A218394 Numbers such that sum(i<=n) binomial(n,i)*binomial(2*n-2*i, n-i) is not divisible by 5.

1, 5, 7, 11, 25, 27, 31, 35, 37, 51, 55, 57, 61, 125, 127, 131, 135, 137, 151, 155, 157, 161, 175, 177, 181, 185, 187, 251, 255, 257, 261, 275, 277, 281, 285, 287, 301, 305, 307, 311, 625, 627, 631, 635, 637, 651, 655, 657, 661, 675, 677, 681, 685, 687, 751

Offset: 1

Author

Michel Marcus, Oct 28 2012

Keywords

• nonn

Comments

a(n) = A037453(2*n-1) (proved by Schur, see link).

Links

• W. Shur, The last digit of C(2*n,n) and Sigma C(n,i)*C(2*n-2*i,n-i), The Electronic Journal of Combinatorics, #R16, Volume 4, Issue 2 (1997).

Crossrefs

Cf. A037453.

Programs

PARI

lista(nb) = {for (n=1, nb, if (sum(i=1,n, binomial(n, i)*binomial(2*n-2*i,n-i)) % 5 != 0, print1(n, ", ")););}

PARI

a(n) = {2*n-1+2*sum(i=1,n, 5^(i-1)*floor((2*n-1)/3^i))}

Python

from gmpy2 import digits
def A218394(n): return int(digits((n<<1)-1,3),5) # Chai Wah Wu, Aug 10 2025

Formula

a(n) = 2*n - 1 + 2*sum{i=1,n} 5^(i-1)*floor((2*n-1)/3^i).