This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181635 #24 Feb 10 2025 05:49:50
%S A181635 0,36,0,1296,15552,46656,559872,5038848,20155392,181398528,1451188224,
%T A181635 6530347008,52242776064,391820820480,1880739938304,14105549537280,
%U A181635 101559956668416,507799783342080,3656158440062976,25593109080440832,131621703842267136,921351926895869952,6317841784428822528,33168669368251318272,227442304239437611008
%N A181635 Expansion of 36*x^2*(1+36*x^2-6*x) / ((36*x^2+6*x+1)*(1-6*x)^2).
%C A181635 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
%C A181635 In the following q and a represent any of the 6 quarks or antiquarks.
%C A181635 For n = 1, we have no combination.
%C A181635 For combinations of 2 we have: qa, [mesons and antimesons]; the number of all possible combinations will be 6^2 = 36.
%C A181635 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.
%C A181635 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
%C A181635 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.
%C A181635 For combinations of n=10 we have: qqqqqqqqaa, qqqqqaaaaa, qqaaaaaaaa; the number of all possible combinations will be 3*6^10 = 181398528.
%C A181635 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.
%C A181635 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.
%H A181635 Colin Barker, <a href="/A181635/b181635.txt">Table of n, a(n) for n = 1..1000</a>
%H A181635 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,0,216,-1296).
%F A181635 a(n) = Sum_{q>0, q'>0, q+q'=n, q==q' (mod 3)} 6^(q+q').
%F A181635 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
%F A181635 From _Colin Barker_, May 14 2016: (Start)
%F A181635 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
%F A181635 a(n) = 6*a(n-1)+216*a(n-3)-1296*a(n-4) for n>4.
%F A181635 (End)
%F A181635 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
%F A181635 a(n) = 6^n*A008611(n-2). - _R. J. Mathar_, May 14 2016
%p A181635 A181635 := proc(n)
%p A181635 res := 0 ;
%p A181635 for q from 1 to n-1 do
%p A181635 a := n-q ;
%p A181635 if modp(a,3) = modp(q,3) then
%p A181635 res := res+6^n;
%p A181635 end if;
%p A181635 end do:
%p A181635 res;
%p A181635 end proc:
%p A181635 seq(A181635(n),n=1..40) ; # _R. J. Mathar_, May 13 2016
%t A181635 LinearRecurrence[{6,0,216,-1296},{0,36,0,1296},40] (* _Harvey P. Dale_, Jul 18 2024 *)
%o A181635 (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
%Y A181635 Cf. A181633, A181685.
%K A181635 nonn,easy
%O A181635 1,2
%A A181635 Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010
%E A181635 Edited by _R. J. Mathar_, May 13 2016
%E A181635 Name changed by _Colin Barker_, May 14 2016