A110380 a(n) = min{p + q + r + ...} where p,q,r,... are distinct unary numbers - containing only ones, i.e., of the form (10^k - 1)/9 - formed by using a total of n ones.

1, 11, 12, 112, 122, 123, 1123, 1223, 1233, 1234, 11234, 12234, 12334, 12344, 12345, 112345, 122345, 123345, 123445, 123455, 123456, 1123456, 1223456, 1233456, 1234456, 1234556, 1234566, 1234567, 11234567, 12234567, 12334567, 12344567, 12345567, 12345667, 12345677

Offset: 1

Author

Amarnath Murthy, Jul 25 2005

Keywords

• base
• easy
• nonn

Comments

The n-th term is the sum of m = A003056(n) repunits A000042, with the last n - m(m+1)/2 terms having one digit more than their index in that sum: see formula. The initial terms can also be described as numbers made of all of the digits 1 through m (m = 1, ..., 9) in increasing order, and at most one of these digits occurring twice in a row. - M. F. Hasler, Aug 08 2020

Examples

			Using n ones and only the + sign we get the following sums:
  a(1) = 1;
  a(2) = 11;
  a(3) = 12 = 1 + 11;
  a(4) = 112 = 1 + 111;
  a(5) = 122 = 11 + 111;
  a(6) = 123 = 1 + 11 + 111;
  a(7) = 1123 = 1 + 11 + 1111;
  a(8) = 1223 = 1 + 111 + 1111;
  a(9) = 1233 = 11 + 111 + 1111.
		

Links

• Fred Schneider, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000042, A003056.

Programs

Haskell

a110380 = drop 1 fn
          where fn    = 0 : 1 : concat (fn' 2)
                fn' n = (map (+ones) (drop nv $ take (n + nv) fn)) : (fn' (n+1))
                        where ones = div (10^n -1) 9
                              nv   = div ((n-1)*(n-2)) 2
-- Fred Schneider, Sep 04 2016

PARI

apply( {A110380(n,m=A003056(n))=sum(k=1,m,10^((n+k>(m+3)*m\2)+k)\9)}, [1..40]) \\ with {A003056(n)=(sqrtint(n*8+1)-1)\2}. M. F. Hasler, Aug 08 2020

Formula

a(n) = Sum_{k=1..m} A000042(k + [m(m+3)/2 < n+k]), with m = A003056(n). - M. F. Hasler, Aug 08 2020

Extensions

More terms from Joshua Zucker, May 08 2006