login
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).
1
2, 4, 6, 10, 10, 18, 18, 22, 22, 30, 30, 34, 42, 42, 78, 78, 78, 78, 102, 102, 114, 114, 114, 114, 142, 142, 142, 142, 214, 214, 214, 214, 214, 214, 214, 222, 274, 274, 274, 274, 274, 354, 354, 354, 354, 354, 354, 642, 642, 642, 642, 642, 642, 642, 642
OFFSET
2,1
COMMENTS
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.
A375232 is the sequence generated when n=10.
EXAMPLE
For n = 5:
b_5(1) = 0; as this contains the digit 0, b_5(2), b_5(3) etc. must also contain a 0
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
b_5(3) = 20; must have 0 but not 1
b_5(4) = 100; must have 0 and 1 but not 2
b_5(5) = 30; must have 0 but not 1 or 2
b_5(6) = 102; must have 0, 1, and 2, but not 3
b_5(7) = 40; must have 0 but not 1, 2, or 3
b_5(8) = 101; must have 0 and 1 but not 2, 3, or 4
b_5(9) = 203; must have 0, 2, and 3, but not 1 or 4
b_5(10) = 110; must have 0 and 1 but not 2, 3, or 4
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
CROSSREFS
Cf. A375232.
Sequence in context: A187333 A321805 A333412 * A089227 A204660 A275956
KEYWORD
nonn,base
AUTHOR
Jake Bird, Sep 12 2024
STATUS
approved