A352502 - OEIS

%I #108 May 01 2022 11:42:48

%S 1,2,2,4,4,2,4,4,6,10,8,6,10,8,2,4,4,6,10,8,6,10,8,10,16,12,18,28,20,

%T 14,22,16,10,16,12,18,28,20,14,22,16,2,4,4,6,10,8,6,10,8,10,16,12,18,

%U 28,20,14,22,16,10,16,12,18,28,20,14,22,16,18,28,20,30

%N a(n) is the number of integers k in the interval 0..n such that k and n-k can be added without carries in balanced ternary.

%C Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

%C This sequence has connections with Gould's sequence (A001316); here we work with balanced ternary, there with binary.

%H Rémy Sigrist, <a href="/A352502/b352502.txt">Table of n, a(n) for n = 0..6561</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Balanced_ternary">Balanced ternary</a>

%F a(n) <= n+1 with equality iff n belongs to A140429.

%F a(3*n) = 3*a(n) - 2.

%F a(3*n+1) = a(3*n-1) + 2.

%e For n = 8:

%e - we consider the following cases:

%e               k|    0    1    2    3    4    5    6    7    8

%e       ---------+---------------------------------------------

%e         bter(k)|    0    1   1T   10   11  1TT  1T0  1T1  10T

%e       bter(8-k)|  10T  1T1  1T0  1TT   11   10   1T    1    0

%e        carries?|  no   yes  no   no   yes  no   no   yes  no

%e - so a(8) = 6.

%o (PARI) ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }

%o a(n) = sum(k=0, n, ok(n-k, k))

%Y Cf. A001316, A059095, A140429, A353174 (corresponding k's).

%K nonn,base

%O 0,2

%A _Rémy Sigrist_, Apr 28 2022