A110380 - OEIS

%I #27 Aug 12 2020 11:44:56

%S 1,11,12,112,122,123,1123,1223,1233,1234,11234,12234,12334,12344,

%T 12345,112345,122345,123345,123445,123455,123456,1123456,1223456,

%U 1233456,1234456,1234556,1234566,1234567,11234567,12234567,12334567,12344567,12345567,12345667,12345677

%N 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.

%C 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

%H Fred Schneider, <a href="/A110380/b110380.txt">Table of n, a(n) for n = 1..5000</a>

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

%e Using n ones and only the + sign we get the following sums:

%e   a(1) = 1;

%e   a(2) = 11;

%e   a(3) = 12 = 1 + 11;

%e   a(4) = 112 = 1 + 111;

%e   a(5) = 122 = 11 + 111;

%e   a(6) = 123 = 1 + 11 + 111;

%e   a(7) = 1123 = 1 + 11 + 1111;

%e   a(8) = 1223 = 1 + 111 + 1111;

%e   a(9) = 1233 = 11 + 111 + 1111.

%o (Haskell)

%o a110380 = drop 1 fn

%o           where fn    = 0 : 1 : concat (fn' 2)

%o                 fn' n = (map (+ones) (drop nv $ take (n + nv) fn)) : (fn' (n+1))

%o                         where ones = div (10^n -1) 9

%o                               nv   = div ((n-1)*(n-2)) 2

%o -- _Fred Schneider_, Sep 04 2016

%o (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

%Y Cf. A000042, A003056.

%K base,easy,nonn

%O 1,2

%A _Amarnath Murthy_, Jul 25 2005

%E More terms from _Joshua Zucker_, May 08 2006