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A207644 a(n) = 1 + (n-1) + (n-2)*[(n-3)/2] + (n-3)*[(n-4)/2]*[(n-5)/3] + (n-4)*[(n-5)/2]*[(n-6)/3]*[(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.

%I A207644 #24 Mar 06 2013 05:19:15
%S A207644 1,1,2,3,4,8,10,17,30,42,55,116,172,220,391,683,1024,1616,2050,3675,
%T A207644 6520,9504,12505,22421,35572,56918,85701,138110,202765,326231,503632,
%U A207644 860497,1376870,1927446,2818531,4892966,7784671,11432772,17287295,30423457,46453786,71810414
%N A207644 a(n) = 1 + (n-1) + (n-2)*[(n-3)/2] + (n-3)*[(n-4)/2]*[(n-5)/3] + (n-4)*[(n-5)/2]*[(n-6)/3]*[(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.
%C A207644 Radius of convergence of g.f. A(x) is near 0.637..., with A(1/phi) = 23.059250143... where phi = (sqrt(5)+1)/2.
%C A207644 Compare the definition of a(n) with the binomial sum:
%C A207644 Fibonacci(n) = 1 + (n-1) + (n-2)*((n-3)/2) + (n-3)*((n-4)/2)*((n-5)/3) + (n-4)*((n-5)/2)*((n-6)/3)*((n-7)/4) +...
%C A207644 with summation extending over the initial [n/2+1] products only.
%H A207644 Paul D. Hanna, <a href="/A207644/b207644.txt">Table of n, a(n) for n = 0..500</a>
%F A207644 a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-k-j+1) / j ).
%F A207644 Equals the antidiagonal sums of triangle A207645.
%e A207644 a(3) = 1 + 2 = 3;
%e A207644 a(4) = 1 + 3 + 2*[1/2] = 4;
%e A207644 a(5) = 1 + 4 + 3*[2/2] = 8;
%e A207644 a(6) = 1 + 5 + 4*[3/2] + 3*[2/2]*[1/3] = 10;
%e A207644 a(7) = 1 + 6 + 5*[4/2] + 4*[3/2]*[2/3] = 17;
%e A207644 a(8) = 1 + 7 + 6*[5/2] + 5*[4/2]*[3/3] + 4*[3/2]*[2/3]*[1/4] = 30;
%e A207644 a(9) = 1 + 8 + 7*[6/2] + 6*[5/2]*[4/3] + 5*[4/2]*[3/3]*[2/4] = 42; ...
%t A207644 a[n_] := 1 + Sum[ Product[ Floor[ (n-k-j+1)/j ], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 41}] (* _Jean-François Alcover_, Mar 06 2013 *)
%o A207644 (PARI) {a(n)=1+sum(k=1,n\2,prod(j=1,k,floor((n-k-j+1)/j)))}
%o A207644 for(n=0,60,print1(a(n),", "))
%Y A207644 Cf. A075885, A207643, A207645, A207646, A207647.
%K A207644 nonn,nice
%O A207644 0,3
%A A207644 _Paul D. Hanna_, Feb 19 2012