This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320919 #25 Oct 25 2018 05:42:43
%S A320919 1,2,9,10,18,20,28,29,36,37,38,45,46,54,56,64,65,72,73,74,81,82,90,92,
%T A320919 100,101,108,109,110,117,118,126,128,136,137,144,145,146,153,154,162,
%U A320919 164,172,173,180,181,182,189,190,198,200
%N A320919 Positive integers k such that binomial(k, 3) is divisible by 6.
%C A320919 When taken modulo 36 this sequence is periodic with period is 9.
%C A320919 These are numbers for which a 3-symmetric permutation of size n might exist.
%C A320919 Numbers for which a 2-symmetric permutations might exist are numbers n such that n choose 2 is even. Equivalently, these are numbers that have remainder 0 or 1 modulo 4. This is sequence A042948.
%H A320919 Colin Barker, <a href="/A320919/b320919.txt">Table of n, a(n) for n = 1..1000</a>
%H A320919 Tanya Khovanova, <a href="https://blog.tanyakhovanova.com/2018/10/3-symmetric-permutations/#comment-12716">3-Symmetric Permutations</a>
%H A320919 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,1,-1).
%F A320919 From _Colin Barker_, Oct 24 2018: (Start)
%F A320919 G.f.: x*(1 + x + 7*x^2 + x^3 + 8*x^4 + 2*x^5 + 8*x^6 + x^7 + 7*x^8) / ((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).
%F A320919 a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.
%F A320919 (End)
%e A320919 For k=8, binomial(8,3) = 56, which is not divisible by 6. Therefore 8 is not in the sequence.
%e A320919 For k=9, binomial(9,3) = 84, which is divisible by 6, so 9 is a term of the sequence.
%p A320919 select(k->modp(binomial(k,3),6)=0,[$1..200]); # _Muniru A Asiru_, Oct 24 2018
%t A320919 Transpose[Select[Table[{n, IntegerQ[Binomial[n, 3]/3!]}, {n, 200}], #[[2]] == True &]][[1]]
%o A320919 (PARI) select(n->binomial(n, 3)%6 == 0, vector(100, n, n)) \\ _Colin Barker_, Oct 24 2018
%o A320919 (PARI) Vec(x*(1 + x + 7*x^2 + x^3 + 8*x^4 + 2*x^5 + 8*x^6 + x^7 + 7*x^8) / ((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)) + O(x^40)) \\ _Colin Barker_, Oct 24 2018
%o A320919 (GAP) Filtered([1..200],k->Binomial(k,3) mod 6 = 0); # _Muniru A Asiru_, Oct 24 2018
%Y A320919 Cf. A042948, A316775.
%K A320919 nonn,easy
%O A320919 1,2
%A A320919 _Tanya Khovanova_, Oct 24 2018