a(1) = 0 as the single number 1 has no neighbor to multiply with.
a(2) = 2 as the numbers 1 and 2 can be placed next to each other in one way, and the pair product is 1*2 = 2.
a(3) = 9. The numbers 1,2,3 can be placed next to each other in six ways: 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. The combinations with the largest pair products are 1-3-2 and 2-3-1, the product sum being (1*3)+(3*2) = 9. This is the largest sum as 3 is placed so that it is in two pairs and thus its products contributes twice to the sum.
a(4) = 25. The best way to arrange the numbers is in a 2 X 2 square where 4 is adjacent to 2 and 3:
.
4 3
2 1
.
The sum is then (4*3)+(4*2)+(3*1)+(2*1) = 25.
a(5) = 54. The best way to arrange the numbers is for 2,3,4,5 to be in a 2 X 2 square with 5 adjacent to 4 and 3, and for 1 to be placed next to 5:
.
1 5 4
3 2
.
The product sum is then (5*4)+(5*3)+(4*2)+(3*2)+(5*1) = 54.
a(6) = 100. The best way to arrange the numbers is in a 2 X 3 block where 5 and 6 are in the middle of the long edge so that they both appear in three pairs. The 2 and 4 are placed either side of the 6 while the 1 and 3 are placed either side of 5:
.
2 6 4
1 5 3
.
The product sum is then (2*6)+(6*4)+(1*5)+(5*3)+(2*1)+(6*5)+(4*3) = 100.
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