A320919 Positive integers k such that binomial(k, 3) is divisible by 6.

1, 2, 9, 10, 18, 20, 28, 29, 36, 37, 38, 45, 46, 54, 56, 64, 65, 72, 73, 74, 81, 82, 90, 92, 100, 101, 108, 109, 110, 117, 118, 126, 128, 136, 137, 144, 145, 146, 153, 154, 162, 164, 172, 173, 180, 181, 182, 189, 190, 198, 200

Offset

1,2

Comments

When taken modulo 36 this sequence is periodic with period is 9.

These are numbers for which a 3-symmetric permutation of size n might exist.

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Tanya Khovanova, 3-Symmetric Permutations

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Formula

From Colin Barker, Oct 24 2018: (Start)

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)).

a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.

(End)

Example

For k=8, binomial(8,3) = 56, which is not divisible by 6. Therefore 8 is not in the sequence.
For k=9, binomial(9,3) = 84, which is divisible by 6, so 9 is a term of the sequence.

Maple

select(k->modp(binomial(k, 3), 6)=0, [$1..200]); # Muniru A Asiru, Oct 24 2018

Mathematica

Transpose[Select[Table[{n, IntegerQ[Binomial[n, 3]/3!]}, {n, 200}], #[[2]] == True &]][[1]]

Prog

(PARI) select(n->binomial(n, 3)%6 == 0, vector(100, n, n)) \\ Colin Barker, Oct 24 2018
(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
(GAP) Filtered([1..200], k->Binomial(k, 3) mod 6 = 0); # Muniru A Asiru, Oct 24 2018

Crossrefs

Cf. A042948, A316775.

Sequence in context: A047468 A032929 A226832 * A046975 A304441 A005695

Adjacent sequences: A320916 A320917 A320918 * A320920 A320921 A320922

Keyword

nonn,easy

Author

Tanya Khovanova, Oct 24 2018

Status

approved