A005887 Theta series of f.c.c. lattice with respect to octahedral hole.
6, 8, 24, 0, 30, 24, 24, 0, 48, 24, 48, 0, 30, 32, 72, 0, 48, 48, 24, 0, 96, 24, 72, 0, 54, 48, 72, 0, 48, 72, 72, 0, 96, 24, 96, 0, 48, 56, 96, 0, 102, 72, 48, 0, 144, 48, 48, 0, 48, 72, 168, 0, 96, 72, 72, 0, 96, 48, 120, 0, 78, 48, 144, 0, 144, 120, 48, 0, 96, 72, 96, 0, 96, 56, 168
Offset: 0
Keywords
Examples
6 + 8*x + 24*x^2 + 30*x^4 + 24*x^5 + 24*x^6 + 48*x^8 + 24*x^9 + 48*x^ 10 + ... 6*q + 8*q^3 + 24*q^5 + 30*q^9 + 24*q^11 + 24*q^13 + 48*q^17 + 24*q^19 + ...
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..9999
- G. Nebe and N. J. A. Sloane, Home page for this lattice
- N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- Index entries for sequences related to f.c.c. lattice
Crossrefs
Cf. A005875.
Programs
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Maple
maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd): th4:=series(subs(q=-q,th3),q,maxd): t1:=series((th3^3-th4^3)/(2*q),q,maxd): t1:=series(subs(q=sqrt(q),t1),q,floor(maxd/2)): t2:=seriestolist(t1): for n from 1 to nops(t2) do lprint(n-1, t2[n]); od:
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Mathematica
s = (EllipticTheta[3, 0, q]^3 - EllipticTheta[3, 0, -q]^3)/(2q) + O[q]^200; CoefficientList[s, q^2] (* Jean-François Alcover, Sep 19 2016 *)
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PARI
{a(n) = if( n<0, 0, n = 2*n + 1; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n))^3, n))} /* Michael Somos, Aug 17 2009 */
Formula
Expansion of q^(-1) * (phi^3(q) - phi^3(-q)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Aug 17 2009
A005875(2*n + 1) = a(n). - Michael Somos, Aug 17 2009
Comments