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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.

%I A110380 #27 Aug 12 2020 11:44:56
%S A110380 1,11,12,112,122,123,1123,1223,1233,1234,11234,12234,12334,12344,
%T A110380 12345,112345,122345,123345,123445,123455,123456,1123456,1223456,
%U A110380 1233456,1234456,1234556,1234566,1234567,11234567,12234567,12334567,12344567,12345567,12345667,12345677
%N 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.
%C A110380 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 A110380 Fred Schneider, <a href="/A110380/b110380.txt">Table of n, a(n) for n = 1..5000</a>
%F A110380 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 A110380 Using n ones and only the + sign we get the following sums:
%e A110380   a(1) = 1;
%e A110380   a(2) = 11;
%e A110380   a(3) = 12 = 1 + 11;
%e A110380   a(4) = 112 = 1 + 111;
%e A110380   a(5) = 122 = 11 + 111;
%e A110380   a(6) = 123 = 1 + 11 + 111;
%e A110380   a(7) = 1123 = 1 + 11 + 1111;
%e A110380   a(8) = 1223 = 1 + 111 + 1111;
%e A110380   a(9) = 1233 = 11 + 111 + 1111.
%o A110380 (Haskell)
%o A110380 a110380 = drop 1 fn
%o A110380           where fn    = 0 : 1 : concat (fn' 2)
%o A110380                 fn' n = (map (+ones) (drop nv $ take (n + nv) fn)) : (fn' (n+1))
%o A110380                         where ones = div (10^n -1) 9
%o A110380                               nv   = div ((n-1)*(n-2)) 2
%o A110380 -- _Fred Schneider_, Sep 04 2016
%o A110380 (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 A110380 Cf. A000042, A003056.
%K A110380 base,easy,nonn
%O A110380 1,2
%A A110380 _Amarnath Murthy_, Jul 25 2005
%E A110380 More terms from _Joshua Zucker_, May 08 2006