A045771 Number of similar sublattices of index n^2 in root lattice D_4.
1, 1, 8, 1, 12, 8, 16, 1, 41, 12, 24, 8, 28, 16, 96, 1, 36, 41, 40, 12, 128, 24, 48, 8, 97, 28, 176, 16, 60, 96, 64, 1, 192, 36, 192, 41, 76, 40, 224, 12, 84, 128, 88, 24, 492, 48, 96, 8, 177, 97, 288, 28, 108, 176, 288, 16, 320, 60, 120, 96, 124, 64, 656, 1
Offset: 1
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..16384
- Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math., Vol. 51, No. 6 (1999), 1258-1276.
- Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.
- J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
- Index entries for sequences related to D_4 lattice.
- Index entries for sequences related to sublattices.
Programs
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Mathematica
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[1 <= p <= 2, 1, (e + 1) p^e + (2 (1 + (e p - e - 1)*p^e))/((p - 1)^2)]] &, 64] (* Michael De Vlieger, Mar 02 2018 *) -
PARI
fp(p, e) = if (p % 2, (e+1)*p^e + 2*(1-(e+1)*p^e+e*p^(e+1))/(p-1)^2, 1); a(n) = { my(f = factor(n)); prod(i=1, #f~, fp(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014
Formula
Multiplicative with a(2^p) = 1, a(p^e) = (e+1)*p^e + (2*(1+(e*p-e-1)*p^e))/((p-1)^2), p>2. - Christian G. Bower, May 21 2005
From Amiram Eldar, May 26 2025: (Start)
Dirichlet g.f.: (zeta(s-1)^2 * zeta(s)^2 / zeta(2*s)) * (1 - 1/2^(s-1))^2/(1 + 1/2^s).
Sum_{k=1..n} a(k) ~ (n^2/4)*(log(n) + 2*gamma - 1/2 + 11*log(2)/5 + 2*zeta'(2)/zeta(2) - 2*zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)
Extensions
More terms from Michel Marcus, Mar 03 2014