OFFSET
0,4
LINKS
Peter J. Grabner and Clemens Heuberger, On the number of optimal base 2 representations of integers, Des. Codes Cryptogr. 40 (2006), no. 1, 25-39.
S. Kropf, S. Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016.
FORMULA
See Maple code for the five recurrences.
MAPLE
for p from 1 to 5 do u[p][0]:=1; od;
u[1][1]:=1; u[2][1]:=1; u[3][1]:=0; u[4][1]:=0; u[5][1]:=0;
for n from 2 to 100 do
if n mod 2 = 0 then
u[1][n]:=u[1][n/2]; u[2][n]:=u[1][n/2]; u[3][n]:=u[2][n/2]; u[4][n]:=u[1][n/2]; u[5][n]:=u[4][n/2];
else
u[1][n]:=u[2][(n-1)/2]+u[4][(n+1)/2]; u[2][n]:=u[3][(n-1)/2]; u[3][n]:=0; u[4][n]:=u[5][(n-1)/2]; u[5][n]:=0;
fi;
od:
[seq(u[1][n], n=0..100)]; # A280747
[seq(u[2][n], n=0..100)]; # A280748
[seq(u[3][n], n=0..100)]; # A280749
[seq(u[4][n], n=0..100)]; # A280750
[seq(u[5][n], n=0..100)]; # A280751
MATHEMATICA
For[p = 1, p <= 5, p++, u[p][0] = 1]; u[1][1] = 1; u[2][1] = 1; u[3][1] = 0; u[4][1] = 0; u[5][1] = 0;
For[n = 2, n <= 100, n++, If[Mod[n, 2] == 0, u[1][n] = u[1][n/2]; u[2][n] = u[1][n/2]; u[3][n] = u[2][n/2]; u[4][n] = u[1][n/2]; u[5][n] = u[4][n/2], u[1][n] = u[2][(n-1)/2] + u[4][(n+1)/2]; u[2][n] = u[3][(n-1)/2]; u[3][n] = 0; u[4][n] = u[5][(n-1)/2]; u[5][n] = 0]];
Table[u[1][n], {n, 0, 100}] (* A280747 *)
Table[u[2][n], {n, 0, 100}] (* A280748 *)
Table[u[3][n], {n, 0, 100}] (* A280749 *)
Table[u[4][n], {n, 0, 100}] (* A280750 *)
Table[u[5][n], {n, 0, 100}] (* A280751 *) (* Jean-François Alcover, Sep 02 2018, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 14 2017
STATUS
approved