+2, n) for a in range(2, 14) for n in range(2, a+1)]) # Michael S. Branicky, Aug 02 2022

Formula

T(a, n) = (1/(a-1))*Sum_{j>=1} floor((n+2^(j-1))/2^j) * ((a-2)*a^(j-1) + 1).

T(a, n) = (1/(a-1))*Sum_{j=1..n} ((a-2)*a^A007814(j) + 1).

G.f. of a-th row: (1/(1-x)) * Sum_{k>=0} a^k*x^2^k/(1+x^2^k).

Recurrence: T(a, 2n) = a*T(a, n), T(a, 2n+1) = a*T(a, n) + 1, T(a, 0) = 0.