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

1,2

Comments

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

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).

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A320126 A234520 A275789 * A274790 A179852 A281020

Adjacent sequences: A031356 A031357 A031358 * A031360 A031361 A031362

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Better description from Vladeta Jovovic, Jan 25 2002

More terms from Sascha Kurz, Mar 24 2002

Status

approved