A295010 Self-slideable numbers: numbers which can reproduce themselves by sliding each digit d by d places either to the left or to the right.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111, 202, 222, 333, 444, 555, 666, 777, 888, 999, 1100, 1111, 2020, 2222, 3003, 3113, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11000, 11011, 11110, 11111, 11202, 12222, 20200, 20211

Offset: 1

Author

Eric Angelini and M. F. Hasler, Dec 14 2017

Keywords

• nonn
• base

Comments

The result of the "slide" operation does not need to be in the initial position, e.g., the number 1 reproduces itself displaced by 1 position.

The first terms which can be "slided" into themselves without changing place are 0, 11, 110, 202, 1100, 1111, 2020, 2222, 3003, 3113, 11000, 11011, 11110, 11202, 20200, 20211, ...

Any concatenation of such fixed-position self-slideable numbers is again one. Primitive terms (not concatenation of smaller terms) are 0, 11, 202, 2222, 3003, 3113, 23203, 30232, 33033, 40004, 40114, 41104, 42024, ... However, even though they are not concatenation of smaller terms, 2222, 3113 and all the 5-digit terms except 40004 are a "non-interfering superposition" of earlier terms, i.e., the nonzero digits take the place of zero digits of earlier terms.

Theorem: Any fixed-position self-slideable number is a non-interfering superposition of terms of the form d*10^d+d.

Actually, most of the terms are of that form: among the 131 terms < 10^6, there are 45 repdigits, 80 fixed-position self-slideable numbers of the above form, and only 6 other terms, { 12222, 22221, 31313, 122221, 131313, 313131 }.

Examples

			12222 is in the sequence because one can slide the 1 and three of the 2's to the right (by one resp. two places), and the last 2 by two places to the left, and get back the same number, shifted one place to the right.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..131

Crossrefs

Cf. A296242 (slideable numbers), A010785 (repdigit numbers: a subsequence).

Programs

PARI

is_A295010(n,d=matdiagonal(n=digits(n)),v=[1..#n]+n)={!n||forvec( s=vector(#n,i,[0,1]),vecmax(p=v-2*s*d)+1==vecmin(p)+#p&&#p==#Set(p)&&sum(i=1,#p,10^(vecmax(p)-p[i])*n[i])==fromdigits(n,10)&&return(1))}