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 A181635 Expansion of 36*x^2*(1+36*x^2-6*x) / ((36*x^2+6*x+1)*(1-6*x)^2). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 15 00:37 EDT 2024. Contains 374323 sequences. (Running on oeis4.)