A140358 Smallest nonnegative integer k such that n = +-1+-2+-...+-k for some choice of +'s and -'s.
0, 1, 3, 2, 3, 5, 3, 5, 4, 5, 4, 5, 7, 5, 7, 5, 7, 6, 7, 6, 7, 6, 7, 9, 7, 9, 7, 9, 7, 9, 8, 9, 8, 9, 8, 9, 8, 9, 11, 9, 11, 9, 11, 9, 11, 9, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 15, 13, 15, 13
Offset: 0
Examples
From _Seiichi Manyama_, Aug 18 2020: (Start) Illustration of initial terms: 0 = 0 (empty sum). 1 = 1. 2 = 1 - 2 + 3. 3 = 1 + 2. 4 = -1 + 2 + 3. 5 = 1 + 2 + 3 + 4 - 5. 6 = 1 + 2 + 3. 7 = 1 + 2 + 3 - 4 + 5. 8 = -1 + 2 + 3 + 4. 9 = 1 + 2 - 3 + 4 + 5. 10 = 1 + 2 + 3 + 4. ... (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..20000
- Rishi Advani, Formula for sequence on Mathematics Stack Exchange
Programs
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Maple
b:= proc(n,i) option remember; (n=0 and i=0) or n<=i*(i+1)/2 and (b(abs(n-i), i-1) or b(n+i, i-1)) end: a:= proc(n) local k; for k from 0 while not b(n,k) do od; k end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 19 2011 -
Mathematica
b[n_, i_] := b[n, i] = (n==0 && i==0) || Abs[n] <= i(i+1)/2 && (b[n-i, i-1] || b[n+i, i-1]); a[n_] := Module[{k}, For[k = 0, !b[n, k], k++]; k]; a /@ Range[0, 100] (* Jean-François Alcover, Nov 15 2020, after Alois P. Heinz *)
Formula
Conjecture when n is greater than 0. Choose k so that t(k)<=n
a(n) = a(-n) for all n in Z. - Seiichi Manyama, Aug 18 2020
Let k be the least integer such that t(k) >= n. If t(k) and n have the same parity then a(n) = k. Otherwise a(n) is equal to the least odd integer greater than k. - Rishi Advani, Jan 24 2021
Extensions
a(0)=3 prepended by Seiichi Manyama, Aug 17 2020
Edited and a(0)=0 from Alois P. Heinz, Aug 18 2020