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COMMENT
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The sequence 0,1,-2,0,8,-16,... has G.f. 1/(1+2x-4x^2), a(n)=2^n*sin(2n*pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)x)/sqrt(3):0,1,-3,0,9,...
a(n+1) is the Hankel transform of A100192. - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
a(n+1) is the Trinomial transform of A010892: a(n+1) = Sum[Trinomial[n,k]A010892[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907); - Paul Barry (pbarry(AT)wit.ie), Sep 10 2007
a(n+1) is the Hankel transform of A100067. [From Paul Barry (pbarry(AT)wit.ie), Jun 16 2009]
1) a(n)=A131577*A128834 (A128834=0,1,1,0,-1,-1, discovered,1995,by recurrence a(n)=3a(n-1)-3a(n-2)+2a(n-3),see A024495. 2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244. 3) Principal sequence for recurrence a(n)=2a(n-1)-4a(n-2).This recurrence is valuable for every difference:first is A138230 (see A128018). 4) Sequence is identical to every 2n-th differences divided by (-3)^n. 5) a(3n)+a(3n+1)+a(3n+2)=3,-24,192,=3*A001018 signed.
6) For missing terms in a(n) see A013731=4*A001018. 7) Recurrences valuable for every differences,among many: a) a(n)=3a(n-1)-3a(n-2)+2a(n-3), principal sequence is A024495 (0,0, introduced Aug 1 2007); b) a(n)=a(n-1)+a(n-2)+2a(n-3) ,principal sequence is 0,0,1,1,2,5,9,=A077947; c) a(n)=2a(n-1)-a(n-2)+2a(n-3), principal sequence is 0,0,1,2,3,6,13,=0,0,A007910; d) a(n)=a(n-1)+2a(n-2)-a(n-3)-a(n-4) ,see A014217, principal sequence is 0,0,0,1,1,3,4,8,12,=0,0,A074331=0,0,0,A052952. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 04 2009]
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