A369817 The fifth term of the greedy B_n set of natural numbers.

4, 12, 32, 55, 108, 154, 256, 333, 500, 616, 864, 1027, 1372, 1590, 2048, 2329, 2916, 3268, 4000, 4431, 5324, 5842, 6912, 7525, 8788, 9504, 10976, 11803, 13500, 14446, 16384, 17457, 19652, 20860, 23328, 24679, 27436, 28938, 32000, 33661, 37044, 38872, 42592, 44595, 48668, 50854, 55296, 57673, 62500, 65076

Offset: 1

Author

Kevin O'Bryant, Feb 02 2024

Keywords

• nonn
• easy

Comments

{0, 1, n+1, n^2+n+1, a(n)} is the lexicographically first set of 5 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.

Examples

			a(2) = 12, as all 15 nonincreasing sums from {0,1,3,7,12}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 12+12, are distinct, and all other 5-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12}.
		

Links

• 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

Column 5 of A365515.

Programs

Mathematica

a[n_] := Floor[(n + 3)/2] n^2 + Floor[(3 n + 2)/2]

Python

def A369817(n): return (n+3>>1)*n**2+(3*n+2>>1) # Chai Wah Wu, Feb 28 2024

Formula

a(n) = floor((n + 3)/2) * n^2 + floor((3*n + 2)/2), proved in arXiv:2311.14021.

G.f.: x*(-x^6 + x^5 + 5*x^4 - x^3 + 8*x^2 + 8*x + 4)/((x - 1)*(x^2 - 1)^3). - Chai Wah Wu, Feb 28 2024

E.g.f.: ((2 + 7*x + 5*x^2 + x^3)*cosh(x) + (1 + 6*x + 6*x^2 + x^3)*sinh(x) - 2)/2. - Stefano Spezia, Mar 09 2024