login
Balanced septenary enumeration (or balanced septenary representation) of integers; write n in septenary and then replace 4's with (-3),s, 5's with (-2)'s, and 6's with (-1)'s.
3

%I #23 Aug 29 2019 20:22:39

%S 0,1,2,3,-3,-2,-1,7,8,9,10,4,5,6,14,15,16,17,11,12,13,21,22,23,24,18,

%T 19,20,-21,-20,-19,-18,-24,-23,-22,-14,-13,-12,-11,-17,-16,-15,-7,-6,

%U -5,-4,-10,-9,-8,49,50,51,52,46,47,48,56,57,58,59,53,54,55,63

%N Balanced septenary enumeration (or balanced septenary representation) of integers; write n in septenary and then replace 4's with (-3),s, 5's with (-2)'s, and 6's with (-1)'s.

%C This sequence, like the balanced ternary and quinary sequences, includes every integer exactly once.

%H Alois P. Heinz, <a href="/A309995/b309995.txt">Table of n, a(n) for n = 0..16806</a>

%e As 54_10 = 105_7, the digits of 54 in base 7 are 1, 0 and 5. 5 > 3 so it's replaced by -2. The digits then are 1, 0 and -2 giving a(54) = 1*7^2 + 0 * 7^1 + (-2) * 7^0 = 49 + 0 - 2 = 47. - _David A. Corneth_, Aug 26 2019

%p a:= proc(n) option remember; `if`(n=0, 0,

%p 7*a(iquo(n, 7))+mods(n, 7))

%p end:

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Aug 26 2019

%o (PARI) a(n,b=7) = fromdigits(apply(d -> if (d<b/2, d, d-b), digits(n, b)), b) \\ _Rémy Sigrist_, Aug 26 2019

%o (PARI) a(n) = my(d = digits(n, 7)); for(i = 1, #d, if(d[i] > 3, d[i]-=7)); fromdigits(d, 7) \\ _David A. Corneth_, Aug 26 2019

%Y Cf. A007093, A117966, A309991.

%Y Column k=3 of A319047.

%K base,sign

%O 0,3

%A _Jackson Haselhorst_, Aug 26 2019