A335854 The digital-root sandwiches sequence (see Comments lines for definition).

11, 9, 99, 999, 9999, 9990, 1, 112, 12, 3, 125, 33, 4, 15, 8, 337, 44, 5, 154, 88, 2, 37, 24, 49, 55, 6, 14, 38, 81, 22, 53, 7, 92, 48, 495, 552, 66, 71, 47, 387, 813, 227, 531, 77, 79, 26, 483, 45, 152, 86, 64, 715, 471, 376, 83, 52, 73, 51, 87, 75, 792, 261, 43, 74, 56, 121, 863, 642, 759, 41, 436, 58, 385, 29

Offset: 1

Author

Carole Dubois and Eric Angelini, Jun 26 2020

Keywords

• base
• nonn

Comments

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. A digital-root sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the digital root of their sum. The pair [1951, 2020] would then produce the DR-sandwich 132. Please note that the pair [2020, 1951] would produce the genuine DR-sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

Examples

			The first successive sandwiches are: 119, 999, 999, 999, 999, 011, 121, 231, ...
The first one (119) is visible between a(1) = 11 and a(2) = 9; we get the sandwich by inserting the digital root of the sum 1 + 9 = 10 (which is 1) between 1 and 9.
The second sandwich (999) is visible between a(2) = 9 and a(3) = 99; we get the sandwich by inserting the digital root of the sum 9 + 9 = 18 (which is 9) between 9 and 9.
The third sandwich (999) is visible between a(3) = 99 and a(3) = 999; we get the sandwich by inserting the digital root of the sum 9 + 9 = 18 (which is 9) between 9 and 9.
(...)
The sixth sandwich (011) is visible between a(6) = 9990 and a(7) = 1; we get the sandwich by inserting the digital root of the sum 0 + 1 = 1 (which is 1) between 0 and 1; etc.
The successive sandwiches rebuild, digit after digit, the starting sequence.
		

Links

• Carole Dubois, Table of n, a(n) for n = 1..428

Crossrefs

Cf. A335600 (first definition of such a sandwich) and A010888 (digital root of n).