%I #72 Aug 13 2022 20:48:21
%S 0,1,5,6,7,11,12,13,17,18,19,23,24,25,29,30,31,35,36,37,41,42,43,47,
%T 48,49,53,54,55,59,60,61,65,66,67,71,72,73,77,78,79,83,84,85,89,90,91,
%U 95,96,97,101,102,103,107,108,109,113,114,115,119,120,121,125
%N Numbers that are congruent to {0, 1, 5} mod 6.
%C a(n+3) is the Hankel transform of A005773(n+3). - _Paul Barry_, Nov 04 2008
%C The numbers m == 0, 2 or 10 mod 12 (the doubles of this sequence, that is, 10, 12, 14, 22, 24, 26, 34, ...) have the property that exactly 1/4 of the 3-part partitions of m form the sides of a triangle. See Mathematics Stack Exchange, 2013, link. - _Ed Pegg Jr_, Dec 19 2013
%C Row sum of a triangle where two rules build the triangle. #1 Start with the value "1" at the top of the triangle. #2 Require every "triple" to contain the values 1,2,3 (see link below). Compare with A136289 that has "3" at the apex. - _Craig Knecht_, Oct 17 2015
%C Nonnegative m such that floor(k*m^2/6) = k*floor(m^2/6), where k = 2, 3, 4 or 5. - _Bruno Berselli_, Dec 03 2015
%H Craig Knecht, <a href="/A047266/a047266.jpg">Triangular row sum of A204259</a>.
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/613202/prove-exactly-a-quarter-of-3-part-partitions-of-numbers-2-equal-to-0-2-10-mo">Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle</a>, 2013.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: x^2*(1+4*x+x^2) / ((1+x+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011
%F a(n) = 2*(n-1) + A057078(n). - _Robert Israel_, Dec 01 2014
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - _Wesley Ivan Hurt_, Nov 09 2015
%F From _Wesley Ivan Hurt_, Jun 13 2016: (Start)
%F a(n) = 2*n-2+cos(2*n*Pi/3)+sin(2*n*Pi/3)/sqrt(3).
%F a(3k) = 6k-1, a(3k-1) = 6k-5, a(3k-2) = 6k-6. (End)
%F Sum_{n>=2} (-1)^n/a(n) = log(2)/6 + log(2 + sqrt(3))/sqrt(3). - _Amiram Eldar_, Dec 14 2021
%p seq(seq(6*s+j, j=[0,1,5]), s=0..100); # _Robert Israel_, Dec 01 2014
%t Select[Range[0, 200], Mod[#, 6] == 0 || Mod[#, 6] == 1 || Mod[#, 6] == 5 &] (* _Vladimir Joseph Stephan Orlovsky_, Jul 07 2011 *)
%o (PARI) concat(0, Vec(x^2*(1+4*x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^100))) \\ _Altug Alkan_, Oct 17 2015
%o (Magma) [n : n in [0..150] | n mod 6 in [0, 1, 5]]; // _Wesley Ivan Hurt_, Jun 13 2016
%Y Cf. A005773, A047240, A047242, A057078, A136289, A204259.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_