A188430 a(n) is the maximum of the largest elements of all n-full sets, or 0 if no such set exists.

1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38

Offset: 1

Author

Madjid Mirzavaziri, Mar 31 2011

Keywords

• nonn
• easy

Comments

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. The number a(n) denotes the maximum of the set {max A: (sum A)=[n]}, or 0 if there is no n-full set.

Links

• Mohammad Saleh Dinparvar, Python program (github)
• L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A188429, A188431.

Cf. A008619.

Programs

Haskell

a188430 n = a188430_list !! (n-1)
a188430_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9] ++
               (drop 19 a008619_list)
-- Reinhard Zumkeller, Aug 06 2015

Mathematica

LinearRecurrence[{1,1,-1},{1,0,2,0,0,3,4,0,0,4,5,6,7,7,8,6,7,8,9,10,11,11},80] (* Harvey P. Dale, Jul 24 2021 *)

PARI

Vec(x*(1 - x + x^2 - x^3 - 2*x^4 + 5*x^5 + x^6 - 7*x^7 - x^8 + 8*x^9 + x^10 - 3*x^11 - x^13 - 2*x^15 + 3*x^17 - x^21) / ((1 - x)^2*(1 + x)) + O(x^80)) \\ Colin Barker, May 11 2020

Formula

a(n) = ceiling(n/2) for n >= 20.

From Colin Barker, May 11 2020: (Start)

G.f.: x*(1 - x + x^2 - x^3 - 2*x^4 + 5*x^5 + x^6 - 7*x^7 - x^8 + 8*x^9 + x^10 - 3*x^11 - x^13 - 2*x^15 + 3*x^17 - x^21) / ((1 - x)^2*(1 + x)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>22.

(End)