cp's OEIS Frontend

A376150 Define b_n(k) to be the lexicographically earliest sequence of distinct nonnegative integers with the property that two terms that contain the digit "d" are always separated by exactly "d" terms that do not contain the digit "d", in base n. a(n) is the number of terms in b_n(k).

%I A376150 #15 Nov 16 2024 16:11:49
%S A376150 2,4,6,10,10,18,18,22,22,30,30,34,42,42,78,78,78,78,102,102,114,114,
%T A376150 114,114,142,142,142,142,214,214,214,214,214,214,214,222,274,274,274,
%U A376150 274,274,354,354,354,354,354,354,642,642,642,642,642,642,642,642
%N A376150 Define b_n(k) to be the lexicographically earliest sequence of distinct nonnegative integers with the property that two terms that contain the digit "d" are always separated by exactly "d" terms that do not contain the digit "d", in base n. a(n) is the number of terms in b_n(k).
%C A376150 This process terminates only when all nonzero digits are prohibited by the restrictions in place for the next term; as b_n(2) = "10" for all n, the digit 1 is only prohibited for odd numbered terms, and as such a(n) must be even for all n. Similar logic can be applied to the digit 3 to show that for all n>3, a(n) is not divisible by 4.
%C A376150 A375232 is the sequence generated when n=10.
%H A376150 Jake Bird, <a href="/A376150/b376150.txt">Table of n, a(n) for n = 2..88</a>
%e A376150 For n = 5:
%e A376150 b_5(1) = 0; as this contains the digit 0, b_5(2), b_5(3) etc. must also contain a 0
%e A376150 b_5(2) = 10 (= 5 in decimal); must contain a 0 from b_5(1); as this contains the digit 1, b_5(4), b_5(6) etc. must also contain a 1, and all other terms must not contain a 1
%e A376150 b_5(3) = 20; must have 0 but not 1
%e A376150 b_5(4) = 100; must have 0 and 1 but not 2
%e A376150 b_5(5) = 30; must have 0 but not 1 or 2
%e A376150 b_5(6) = 102; must have 0, 1, and 2, but not 3
%e A376150 b_5(7) = 40; must have 0 but not 1, 2, or 3
%e A376150 b_5(8) = 101; must have 0 and 1 but not 2, 3, or 4
%e A376150 b_5(9) = 203; must have 0, 2, and 3, but not 1 or 4
%e A376150 b_5(10) = 110; must have 0 and 1 but not 2, 3, or 4
%e A376150 b_5(11) = ---; must have 0 but not 1, 2, 3, or 4 - the only number that fills this condition is 0, but 0 already appears in the sequence, so the sequence terminates after ten terms, and a(5) = 10
%Y A376150 Cf. A375232.
%K A376150 nonn,base
%O A376150 2,1
%A A376150 _Jake Bird_, Sep 12 2024