A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

1, 2, 2, 3, 4, 2, 3, 4, 4, 3, 4, 4, 5, 6, 4, 5, 6, 6, 7, 6, 2, 3, 4, 4, 5, 6, 4, 3, 4, 5, 5, 6, 6, 5, 6, 4, 5, 6, 6, 7, 8, 4, 5, 6, 4, 5, 6, 6, 5, 6, 6, 7, 8, 8, 3, 4, 5, 5, 6, 7, 5, 6, 6, 7, 6, 4, 5, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 4, 5, 6, 6, 7, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 7, 7, 8, 9, 7, 8, 6, 7, 8, 8

Offset

1,2

Comments

This problem has the optimal substructure property.

References

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, 1971.

R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]

R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.

Ed Pegg, Jr., Integer Complexity.

Eric Weisstein's World of Mathematics, Integer Complexity.

Wikipedia, Optimal substructure.

Example

a(1) = 1 because "1" has a single 1.
a(2) = 2 because "1+1" has two 1's.
a(3) = 2 because 3 = T(1+1) has two 1's.
a(6) = 2 because 6 = T(T(1+1)).
a(10) = 3 because 10 = T(T(1+1)+1).
a(12) = 4 because 12 = T(T(1+1)) + T(T(1+1)).
a(15) = 4 because 15 = T(T(1+1)+1+1).
a(21) = 2 because 21 = T(T(T(1+1))).
a(28) = 3 because 28 = T(T(T(1+1))+1).
a(55) = 3 because 55 = T(T(T(1+1)+1)).

Maple

a:= proc(n) option remember; local m; m:= floor (sqrt (n*2));
      if n<3 then n
    elif n=m*(m+1)/2 then a(m)
    else min (seq (a(i)+a(n-i), i=1..floor(n/2)))
      fi
    end:
seq (a(n), n=1..110);  # Alois P. Heinz, Jan 05 2011

Mathematica

a[n_] := a[n] = Module[{m = Floor[Sqrt[n*2]]}, If[n < 3, n, If[n == m*(m + 1)/2, a[m], Min[Table[a[i] + a[n - i], {i, 1, Floor[n/2]}]]]]];
Array[a, 110] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

See also A023361 = number of compositions into sums of triangular numbers, A053614 = numbers that are not the sum of triangular numbers. Iterated triangular numbers: A050536, A050542, A050548, A050909, A007501.

Cf. A000217, A005245, A005520, A003313, A076142, A076091, A061373, A005421, A023361, A053614, A064097, A025280, A003037, A099129, A050536, A050542, A050548, A050909, A007501.

Sequence in context: A351080 A187200 A352340 * A236241 A127731 A159978

Adjacent sequences: A117629 A117630 A117631 * A117633 A117634 A117635

Keyword

nonn

Author

Jonathan Vos Post, Apr 08 2006

Extensions

I do not know how many of these entries have been proved to be minimal. - N. J. A. Sloane, Apr 15 2006

Corrected and extended by Alois P. Heinz, Jan 05 2011

Status

approved