A181635 Expansion of 36*x^2*(1+36*x^2-6*x) / ((36*x^2+6*x+1)*(1-6*x)^2).

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

Offset

1,2

Comments

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,0,216,-1296).

Formula

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

From Colin Barker, May 14 2016: (Start)

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

a(n) = 6^n*A008611(n-2). - R. J. Mathar, May 14 2016

Maple

A181635 := proc(n)
    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:
seq(A181635(n), n=1..40) ; # R. J. Mathar, May 13 2016

Prog

(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

Crossrefs

Cf. A181633, A181685.

Sequence in context: A262795 A309379 A022071 * A174673 A203277 A156645

Adjacent sequences: A181632 A181633 A181634 * A181636 A181637 A181638

Keyword

nonn,easy

Author

Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010

Extensions

Edited by R. J. Mathar, May 13 2016

Name changed by Colin Barker, May 14 2016

Status

approved