A102785 - OEIS

%I #26 Mar 08 2024 12:11:16

%S 1,1,0,1,5,8,9,17,40,73,117,208,401,737,1296,2321,4261,7768,13977,

%T 25201,45752,83033,150165,271520,491809,891073,1613088,2919457,

%U 5285957,9572264,17330985,31375313,56805448

%N G.f.: (x-1)/(-2*x^2 + 3*x^3 + 2*x - 1).

%C Inverse binomial transform of A078017. Inversion of A052102.

%C Floretion Algebra Multiplication Program, FAMP Code: 4jbasekseq[ (+ 'ii' + 'jj' + 'ij' + 'ji' + e)*x) ] where x is defined as 1/4 times the sum of all 16 floretion basis vectors.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,3).

%F a(n+3) = 2a(n+2) - 2a(n+1) + 3a(n), a(0) = 1, a(1) = 1, a(2) = 0

%F a(n) = Sum(k=1..n, Sum(i=k..n, (Sum(j=0..k, binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j)))*binomial(n+k-i-1,k-1))), n > 0, a(0)=1. - _Vladimir Kruchinin_, May 05 2011

%o (Maxima)

%o a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j),j,0,k))*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* _Vladimir Kruchinin_, May 05 2011 */

%o (Maxima) makelist(coeff(taylor((x-1)/(-2*x^2+3*x^3+2*x-1), x, 0, n), x, n), n, 0, 32); /* _Bruno Berselli_, May 30 2011 */

%Y Cf. A078017, A052102, A077952.

%K nonn

%O 0,5

%A _Creighton Dement_, Feb 11 2005