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

1,3

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.

Formula

a(n) = Sum_{k1<k2, lcm(k1, k2)=n} phi(k1)*phi(k2) + phi(n)*(phi(n)-1)/2, for n>= 2. See link.

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

Prog

(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)))));

Crossrefs

Cf. A102309.

Sequence in context: A101130 A191513 A254540 * A102309 A067230 A321793

Adjacent sequences: A326416 A326417 A326418 * A326420 A326421 A326422

Keyword

nonn

Author

Michel Marcus, Sep 30 2019

Status

approved