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%I A079289 #44 Sep 08 2022 08:45:08
%S A079289 1,1,2,3,6,9,17,26,47,73,128,201,345,546,923,1469,2456,3925,6509,
%T A079289 10434,17199,27633,45344,72977,119345,192322,313715,506037,823848,
%U A079289 1329885,2161925,3491810,5670119,9161929,14864816,24026745,38957097,62983842
%N A079289 For even n, a(n) = a(n-2) + a(n-1) + 2^(n/2-2), n>2. For odd n, a(n) = a(n-2) + a(n-1).
%C A079289 Generalized Fibonacci sequence: a(n) = a(n-2) + a(n-1), and for even n a row sum of Pascal's triangle (a power of two) is added.
%C A079289 Call a multiset of nonzero integers good if the sum of the cubes is the square of the sum. The number of ascending chains of good multisets starting from the empty set by adding one element at a time is a(n). - _Michael Somos_, Apr 14 2005
%C A079289 a(n) is the number of compositions of n which consist of an initial (possibly empty) subsequence of even summands and a remaining (possibly empty) sequence of odd summands.
%H A079289 Vincenzo Librandi, <a href="/A079289/b079289.txt">Table of n, a(n) for n = 0..1000</a>
%H A079289 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-2).
%F A079289 a(n) = a(n-2) + a(n-1) + floor(2^(n/2-2))*(1-(-1)^(n+1))/2 for n>1.
%F A079289 G.f.: (1-x^2)^2/((1-x-x^2)*(1-2*x^2)).
%F A079289 a(n) = -A016116(n+1)/2 +A000045(n+2), n>0. - _R. J. Mathar_, Sep 27 2012
%F A079289 From _Gregory L. Simay_, Jul 25 2016: (Start)
%F A079289 If n = 2k+1, a(n) = the convolution Sum_{j=0,..k} c(j)*F(n-2j), where c(j) = A011782(j) = 2^(j-1) and f(j)= A000045(j).
%F A079289 If n = 2k, a(n) = c(k) + the convolution Sum_{j=0,..(k-1)} c(j)*F(n-2j), where c(j)=A011782(j)=2^(j-1) and f(j)= A000045(j). (End)
%e A079289 a(4) = 6 from the good multisets {-1,-1,1,1}, {-1,1,1,2}, {-2,-1,1,2}, {-2,1,2,2}, {-3,1,2,3}, {1,2,3,4}.
%e A079289 a(4) = 6 because there are six compositions of four, in which the initial parts are all even and the final parts are all odd: 4, 3+1, 1+3, 2+2, 2+1+1, 1+1+1+1.
%t A079289 CoefficientList[Series[(1-x^2)^2/(1-x-x^2)/(1-2x^2),{x,0,37}],x]
%t A079289 LinearRecurrence[{1,3,-2,-2}, {1,1,2,3,6}, 25] (* _G. C. Greubel_, Aug 16 2016; corrected by _Georg Fischer_, Apr 02 2019 *)
%t A079289 nxt[{n_,a_,b_}]:={n+1,b,If[EvenQ[n],a+b,a+b+2^((n+1)/2-2)]}; Join[{1}, NestList[ nxt,{2,1,2},40][[All,2]]] (* _Harvey P. Dale_, Jul 13 2019 *)
%o A079289 (PARI) {a(n)=local(A); if(n<3,(n>=0)+(n>1), A=vector(n,i,i); for(i=3,n,A[i]=A[i-1]+A[i-2]+ if(i%2==0,2^(i/2-2))); A[n])} /* _Michael Somos_, Apr 14 2005 */
%o A079289 (Magma) I:=[1,1,2,3,6]; [n le 5 select I[n] else Self(n-1)+3*Self(n-2) -2*Self(n-3)-2*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Aug 05 2013
%Y A079289 Cf. A000045, A005674, A007318, A011782, A061667 (bisection).
%K A079289 easy,nonn
%O A079289 0,3
%A A079289 _Paul Barry_, Feb 08 2003

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