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A031361
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Number of coincidence site lattices of index n in lattice Z^4.
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0
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1, 2, 16, 0, 36, 32, 64, 0, 168, 72, 144, 0, 196, 128, 576, 0, 324, 336, 400, 0, 1024, 288, 576, 0, 960, 392, 1584, 0, 900, 1152, 1024, 0, 2304, 648, 2304, 0, 1444, 800, 3136, 0, 1764, 2048, 1936, 0, 6048, 1152, 2304, 0, 3248, 1920, 5184, 0, 2916, 3168, 5184, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Dirichlet product of 1+2/2^s with sum_{n>=1} A031360(n)/(2n-1)^s. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 16 2010]
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REFERENCES
| M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
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FORMULA
| Dirichlet series: (1+2^(1-s))* Product (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))); p != 2.
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MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 16 2010: (Start)
read("transforms") : maxOrd := 120 :
ZetaNum := proc(p, nmax, f) local n ; L := [1, seq(0, n=2..p-1), f, seq(0, n=p+1..nmax)] ; end proc:
Zeta := proc(p, nmax, f) local L, e; L := [1, seq(0, n=2..nmax)] ; for e from 1 do if p^e > nmax then break; else L := subsop(p^e=f^e, L) ; end if; end do: L ; end proc:
Zetap := ZetaNum(2, maxOrd, 2): for e from 3 to maxOrd do if isprime(e) then ZetaNum(e, maxOrd, 1) ; Zetap := DIRICHLET(Zetap, %) ; ZetaNum(e, maxOrd, e) ; Zetap := DIRICHLET(Zetap, %) ; Zeta(e, maxOrd, e) ; Zetap := DIRICHLET(Zetap, %) ; Zeta(e, maxOrd, e^2) ; Zetap := DIRICHLET(Zetap, %) ; end if; end do: Zetap ; (End)
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CROSSREFS
| Sequence in context: A056072 A175982 A136666 * A155955 A145684 A197226
Adjacent sequences: A031358 A031359 A031360 * A031362 A031363 A031364
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 16 2010
Typo in formula (exclamation mark for 1) corrected - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2010
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