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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 }.

LINKS

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

EXAMPLE

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.

PROG

(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))}

CROSSREFS

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

Sequence in context: A175688 A048321 A071267 * A257054 A110784 A193408

Adjacent sequences:  A295007 A295008 A295009 * A295011 A295012 A295013

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

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Last modified February 19 19:30 EST 2020. Contains 332047 sequences. (Running on oeis4.)