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a(n) is the number of 0's in the maximal tribonacci representation of n (A352103).
2

%I #12 Sep 05 2022 05:24:32

%S 1,0,1,0,2,1,1,0,2,2,1,2,1,1,0,3,2,3,2,2,1,2,2,1,2,1,1,0,4,3,3,2,3,3,

%T 2,3,2,2,1,3,2,3,2,2,1,2,2,1,2,1,1,0,4,4,3,4,3,3,2,4,3,4,3,3,2,3,3,2,

%U 3,2,2,1,4,3,3,2,3,3,2,3,2,2,1,3,2,3,2

%N a(n) is the number of 0's in the maximal tribonacci representation of n (A352103).

%H Amiram Eldar, <a href="/A356894/b356894.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A356895(n) - A352104(n).

%e n a(n) A352103(n)

%e - ---- ----------

%e 0 1 0

%e 1 0 1

%e 2 1 10

%e 3 0 11

%e 4 2 100

%e 5 1 101

%e 6 1 110

%e 7 0 111

%e 8 2 1001

%e 9 2 1010

%t t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; a[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 1, Count[v[[i[[1, 1]] ;; -1]], 0]]]; Array[a, 100, 0]

%Y Cf. A000073, A352103, A352104, A356895.

%Y Similar sequences: A023416, A102364, A117479, A278042.

%K nonn,base

%O 0,5

%A _Amiram Eldar_, Sep 03 2022