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A185173 Minimum number of distinct sums from consecutive terms in a circular permutation. 0
1, 3, 6, 9, 13, 17, 22, 28, 35, 41, 49, 57, 65, 73, 82, 93 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) <= n(n+1)/2, but this apparently is impossible for n >= 4. - N. J. A. Sloane, Mar 14 2012

LINKS

Table of n, a(n) for n=1..16.

EXAMPLE

a(4)=9 because the circular permutation 1243 has no way to get 5 as a sum of consecutive terms.

a(5)=13 because the circular permutation 12534 has no way to get 6 or 9 as a sum of consecutive terms.

From Bert Dobbelaere, Jun 24 2019: (Start)

Permutations achieving the minimum number of distinct sums:

a(1) = 1: {1}

a(2) = 3: {1, 2}

a(3) = 6: {1, 2, 3}

a(4) = 9: {1, 2, 4, 3}

a(5) = 13: {1, 2, 5, 3, 4}

a(6) = 17: {1, 3, 2, 4, 6, 5}

a(7) = 22: {1, 3, 2, 5, 7, 4, 6}

a(8) = 28: {1, 4, 3, 7, 6, 2, 8, 5}

a(9) = 35: {1, 3, 2, 4, 5, 8, 9, 6, 7}

a(10) = 41: {1, 3, 10, 9, 4, 6, 7, 2, 8, 5}

a(11) = 49: {1, 3, 5, 2, 8, 7, 4, 11, 10, 9, 6}

a(12) = 57: {1, 2, 6, 11, 10, 7, 4, 8, 9, 12, 5, 3}

a(13) = 65: {1, 2, 10, 12, 11, 13, 7, 5, 8, 3, 9, 4, 6}

a(14) = 73: {1, 4, 7, 2, 9, 14, 13, 11, 12, 10, 3, 6, 5, 8}

a(15) = 82: {1, 4, 5, 2, 3, 8, 14, 11, 12, 10, 15, 7, 6, 9, 13}

a(16) = 93: {1, 3, 8, 5, 10, 13, 4, 11, 16, 12, 14, 7, 6, 15, 2, 9}

(End)

PROG

(Sage)

# a(n)=distinct_sum_count(n)

def distinct_sum_count(n):

    min_sum_count=n*(n+1)/2

    for p in Permutations(n=n):

        if p[0]==1 and p[1]<p[-1]:  # remove cyclic shifts/reflections

            sums=[]

            for m in range(1, n+1):

                for i in range(n):

                    q=0

                    for j in range(m):

                        q+=p[(i+j)%n]

                    if not q in sums:

                        sums.append(q)

            if len(sums)<min_sum_count:

                min_sum_count=len(sums)

    return min_sum_count

CROSSREFS

Sequence in context: A004116 A004129 A219646 * A171662 A302292 A235269

Adjacent sequences:  A185170 A185171 A185172 * A185174 A185175 A185176

KEYWORD

nonn,nice,more

AUTHOR

Steve Butler, Mar 12 2012

EXTENSIONS

a(12)-a(16) from Bert Dobbelaere, Jun 24 2019

STATUS

approved

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Last modified October 17 16:42 EDT 2019. Contains 328120 sequences. (Running on oeis4.)