A349544 Smallest possible value of |Sum_{k=0..n} (+-) 2^k * 3^(n-k)|, where each (+-) can be either plus or minus sign, independently for each term in the sum.
1, 1, 1, 5, 1, 19, 7, 5, 65, 61, 73, 227, 257, 5, 439, 1253, 2425, 2035, 833, 2677, 10591, 6509, 32071, 41173, 77263, 114323, 18145, 129685, 321151, 15757, 645449, 113957, 50735, 477653, 24295, 5089013, 3743881, 4809115, 12209455, 8216179, 32894927, 80299843, 45673913
Offset: 0
Examples
For n = 3, there are 2^3 = 8 possible choices of signs: 3^3 + 2*3^2 + 2^2*3 + 2^3 = 65, 3^3 + 2*3^2 + 2^2*3 - 2^3 = 49, 3^3 + 2*3^2 - 2^2*3 + 2^3 = 41, 3^3 + 2*3^2 - 2^2*3 - 2^3 = 25, 3^3 - 2*3^2 + 2^2*3 + 2^3 = 29, 3^3 - 2*3^2 + 2^2*3 - 2^3 = 13, 3^3 - 2*3^2 - 2^2*3 + 2^3 = 5, and 3^3 - 2*3^2 - 2^2*3 - 2^3 = -11. The smallest absolute value is 5, so a(3) = 5.
Links
- Math StackExchange, Minimizing a generalized partial sum of a geometric progression
Programs
-
Maple
b:= proc(k, n) option remember; `if`(k<0, {0}, map(x-> (t-> [x+t, abs(x-t)][])(2^(n-k)*3^k), b(k-1, n))) end: a:= n-> min(b(n$2)): seq(a(n), n=0..18); # Alois P. Heinz, Nov 21 2021 -
Mathematica
Min@*Abs/@FoldList[Join[3 #1 + 2^#2, 3 #1 - 2^#2] &, {1}, Range[25]] -
Python
def f(k,n): if k == 0 and n == 0: return (x for x in (1,)) if k < n: return (y*3 for y in f(k,n-1)) return (abs(x+y) for x in f(k-1,n) for y in (2**n,-2**n)) def A349544(n): return min(f(n,n)) # Chai Wah Wu, Nov 24 2021
Extensions
a(33)-a(35) from Chai Wah Wu, Nov 24 2021
a(36)-a(42) from Martin Ehrenstein, Nov 26 2021
Comments