A299952 -id:A299952 - cp's OEIS frontend

A299957 The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.

0, 1, 9, 2, 8, 3, 7, 4, 6, 5, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 49, 12, 19, 22, 29, 32, 39, 42, 58, 13, 18, 23, 28, 33, 38, 43, 48, 52, 53, 47, 14, 17, 24, 27, 34, 37, 44, 56, 15, 16, 25, 26, 35, 36, 45, 46, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 0

Eric Angelini, Feb 22 2018

The sequence starts with a(0) = 0 and is always extended with the smallest integer not yet present that does not lead to a contradiction. The sequence is a permutation of the natural numbers.

Originally the sequence was defined starting with a(1) = 1 and using only positive integers. This leads to the same sequence restricted to positive indices, which yields a permutation of the positive integers. - M. F. Hasler, Feb 28 2018

			1 + 9 = 10; 9 + 2 = 11; 2 + 8 = 10; 8 + 3 = 11; 3 + 7 = 10; 7 + 4 = 11; 4 + 6 = 10; 6 + 5 = 11; etc.

M. F. Hasler, Table of n, a(n) for n = 0..10000

Cf. A299952 (different constraint: a(n) + a(n+1) must be substring of concatenation of a(1..n+1)).
Cf. A299970, A299982, ..., A299988, A299969  (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[k + #[[-1]], 10, 1] > 0], k++]; k]] &, {1}, 98] (* Michael De Vlieger, Feb 22 2018 *)

a(n, f=1, a=0, u=[a])={for(n=a+1, n, f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1, oo, setsearch(u, k)&&next; setsearch(Set(digits(a+k)),1)&&(a=k)&&break); u=setunion(u, [a]); u[2]==u[1]+1&&u=u[^1]); a} \\ M. F. Hasler, Feb 22 2018

Extended to a(0) = 0 by M. F. Hasler, Feb 28 2018

A363872 Lexicographically earliest sequence of distinct terms > 0 such that n is a substring of a(n) + a(n+1).

Original entry on oeis.org

1, 9, 3, 10, 4, 11, 5, 2, 6, 13, 87, 23, 89, 24, 90, 25, 91, 26, 92, 27, 93, 28, 94, 29, 95, 30, 96, 31, 97, 32, 98, 33, 99, 34, 100, 35, 101, 36, 102, 37, 103, 38, 104, 39, 105, 40, 106, 41, 7, 42, 8, 43, 109, 44, 110, 45, 111, 46, 12, 47, 113, 48, 14, 49, 15

Offset: 1

Eric Angelini, Jul 03 2023

			n = 1 is a substring of the sum  1 +  9 = 10
n = 2 is a substring of the sum  9 +  3 = 12
n = 3 is a substring of the sum  3 + 10 = 13
n = 4 is a substring of the sum 10 +  4 = 14
n = 5 is a substring of the sum  4 + 11 = 15
n = 6 is a substring of the sum 11 +  5 = 16
...
n = 10 is a substring of the sum 13 + 87 = 100, etc.

Eric Angelini, Échecs et Maths, Personal blog, bottom of page.

Cf. A299952.

R:= 1: x:= 1: S:= {1}:
for n from 1 to 100 do
  ns:= convert(n,string);
  for y from 1 do
    if member(y,S) then next fi;
    if SearchText(ns,convert(x+y,string)) <> 0 then
      R:= R,y; x:= y; S:= S union {y}; break
    fi
  od
od:
R;  # Robert Israel, Jul 04 2023

a[1] = 1; a[n_] := a[n] = Module[{k = 2}, While[! FreeQ[Array[a, n - 1], k] || ! StringContainsQ[ToString[a[n - 1] + k], ToString[n - 1]], k++]; k]; Array[a, 100] (* Amiram Eldar, Jul 04 2023 *)

cp's OEIS Frontend

A299957 The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.

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