A106464 Antidiagonal sums of number triangle A003989.

1, 1, 2, 3, 3, 4, 6, 6, 5, 11, 6, 9, 15, 12, 8, 18, 9, 21, 22, 15, 11, 32, 20, 18, 27, 31, 14, 45, 15, 32, 36, 24, 41, 57, 18, 27, 43, 60, 20, 66, 21, 51, 72, 33, 23, 84, 42, 60, 57, 61, 26, 81, 67, 88, 64, 42, 29, 135, 30, 45, 105

Offset: 0

Author

Paul Barry, May 03 2005

Keywords

• easy
• nonn

Comments

Consider the triangle T(n, k) = A003989(n, k) = gcd(n-k+1, k), n >= 1, k = 1..n. Then a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1), for n >= 0. - R. J. Mathar, May 11 2018 [adjusted to the definition of A003989. - Wolfdieter Lang, May 12 2018]

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003989, A106466.

Programs

GAP

Flat(List([0..70],n->Sum([0..Int(n/2)],k->Gcd(n-2*k+1,k+1)))); # Muniru A Asiru, May 15 2018

Maple

f:= n -> add(igcd(n-2*k+1,k+1),k=0..n/2):
map(f, [$0..100]); # Robert Israel, May 11 2018

Mathematica

Array[Sum[GCD[# - 2 k + 1, k + 1], {k, 0, Floor[#/2]}] &, 61, 0] (* Michael De Vlieger, May 14 2018 *)

PARI

a(n) = sum(k=0, n\2, gcd(n-2*k+1, k+1)); \\ Michel Marcus, May 11 2018

Formula

a(n) = Sum_{k=0..floor(n/2)} gcd(n-2*k+1, k+1). [corrected by R. J. Mathar, May 11 2018]

Extensions

Name corrected by R. J. Mathar, May 11 2018