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A369817
The fifth term of the greedy B_n set of natural numbers.
4
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
1,1
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.
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.
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
EXAMPLE
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}.
MATHEMATICA
a[n_] := Floor[(n + 3)/2] n^2 + Floor[(3 n + 2)/2]
PROG
(Python)
def A369817(n): return (n+3>>1)*n**2+(3*n+2>>1) # Chai Wah Wu, Feb 28 2024
CROSSREFS
Column 5 of A365515.
Sequence in context: A324971 A273387 A328240 * A161217 A152527 A005104
KEYWORD
nonn,easy
AUTHOR
Kevin O'Bryant, Feb 02 2024
STATUS
approved