A277845 The digits of the absolute difference |a(n+1)-a(n)| are a subsequence of the digits of a(n): Lexicographically first infinite sequence with this property and without negative or repeated terms.

1, 2, 4, 8, 16, 10, 9, 18, 17, 24, 20, 22, 44, 40, 36, 30, 27, 25, 23, 21, 19, 28, 26, 32, 29, 31, 34, 37, 74, 67, 60, 54, 49, 45, 41, 42, 38, 35, 70, 63, 57, 50, 55, 110, 99, 90, 81, 73, 66, 72, 65, 59, 64, 58, 53, 48, 52, 47, 43, 39, 78, 71, 142, 100, 101, 91, 82, 80, 88, 96, 87, 79, 86, 92, 83, 75, 68, 62, 56

Offset

1,2

Comments

In contrast to A277858, any one or more digits of a(n) can be omitted. (For example, if a(n) = 123, the absolute difference may be any of { 1, 2, 3, 12, 13, 23, 123 }, while 13 would be forbidden in A277858.)

To ensure the sequence is infinite, one has to check whether a potential successor of a term will have a successor itself. For example, a(80) = 51 would most naively be followed by 51 - 5 = 46, which does not occur earlier, but which then would have no possible successor. Therefore one must choose a(81) = 51 + 51 = 102.

This sequence differs from A277858 from a(88) = 120 on (which follows a(87) = 138 = a(88) + 18), while a gap of 18 is not allowed at this point in A277858, where A277858(88) = 125 = a(87) - 13. Thereafter this sequence continues completely differently from A277858, but they have the same set of 12 numbers {3, 5, 6, 7, 11, 12, 13, 14, 15, 33, 46, 61} which never appear.

It is easy to prove that this is not a permutation of the positive integers: indeed, for example, the number 5 can never occur in the sequel, since it has no possible successor (10 being already used). There are several variants of this sequence, some of which are permutations: Aside the stronger condition defining A277858, one could consider the weaker conditions "digits of |a(n+1)-a(n)| are a sub-multiset of those of a(n)" (e.g., a step of 21 would be allowed after 123), or "... are a subset of ..." (a step of 11 would be allowed after 123). The latter variant (which is a permutation) is A276070.

Links

Table of n, a(n) for n=1..79.

Prog

(PARI) /* 'a' subsequence of 'b'?*/ ss(a, b, i=1, k=1)={while(k < #b && #b-k >= #a-i, b[k] == a[i] && (i==#a || ss(a, b, i+1, k+1)) && return(1); k++); k==#b && i==#a && b[k] == a[i]}
A(n, a=1, g=1)={u=[]; for(n=2, n, g&&print1(a", "); u=setunion(u, [a]); /*while(#u>2&&u[2]==u[1]+1, u=u[^1]); */ d=digits(a); for(k=u[1]+1, 2*a, (!setsearch(u, k)&&ss(digits(abs(a-k)), d))||next; p==a&&u=setminus(u, [b]); [p, a]=[a, k]; next(2)); g&&print1(" er, no: delete "a", go back to "); b=a; a=p); a} \\ Implementation of backtracking should be improved

Crossrefs

Cf. A276070, A277858.

Sequence in context: A009289 A070335 A277846 * A277858 A366031 A167421

Adjacent sequences: A277842 A277843 A277844 * A277846 A277847 A277848

Keyword

nonn,base

Author

M. F. Hasler, Nov 05 2016

Status

approved