A370754 a(n) = 2 + n^2*floor((n+3)/2) + floor(3*n/2).

5, 13, 33, 56, 109, 155, 257, 334, 501, 617, 865, 1028, 1373, 1591, 2049, 2330, 2917, 3269, 4001, 4432, 5325, 5843, 6913, 7526, 8789, 9505, 10977, 11804, 13501, 14447, 16385, 17458, 19653, 20861, 23329, 24680, 27437, 28939, 32001, 33662, 37045, 38873, 42593, 44596

Offset: 1

Author

Chai Wah Wu, Feb 29 2024

Keywords

• nonn
• easy

Comments

{1, 2, n+2, n^2+n+2, a(n)} is the lexicographically first set of 5 positive integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
• M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
• Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A347570, A365515, A369817.

Programs

Mathematica

A370754[n_] := 2 + n^2*Floor[(n+3)/2] + Floor[3*n/2]; Array[A370754, 50] (* or *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {5, 13, 33, 56, 109, 155, 257}, 50] (* Paolo Xausa, Mar 08 2024 *)

Formula

Column 5 of A347570.

a(n) = A369817(n) + 1.

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 7.

G.f.: x*(-2*x^6 + x^5 + 8*x^4 - x^3 + 5*x^2 + 8*x + 5)/((x - 1)*(x^2 - 1)^3).