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

0,4

Comments

All terms are positive odd integers.

Links

Table of n, a(n) for n=0..42.

Math StackExchange, Minimizing a generalized partial sum of a geometric progression

Example

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.

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]]

Prog

(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

Crossrefs

Cf. A001047, A036561, A232111, A232112.

Sequence in context: A389775 A055584 A193861 * A193857 A146055 A286232

Adjacent sequences: A349541 A349542 A349543 * A349545 A349546 A349547

Keyword

nonn,hard

Author

Vladimir Reshetnikov, Nov 21 2021

Extensions

a(33)-a(35) from Chai Wah Wu, Nov 24 2021

a(36)-a(42) from Martin Ehrenstein, Nov 26 2021

Status

approved