A326419 a(n) is the number of distinct Horadam sequences of period n.

1, 1, 3, 5, 10, 11, 21, 22, 33, 34, 55, 46, 78, 69, 92, 92, 136, 105, 171, 140, 186, 175, 253, 188, 290, 246, 315, 282, 406, 284, 465, 376, 470, 424, 564, 426, 666, 531, 660, 568, 820, 570, 903, 710, 852, 781, 1081, 760, 1155, 890, 1136, 996, 1378, 963, 1420, 1140

Offset: 1

Author

Michel Marcus, Sep 30 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ovidiu D. Bagdasar and Peter J. Larcombe, On the Number of Complex Horadam Sequences with a Fixed Period, Fib. Q. 51(4), 2013, 339-347.

Crossrefs

Cf. A102309.

Programs

Maple

N:= 200: # for a(1)..a(N)
V:= Vector(N,n -> numtheory:-phi(n)*(numtheory:-phi(n)-1)/2):
for k1 from 1 to N do
  p1:= numtheory:-phi(k1);
  for k2 from k1+1 to N do
     n:= ilcm(k1,k2);
     if n <= N then V[n]:= V[n] + p1*numtheory:-phi(k2) fi;
  od:
od:
V[1]:= 1:
convert(V,list); # Robert Israel, Dec 06 2020

PARI

a(n) = if (n==1, 1, eulerphi(n)*(eulerphi(n)-1)/2 + sum(k2=1, n, sum(k1=1, k2-1, if (lcm(k1, k2)==n, eulerphi(k1)*eulerphi(k2)))));

Python

from math import comb
from sympy import mobius, divisors
def A326419(n): return sum(mobius(d)*comb(n//d,2) for d in divisors(n,generator=True)) if n>1 else 1 # Chai Wah Wu, May 09 2025

Formula

a(n) = Sum_{k1= 2. See link.