A225231 Schur numbers S(3,n).

9, 16, 23, 37, 53, 71, 93, 119, 147, 177, 211, 249, 289, 331, 377, 427, 479, 533, 591, 653, 717, 783, 853, 927, 1003, 1081, 1163, 1249, 1337, 1427, 1521, 1619, 1719, 1821, 1927, 2037, 2149, 2263, 2381, 2503, 2627, 2753, 2883, 3017, 3153, 3291, 3433

Offset: 3

Author

Eric M. Schmidt, May 03 2013

Keywords

• nonn
• easy

Comments

a(n) is, by definition, the least positive m such that if {1,...,m} is written as a disjoint union of sets A and B, then either A contains 3 distinct numbers, one the sum of the other two, or B contains n distinct numbers, one the sum of the other n - 1.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 3..1000
• Tanbir Ahmed, Michael G. Eldredge, Jonathan J. Marler, and Hunter S. Snevily, Strict Schur Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A22, 2013.
• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A030126, A045652, A072842.

Programs

Mathematica

Join[{9,16},LinearRecurrence[{3,-4,4,-3,1},{23,37,53,71,93},45]] (* Ray Chandler, Feb 13 2014 *)

Sage

def A225231(n) : return 9 if n == 3 else 16 if n == 4 else (3*n^2 - 7*n)//2 + [3,3,4,4][n%4]

Formula

For n >= 5, a(n) = 3n^2/2 - 7n/2 + c, where c = 3 if n == 0,1 (mod 4), else c = 4.

G.f.: x^3*(3*x^6-7*x^5+3*x^4+4*x^3-11*x^2+11*x-9) / ((x-1)^3*(x^2+1)). - Colin Barker, May 16 2013