A226465 Numbers n = x0 x1...x9 such that sum of digits (mod 10) - xi == i+1 mod 10.

21, 210, 765, 9876, 2109876, 7654321, 76543210, 4321098765

Offset

1,1

Comments

We consider the system of linear equations using (mod 10):

x1 + x2 + x3 + ... + x9 = 1

x0 + x2 + x3 + ... + x9 = 2

x0 + x1 + x3 + ... + x9 = 3

………………………….

x0 + x1 + ... + x7 + x9 = 9

x0 + x1 + x2 + ... + x8 = 0

where xi are the decimal digits of n.

For n = 2, 3,...,10 we obtain the equation Ax = b where A is the matrix with n rows and n columns, x is the vector [x0, x1,...,xp] where p = n-1, and b is the vector [1, 2,..., n] (mod 10). For n = 2, 4, 8 and 10, the solution is unique. For n = 5, 6 and 9, the system has no solution. For n = 3 and 7, the system has two solutions.

Properties of the sequence:

a(5) is the concatenation of a(2) and a(4);

a(2) is the suffix of a(7);

a(3) is the prefix of a(6) and a(7) and also the suffix of a(8);

a(8) contains the digits of a(4) and a(2).

Links

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

Example

9876 = x0 x1 x2 x3 is in the sequence because:
x1 + x2 + x3 = 8+7+6 = 21 == 1 mod 10;
x0 + x2 + x3 = 9+7+6 = 22 == 2 mod 10;
x0 + x1 + x3 = 9+8+6 = 23 == 3 mod 10;
x0 + x1 + x2 = 9+8+7 = 24 == 4 mod 10.

Maple

with(numtheory):for n from 1 to 1000000 do:x:=convert(n, base, 10):n1:=nops(x): s:=sum('x[i]', 'i'=1..n1):ii:=0:for k from n1 by -1 to 1 do:if irem(n1-k+1, 10)=irem(s-x[k], 10) then ii:=ii+1:else fi:od:if ii=n1 then print(n):else fi:od:

Crossrefs

Sequence in context: A232345 A036205 A160787 * A047646 A010937 A022616

Adjacent sequences: A226462 A226463 A226464 * A226466 A226467 A226468

Keyword

nonn,base,fini,full

Author

Michel Lagneau, Jun 08 2013

Status

approved