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A234512 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. 1
110, 311000, 2301000, 3003000, 3120000, 42100000, 410300000, 430100000 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..8.

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

Cf. A037904, A046043, A108551, A138480.

Sequence in context: A203718 A143750 A192844 * A343181 A028673 A138280

Adjacent sequences:  A234509 A234510 A234511 * A234513 A234514 A234515

KEYWORD

nonn,base,fini

AUTHOR

Michel Lagneau, Dec 27 2013

STATUS

approved

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Last modified June 26 04:58 EDT 2022. Contains 354877 sequences. (Running on oeis4.)