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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.

%I A295010 #15 Dec 15 2017 07:03:12
%S A295010 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,110,111,202,222,333,
%T A295010 444,555,666,777,888,999,1100,1111,2020,2222,3003,3113,3333,4444,5555,
%U A295010 6666,7777,8888,9999,11000,11011,11110,11111,11202,12222,20200,20211
%N 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.
%C A295010 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.
%C A295010 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, ...
%C A295010 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.
%C A295010 Theorem: Any fixed-position self-slideable number is a non-interfering superposition of terms of the form d*10^d+d.
%C A295010 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 }.
%H A295010 M. F. Hasler, <a href="/A295010/b295010.txt">Table of n, a(n) for n = 1..131</a>
%e A295010 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.
%o A295010 (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))}
%Y A295010 Cf. A296242 (slideable numbers), A010785 (repdigit numbers: a subsequence).
%K A295010 nonn,base
%O A295010 1,3
%A A295010 _Eric Angelini_ and _M. F. Hasler_, Dec 14 2017