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%I #6 Dec 28 2013 04:10:34
%S 110,311000,2301000,3003000,3120000,42100000,410300000,430100000
%N Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.
%C In the decimal system a differential autobiographical number is a natural number such that d(0) is the number of differences |d(i)-d(i-1)| = 0, d(1) is the number of differences |d(i)-d(i-1)| = 1, and so on.
%C Property of this sequence: the sum of the decimal digits of a(n) equals length(a(n))-1.
%C It is possible to extend this problem by counting the differences |d(i)-d(i-1)| with the additional difference |d(r)-d(1)|. So we find a new sequence b(n) = 22100, 311100, 3022000, 20402000, 31310000, 40004000, 422010000, 430110000 with the property that the sum of the decimal digits of b(n) equals length(b(n)).
%H Tanya Khovanova, <a href="http://arXiv.org/find/all/1/au:+khovanova/0/1/0/all/0/1">Autobiographical Numbers</a>
%e 311000 is in the sequence because the differential digits are:
%e |1-3| = 2;
%e |1-1| = 0;
%e |0-1| = 1;
%e |0-0| = 0;
%e |0-0| = 0, and
%e 0 appears three times => 3;
%e 1 appears one time => 1;
%e 2 appears one time => 1;
%e 3 appears zero time => 0;
%e 4 appears zero time => 0;
%e 5 appears zero time => 0, hence a(2) = 311000.
%p with(numtheory):for n from 10 to 10^10 do:T:=array(0..9):for k from 0 to 9 do:T[k]:=0:od:x:=convert(n,base,10):n1:=nops(x):for i from 1 to n1-1 do:a:=abs(x[i]-x[i+1]):T[a]:=T[a]+1:od:s:=sum('T[i]*10^(10-i-1)','i'=0..9): for u from 9 by -1 to 1 do:if T[0]<>0 and irem(s,10^u)=0 and s/10^u = n then print(n):else fi:od:od:
%Y Cf. A037904, A046043, A108551, A138480.
%K nonn,base,fini
%O 1,1
%A _Michel Lagneau_, Dec 27 2013
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