1,2

The minimum number m (denoted by LSDT(n)) which can be represented in n different ways as a symmetric unimodal consecutive integer sequence (e.g. 6+7+8+7+6) that sums to the integer m. More precisely, n is the number of ways to arrange m objects into symmetrically-placed, congruent isosceles trapezoids adjoined at overlapping largest bases, and m is the minimum number of objects that allows this number of arrangements.

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

LSDT(k)={min n: SDT(n)=k}, where SDT(n)=((r1+1)*(r2+1)*...)/2, and ((p1^r1)*(p2^r2)*...) is the factorization of 4n-1 into (odd) primes.

Let SDT(n) = the number, k, of symmetric double trapezoidal arrangements of n objects, then SDT(34) = 4, since we have 34 or 11+12+11 or 6+7+8+7+6 or 2+3+4+5+6+5+4+3+2. For SDT(n) = 4, we have n = 34 or 49 or 58 or 64 ..., so that the least value of SDT(n)=4 is LSDT(4) = 34. Also 4*34 - 1 = 135 = (3^3)*(5^1) so that r1=3 and r2=1 (p1=3 and p2=5), resulting in SDT(34) = (3+1)*(1+1)/2 = 4, and 34 is the least value of n which satisfies 4*n-1 so that one half the number of odd divisors equals 4.

The following function determines the number of ways, SDT(n), of arranging n identical objects into symmetric double trapezoidal arrangements: SDT[n_] := (Times @@ Cases[FactorInteger[4 n - 1], {p_, r_} -> r + 1])/2 The program below computes the first few terms of the sequence LSDT(k)=min{n:SDT(n)=k}. The output is in the form {{1, LSDT(1)}, {2, LSDT(2)}, {3, LSDT(3)}, ...}: Union[Sort[{SDT[ # ], #} & /@ Range[1, 100000]], SameTest -> (#1[[1]] == #2[[1]] &)]

Cf. A078703, A038547.

Adjacent sequences: A078711 A078712 A078713 this_sequence A078715 A078716 A078717

Sequence in context: A034713 A101653 A043100 this_sequence A104125 A014727 A044065

nonn

R. L. Coffman, K. W. McLaughlin and R. J. Dawson (robert.l.coffman(AT)uwrf.edu), Dec 19 2002

Sequence continues ?, 12994, 88594, 4650109, 30319, 82924, ?, 46069, ?, 33784, 2583394, 376658809, 797344, 78829, ?, ?, 23250544, 148129, ?, 414619, ?, 6716824, 272869, ?, ?, 168919, 19933594, 1151719 - Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 24 2002: