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A319154
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a(n) is the smallest nonnegative integer not yet in the sequence that starts with the ending digit of a(n-1); a(1)=0; initial zeros are dropped.
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1
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0, 1, 10, 2, 20, 3, 30, 4, 40, 5, 50, 6, 60, 7, 70, 8, 80, 9, 90, 11, 12, 21, 13, 31, 14, 41, 15, 51, 16, 61, 17, 71, 18, 81, 19, 91, 100, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 200, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 300, 44, 45, 54
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OFFSET
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1,3
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COMMENTS
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Theorem: Every nonnegative number appears.
(i) Sequence is infinite (dG, G=giant number, is always available)
(ii) As usual for these "lexicographically earliest distinct term sequences", for any k, there is a threshold n_k such that for all n > n_k, a(n) > k.
(iii) Some final digit (d, say) appears infinitely often. (Otherwise sequence would be finite.) If d=0, go to step (vi).
(iv) All numbers beginning with d appear (If dm were missing, find xd in sequence which is > dm and also > n_{dm}. Then term after xd would be dm, contradiction.)
(v) In particular, all numbers dm0 appear.
(vi) After a number ending in 0, the next number is the smallest missing number. So if x is missing, find dm0 > n_x, then the next term would be (0)x = x, a contradiction. QED
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LINKS
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EXAMPLE
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a(2) = 1 since it is formed from a(1) = 0 as 01 = 1.
a(20) = 11 since it is the smallest number not yet in the sequence that starts with the ending digit 0 of a(19) = 90.
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MATHEMATICA
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Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], If[# == 0, True, First@ IntegerDigits@ k == #] &@ Mod[#[[-1]], 10]], k++]; k]] &, {0}, 69] (* Michael De Vlieger, Oct 15 2018 *)
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PROG
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(PARI) nexta(v, x) = {my(d = x % 10, newa); for (i=0, oo, newa = eval(concat(Str(d), Str(i))); if (! vecsearch(v, newa), return (newa)); ); }
lista(nn) = {lasta = 0; print1(lasta, ", "); va = [lasta]; for (n=1, nn, newa = nexta(va, lasta); print1(newa, ", "); va = vecsort(concat(va, newa)); lasta = newa; ); } \\ Michel Marcus, Oct 15 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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