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

3,1

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).

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

Mathematica

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

Prog

(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]

Crossrefs

Cf. A030126, A045652, A072842.

Sequence in context: A003332 A345793 A091571 * A151973 A102219 A227650

Adjacent sequences: A225228 A225229 A225230 * A225232 A225233 A225234

Keyword

nonn,easy

Author

Eric M. Schmidt, May 03 2013

Status

approved