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A245738 Number of compositions of n into parts 1 and 2 with both parts present.

%I A245738 #31 May 26 2024 12:06:42
%S A245738 2,3,7,11,20,32,54,87,143,231,376,608,986,1595,2583,4179,6764,10944,
%T A245738 17710,28655,46367,75023,121392,196416,317810,514227,832039,1346267,
%U A245738 2178308,3524576,5702886,9227463,14930351,24157815,39088168,63245984,102334154,165580139,267914295,433494435,701408732,1134903168,1836311902
%N A245738 Number of compositions of n into parts 1 and 2 with both parts present.
%H A245738 Colin Barker, <a href="/A245738/b245738.txt">Table of n, a(n) for n = 3..1000</a>
%H A245738 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F A245738 G.f.: 1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2).
%F A245738 a(n) = A052952(n-4)+2*A052952(n-3). - _R. J. Mathar_, Aug 05 2014
%F A245738 From _Colin Barker_, Jul 13 2017: (Start)
%F A245738 a(n) = (-20 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n even.
%F A245738 a(n) = (-10 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n odd.
%F A245738 a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>6. (End)
%F A245738 a(n) = Sum_{i=1..floor((n-1)/2)} C(n-i,i). - _Wesley Ivan Hurt_, Sep 19 2017
%F A245738 a(n) = A000045(n+1) - A000034(n+1). - _J. M. Bergot_ and _Robert Israel_, Oct 11 2021
%e A245738 a(9) = 54. The tuples are (22221) = 5!/4! = 5, (222111) = 6!/3!/3! = 20, (2211111) = 7!/5!/2! = 21, (21111111) = 8!/7! = 8.
%t A245738 LinearRecurrence[{1,2,-1,-1},{2,3,7,11},50] (* _Harvey P. Dale_, Dec 20 2014 *)
%o A245738 (PARI) Vec(1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2)+O(x^66)) \\ _Joerg Arndt_, Aug 04 2014
%Y A245738 Cf. A245332, A245492, A245487, A245527.
%Y A245738 Column k=2 of A373118.
%K A245738 nonn,easy
%O A245738 3,1
%A A245738 _David Neil McGrath_, Jul 31 2014