A031359 Bisection of A001615.

1, 4, 6, 8, 12, 12, 14, 24, 18, 20, 32, 24, 30, 36, 30, 32, 48, 48, 38, 56, 42, 44, 72, 48, 56, 72, 54, 72, 80, 60, 62, 96, 84, 68, 96, 72, 74, 120, 96, 80, 108, 84, 108, 120, 90, 112, 128, 120, 98, 144, 102, 104, 192, 108, 110, 152, 114, 144, 168, 144, 132, 168

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of coincidence site lattices of index 2n-1 in lattice Z^3.

Examples

			G.f. = x + 4*x^2 + 6*x^3 + 8*x^4 + 12*x^5 + 12*x^6 + 14*x^7 + 24*x^8 + ...
G.f. = q + 4*q^3 + 6*q^5 + 8*q^7 + 12*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + ...
		

References

• Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 9-44.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Michael Baake, Solution of coincidence problem in dimensions d<=4, arXiv:math/0605222 [math.MG], 2006.
• Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 715, the Dirichlet g.f. following Eq. (18).

Crossrefs

Cf. A000010, A000203, A001615, A008683, A159634.

Programs

Haskell

a031359 = a001615 . (subtract 1) . (* 2)
-- Reinhard Zumkeller, Jun 03 2013

Maple

A001615 := n -> mul((op(1,i)+1)*op(1,i)^(op(2,i)-1),i=op(2,numtheory[ifactors](n)));
A031359 := n -> A001615(2*n-1); # Peter Luschny, Oct 23 2010

Mathematica

a[n_] := (2n-1)*Sum[ MoebiusMu[d]^2/d, {d, Divisors[2n-1]}]; Table[a[n], {n, 1, 62}] (* Jean-François Alcover, Jan 18 2012, after Michael Somos *)
a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m Sum[ MoebiusMu[ d]^2 / d, {d, Divisors[m]}]]] (* Michael Somos, Nov 22 2013 *)

PARI

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; m * sumdiv( m, d, moebius(d)^2 / d))} /* Michael Somos, Nov 22 2013 */

PARI

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; direuler( p=2, m, (1 + X) / (1 - p*X))[ m])} /* Michael Somos, Nov 22 2013 */

PARI

{a(n) = my(A, p, e); if( n<1, 0, n = 2*n - 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, p^(e-1) * (p + 1)))))} /* Michael Somos, Nov 22 2013 */

Formula

a(n) = b(2*n - 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^(e-1) * (p+1) if p > 2. - Michael Somos, Nov 22 2013

Dirichlet series: Product (1+p^(-s))/(1-p^(1-s)); p != 2.

a(n) = A001615(2*n - 1).

From Peter Bala, Mar 19 2019: (Start)

a(n) = (2*n - 1)*Product_{p|(2*n-1), p prime} (1 + 1/p).

a(n) = Sum_{ d|(2*n-1) } mu(d)^2*(2*n-1)/d, where mu(n) = A008683(n) is the Möbius function.

a(n) = Sum_{ d^2|(2*n-1) } mu(d)*sigma((2*n-1)/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also

a(n) = Sum_{ d|(2*n-1) } 2^omega(d)*phi((2*n-1)/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.

O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^n*(1 + x^(2*n-1))/(1 - x^(2*n-1))^2.

Bisection of A159634. (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = 12/Pi^2 = 1.215854... . - Amiram Eldar, Nov 24 2022

Extensions

Better description from Vladeta Jovovic, Jan 25 2002

More terms from Sascha Kurz, Mar 24 2002