A342713 Partition the integers from 1 to n into three groups with consecutive numbers, then a(n) is the maximum value of the sum of the numbers in the second group multiplied by the minimum of the sum of the numbers in the first and third groups.

2, 9, 21, 54, 90, 144, 234, 350, 504, 714, 950, 1350, 1764, 2156, 2772, 3500, 4374, 5390, 6380, 7812, 9504, 10890, 12740, 14850, 17442, 20475, 23100, 26334, 30444, 34320, 38709, 43146, 48510, 55250, 61047, 66780, 74925, 83600, 92169, 100485, 109350, 121512, 133331, 144000, 156195, 171171

Offset

3,1

Comments

The maximum product is obtained by making the sum of the numbers in the first and third groups as close as possible to each other and to half the sum of the numbers in the second group.

Geometrically the value of a(n) corresponds to the maximum area surrounded by three sides in a square-bottom 'U' shaped figure where the sides are drawn with single steps of incrementing length from 1 to n.

Links

Table of n, a(n) for n=3..48.

Example

a(3) = 2 as the only partition is {1},{2},{3}. The minimum sum of the first and third group is 1, thus a(3) = 2*1 = 2.
a(5) = 21 as the three group partition {1,2},{3,4},{5} has a minimum sum of the first and third groups of 1+2 = 3, thus a(5) = 3*(3+4) = 3*7 = 21.
a(12) = 714 as the three group partition {1,2,3,4,5,6},{7,8,9,10},{11,12} has a minimum sum of the first and third groups of 1+2+3+4+5+6 = 21, thus a(12) = 21*(7+8+9+10) = 21*34 = 714.

Crossrefs

Cf. A000217, A006011, A007294, A008443, A286430.

Sequence in context: A192971 A024850 A237044 * A248116 A259702 A284923

Adjacent sequences: A342710 A342711 A342712 * A342714 A342715 A342716

Keyword

nonn

Author

Scott R. Shannon, Mar 20 2021

Status

approved