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A361350

A variant of A359143 which includes the intermediate terms before digits are deleted (see Comments for precise definition).

11, 112, 1124, 11248, 1124816, 2486, 248620, 4860, 486018, 48601827, 4860182736, 486018273645, 8601827365, 860182736546, 86018273654656, 8601827365465667, 860182736546566780, 601273654656670, 60127365465667064, -1273545704, -127354570438, -12735457043849, -1273545704384962, -127354570438496270, 1273545743849627, 127354574384962777, 12735457438496277791, 273545743849627779

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OFFSET

0,1

COMMENTS

This is essentially the same sequence as A359143 (so this too is a finite sequence), the difference being that it includes the terms before any digits are cancelled. Let S be the digit string of a(n), replacing a minus sign if present by 0.

Let T = S concatenated with the digit-sum of S.

If the leading digit of T is not present in the digit-sum of S, then a(n+1) = A359142(T), as in A359143.

If the leading digit of T is present in the digit-sum of S, then we add two new terms instead of one: a(n+1) = a(n) concatenated with the digit-sum of S, and a(n+2) = A359142(T), as in A359143.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Michael De Vlieger, Scatterplot of log_10(abs(a(n))), n = 1..10^3, showing negative terms in red.

Michael De Vlieger, Scatterplot of log_10(abs(a(n))), n = 1..10^4, showing negative terms in red.

Michael De Vlieger, Scatterplot of log_10(abs(a(n))), showing all terms, with negative terms in red.

EXAMPLE

The digit strings for the initial terms are:

11,

112,

1124,

11248,

1124816,

2486,

248620,

4860,

486018,

48601827,

4860182736,

486018273645,

8601827365,

860182736546,

86018273654656,

8601827365465667,

860182736546566780,

601273654656670,

60127365465667064,

01273545704,

0127354570438,

012735457043849,

01273545704384962,

0127354570438496270,

1273545743849627,

127354574384962777,

12735457438496277791,

273545743849627779, ...

The sequence itself is obtained by replacing the leading zeros by minus signs.

For example, after the term 601273654656670, we first append its digit-sum 64, getting 60127365465667064. Since the leading digit 6 is present in 64, we cancel all the 6's, getting 01273545704. The corresponding term in the sequence is -1273545704.

MATHEMATICA

a[1] = {1, 1}; nn = 28;

Do[Which[ListQ[m], k = m; Clear[m],

FreeQ[#3, #2], Set[k, #1~Join~#3],

True, Set[k, #1~Join~#3];

Set[m, DeleteCases[#1~Join~#3, #2]]] & @@

{#, First[#], IntegerDigits@ Total[#]} &[a[n - 1]];

Set[a[n], k], {n, 2, nn}];

Array[(1 - 2 Boole[First[#] == 0])*FromDigits@ # &@ a[#] &, nn] (* Michael De Vlieger, Mar 16 2023 *)

CROSSREFS

Cf. A359142, A359143.

Sequence in context: A019523 A359143 A375101 * A361501 A372074 A132939