Question

Suppose there is number s=12 , now i want to make sequence with the element a1+a2+…..+an=12.

The criteria is as follows-

1. n must be minimum.
2. a1 and an must be 1;
3. ai can differs a(i-1) by only 1,0 and -1.

for s=12 the result is 6.

So how to find the minimum value of n.

Answer 1

Algorithm for finding n from given s:

1.Find q = FLOOR( SQRT(s-1) )

2.Find r = q^2 + q

3.If s <= r then n = 2q, else n = 2q + 1

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Example: s = 12

1. q = FLOOR( SQRT(12-1) ) = FLOOR(SQRT(11) = 3
2. r = 3^2 + 3 = 12
3. 12 <= 12, therefore n = 2*3 = 6

Example: s = 160

1. q = FLOOR( SQRT(160-1) ) = FLOOR(SQRT(159) = 12
2. r = 12^2 + 12 = 156
3. 159 > 156, therefore n = 2*12 + 1 = 25

and the 25-numbers sequence for

159:    1,2,3,4,5,6,7,8,9,10,10,10,9,10,10,10,9,8,7,6,5,4,3,2,1