|
%I #17 Jan 05 2024 12:29:30
%S 0,0,1,0,1,3,0,1,2,7,0,1,2,5,15,0,1,2,3,8,31,0,1,2,3,7,17,63,0,1,2,3,
%T 4,11,26,127,0,1,2,3,4,9,15,53,255,0,1,2,3,4,5,14,31,80,511,0,1,2,3,4,
%U 5,11,19,47,161,1023,0,1,2,3,4,5,6,17,24,63,242,2047
%N Square array A(n, k), n >= 2, k >= 0, read by antidiagonals upwards; A(n, k) is the least integer with sum of digits k in base n.
%C The expansion of A(n, k) in base n is:
%C q n-1 ... n-1
%C <- p times ->
%C where q = k mod (n-1) and p = floor(k / (n-1)).
%H Andrew Howroyd, <a href="/A356517/b356517.txt">Table of n, a(n) for n = 2..1276</a> (first 50 antidiagonals)
%F A(2, k) = 2^k - 1.
%F A(3, k) = A062318(k+1).
%F A(4, k) = A180516(k+1).
%F A(5, k) = A181287(k+1).
%F A(6, k) = A181288(k+1).
%F A(7, k) = A181303(k+1).
%F A(8, k) = A165804(k+1).
%F A(9, k) = A140576(k+1).
%F A(10, k) = A051885(k).
%F A(n, 0) = 0.
%F A(n, 1) = 1.
%F A(n, k) = k iff k < n.
%F A(n, n) = 2*n - 1.
%F A(n, n+1) = 3*n - 1 for any n > 2.
%e Array A(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12
%e ---+---------------------------------------------------------
%e 2| 0 1 3 7 15 31 63 127 255 511 1023 2047 4095
%e 3| 0 1 2 5 8 17 26 53 80 161 242 485 728
%e 4| 0 1 2 3 7 11 15 31 47 63 127 191 255
%e 5| 0 1 2 3 4 9 14 19 24 49 74 99 124
%e 6| 0 1 2 3 4 5 11 17 23 29 35 71 107
%e 7| 0 1 2 3 4 5 6 13 20 27 34 41 48
%e 8| 0 1 2 3 4 5 6 7 15 23 31 39 47
%e 9| 0 1 2 3 4 5 6 7 8 17 26 35 44
%e 10| 0 1 2 3 4 5 6 7 8 9 19 29 39
%e Array A(n, k) begins (with values given in base n):
%e n\k| 0 1 2 3 4 5 6 7 8 9
%e ---+------------------------------------------------------------------
%e 2| 0 1 11 111 1111 11111 111111 1111111 11111111 111111111
%e 3| 0 1 2 12 22 122 222 1222 2222 12222
%e 4| 0 1 2 3 13 23 33 133 233 333
%e 5| 0 1 2 3 4 14 24 34 44 144
%e 6| 0 1 2 3 4 5 15 25 35 45
%e 7| 0 1 2 3 4 5 6 16 26 36
%e 8| 0 1 2 3 4 5 6 7 17 27
%e 9| 0 1 2 3 4 5 6 7 8 18
%e 10| 0 1 2 3 4 5 6 7 8 9
%o (PARI) A(n,k) = { (1+k%(n-1))*n^(k\(n-1))-1 }
%o (Python) def A(n,k): return (1+(k % (n-1)))*n**(k//(n-1))-1
%Y Cf. A000225, A051885, A062318, A140576, A165804, A180516, A181287, A181288, A181303, A138530, A240236.
%K nonn,tabl,base
%O 2,6
%A _Rémy Sigrist_, Aug 10 2022
|
