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A181635
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Expansion of 36*x^2*(1+36*x^2-6*x) / ((36*x^2+6*x+1)*(1-6*x)^2).
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1
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0, 36, 0, 1296, 15552, 46656, 559872, 5038848, 20155392, 181398528, 1451188224, 6530347008, 52242776064, 391820820480, 1880739938304, 14105549537280, 101559956668416, 507799783342080, 3656158440062976, 25593109080440832, 131621703842267136, 921351926895869952, 6317841784428822528, 33168669368251318272, 227442304239437611008
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OFFSET
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1,2
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COMMENTS
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The previous definition was: Let n=q+q' define any state with q quarks and q' antiquarks, q,q'>0 and q==q' (mod 3). Then a(n) = sum_{q,q'} 6^q*6^q' counts all states allowing q and q' to be any of the 6 quarks or 6 antiquarks. - Colin Barker, May 14 2016
In the following q and a represent any of the 6 quarks or antiquarks.
For n = 1, we have no combination.
For combinations of 2 we have: qa, [mesons and antimesons]; the number of all possible combinations will be 6^2 = 36.
For combinations of n= 7 we have: qqqqqaa, qqaaaaa; the number of all possible combinations will be 6^5*6^2 + 6^2*6^5 =559872.
For combinations of n=8 we have: qqqqaaaa, qqqqqqqa, qaaaaaaa; the number of all possible combinations will be 6^4*6^4 + 6^7*6^1 + 6^1*6^7 = 5038848
For combinations of n=9 we have: qqqqqqaaa, qqqaaaaaa; the number of all possible combinations will be 6^6*6^3 + 6^3*6^6 = 2*6^9 = 20155392.
For combinations of n=10 we have: qqqqqqqqaa, qqqqqaaaaa, qqaaaaaaaa; the number of all possible combinations will be 3*6^10 = 181398528.
If n is even, n=2k, then its pairs are: (k+3p,k-3p), where p is an integer such that both k+3p > 0 and k-3p > 0.
If n is odd, n=2k+1, then its pairs are(k+3p+2,k-3p-1), where p is an integer such that both k+3p+2 > 0 and k-3p-1 > 0.
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LINKS
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FORMULA
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a(n) = sum_{q,q'>0 , q+q'=n, q==q' (mod 3} 6^(q+q').
G.f.: 36*x^2*(1+36*x^2-6*x) / ( (36*x^2+6*x+1)*(1-6*x)^2 ). - Joerg Arndt, Mar 16 2013
a(n) = (-2^n*3^(1+n)+(-3-i*sqrt(3))*(-3-3*i*sqrt(3))^n-3*(-3+3*i*sqrt(3))^n+i*sqrt(3)*(-3+3*i*sqrt(3))^n+2^n*3^(1+n)*n)/9 where i is the imaginary unit. - Colin Barker, May 14 2016
a(n) = 6*a(n-1)+216*a(n-3)-1296*a(n-4) for n>4.
(End)
E.g.f.: 1 + ((18*x - 3)*exp(9*x) - 4*sqrt(3)*cos(Pi/6-3*sqrt(3)*x))*exp(-3*x)/9. - Ilya Gutkovskiy, May 14 2016
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MAPLE
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res := 0 ;
for q from 1 to n-1 do
a := n-q ;
if modp(a, 3) = modp(q, 3) then
res := res+6^n;
end if;
end do:
res;
end proc:
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PROG
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(PARI) a(n) = round((-2^n*3^(1+n)+(-3-I*sqrt(3))*(-3-3*I*sqrt(3))^n-3*(-3+3*I*sqrt(3))^n+I*sqrt(3)*(-3+3*I*sqrt(3))^n+2^n*3^(1+n)*n)/9) \\ Colin Barker, May 14 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010
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EXTENSIONS
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STATUS
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approved
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