A078714 - OEIS

%I #21 Mar 25 2015 01:22:28

%S 1,4,16,34,142,79,1276,289,394,709,103336,1024,930022,6379,3544,2599,

%T 75331762,5119,677985856,9214,31894,516679,54916854316,12994,88594,

%U 4650109,30319,82924,40034386796182,46069,360309481165636,33784,2583394,376658809,797344

%N a(n) = smallest number m which can be obtained in n ways by subtracting twice a triangular number from a perfect square.

%C 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.

%C a(23)-a(50) are ?, 12994, 88594, 4650109, 30319, 82924, ?, 46069, ?, 33784, 2583394, 376658809, 797344, 78829, ?, ?, 23250544, 148129, ?, 414619, ?, 6716824, 272869, ?, ?, 168919, 19933594, 1151719. - _Robert G. Wilson v_, Dec 24 2002

%H Ray Chandler, <a href="/A078714/b078714.txt">Table of n, a(n) for n = 1..2098</a> (a(2099) exceeds 1000 digits).

%H T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

%F 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.

%F a(n) = (A204086(n) + 1)/4. - _Ray Chandler_, Jan 10 2012

%F For odd prime p, a(p) = (3^(p-1)*7 + 1)/4.

%e 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.

%t 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]] &)]

%Y Cf. A078703, A038547, A018782, A204046, A204086.

%K nonn

%O 1,2

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

%E Missing terms noted in Comments and b-file from _Ray Chandler_, Jan 10 2012