A316775 -id:A316775 - cp's OEIS frontend

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

Tanya Khovanova, Oct 24 2018

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.

			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.

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).

Cf. A042948, A316775.

Filtered([1..200],k->Binomial(k,3) mod 6 = 0); # Muniru A Asiru, Oct 24 2018

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

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

select(n->binomial(n, 3)%6 == 0, vector(100, n, n)) \\ Colin Barker, Oct 24 2018

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

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)

A320920 a(n) is the smallest number m such that binomial(m,n) is nonzero and is divisible by n!.

Original entry on oeis.org

1, 4, 9, 33, 28, 165, 54, 1029, 40832, 31752, 28680, 2588680, 2162700, 12996613, 12341252, 4516741125, 500367376, 133207162881, 93770874890, 7043274506259, 40985291653137, 70766492123145, 321901427163142, 58731756479578128, 676814631896875010, 6820060161969750025

Offset: 1

Tanya Khovanova, Oct 24 2018

nonn

a(n) is such that a nontrivial n-symmetric permutation of [1..a(n)] might exist.

			The sequence of binomial coefficients C(n,3) starts as: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, and so on. The smallest nonzero number divisible by 3! is 84, which is C(9,3). Therefore a(3) = 9.

Bert Dobbelaere, Table of n, a(n) for n = 1..34
Bert Dobbelaere, Python program
Tanya Khovanova, 3-Symmetric Permutations

Cf. A042948, A316775, A320919.

a[n_] := Module[{w, m, bc}, {w, m} = {n!, n}; bc[i_] := Binomial[n-1, i] ~Mod~ w; While[True, bc[n] = (bc[n-1] + bc[n]) ~Mod~ w; If[bc[n] == 0, Return[m]]; For[i = n-1, i >= 0, i--, bc[i] = (bc[i-1] + bc[i]) ~Mod~ w]; m++]];
Array[a, 12] (* Jean-François Alcover, May 31 2019, after Chai Wah Wu *)

from sympy import factorial, binomial
def A320920(n):
    w, m = int(factorial(n)), n
    bc = [int(binomial(n-1,i)) % w for i in range(n+1)]
    while True:
        bc[n] = (bc[n-1]+bc[n]) % w
        if bc[n] == 0:
            return m
        for i in range(n-1,0,-1):
            bc[i] = (bc[i-1]+bc[i]) % w
        m += 1 # Chai Wah Wu, Oct 25 2018

a(14)-a(15) from Alois P. Heinz, Oct 24 2018
a(16)-a(17) from Chai Wah Wu, Oct 25 2018
a(18)-a(19) from Giovanni Resta, Oct 26 2018
a(20) from Giovanni Resta, Oct 27 2018
a(21) and beyond from Bert Dobbelaere, Feb 11 2020

cp's OEIS Frontend

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

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