The Fourth Positive Element in the Greedy Bh-Set

Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.3

The Fourth Positive Element in the Greedy B_h-Set

Melvyn B. Nathanson
Department of Mathematics
Lehman College (CUNY)
Bronx, NY 10468
USA
Kevin O'Bryant
Department of Mathematics
College of Staten Island (CUNY)
Staten Island, NY 10314
USA

Abstract:

For a positive integer $h$

$h$

, a -set is a set of integers $A$

$A$

such that every integer $n$

$n$

has at most one representation in the form $n = x_1 + \cdots + x_h$ , where $x_r \in A$ for all $r = 1,\ldots, h$ and $x_1 \le \cdots \le x_h$ . The greedy $B_h$

$B_h$

-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: $a_0(h) = 0$

$a_0(h) = 0$

, and $a_{k+1}(h)$ is least integer greater than $a_k(h)$

$a_k(h)$

for which $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h),a_{k+1}(h) \}$ is a $B_h$

$B_h$

-set. Nathanson gave the formulas $a_1(h) = 1$

$a_1(h) = 1$

,

$a_2(h) = h+1$

, and

$a_3(h) = h^2+h+1$

, valid for all $h$

$h$

. This paper proves that $a_4(h)$

$a_4(h)$

, the fourth positive term of the greedy $B_h$

$B_h$

-set, is $\left( h^3 + 3h^2 + 3h + 1\right) /2$ if $h$

$h$

is odd and $\left( h^3 + 2h^2 + 3h + 2\right) /2$ if $h$

$h$

is even. In particular, $a_4(h)$

$a_4(h)$

is not a polynomial, but is a quasipolynomial.

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(Concerned with sequences A000012 A001477 A002061 A005282 A020725 A051912 A347570 A365300 A365301 A365302 A365303 A365304 A365305 A365515 A369817 A369818 A369819.)

Received March 11 2024; revised version received August 28 2024. Published in Journal of Integer Sequences, August 28 2024.

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