OFFSET
1,1
COMMENTS
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.
Property of this sequence: the sum of the decimal digits of a(n) equals length(a(n))-1.
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)).
LINKS
Tanya Khovanova, Autobiographical Numbers
EXAMPLE
311000 is in the sequence because the differential digits are:
|1-3| = 2;
|1-1| = 0;
|0-1| = 1;
|0-0| = 0;
|0-0| = 0, and
0 appears three times => 3;
1 appears one time => 1;
2 appears one time => 1;
3 appears zero time => 0;
4 appears zero time => 0;
5 appears zero time => 0, hence a(2) = 311000.
MAPLE
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:
CROSSREFS
KEYWORD
nonn,base,fini
AUTHOR
Michel Lagneau, Dec 27 2013
STATUS
approved