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A264038 Convolution of Lucas and Jacobsthal numbers.

%I A264038 #38 Jan 05 2025 19:51:40
%S A264038 0,2,3,10,20,47,98,210,435,902,1848,3775,7670,15542,31403,63330,
%T A264038 127500,256367,514938,1033450,2072675,4154702,8324528,16673535,
%U A264038 33386670,66837422,133778523,267724810,535721060,1071881327,2144473298,4290096450,8582053395,17167117142,34339105128,68686091455,137384934950,274790503142,549614391563,1099282801650
%N A264038 Convolution of Lucas and Jacobsthal numbers.
%C A264038 The main theorem of the Griffith-Bramham paper found in the LINKS section is the equivalence of the following alternate definitions for a(n). (I) a(n) equals the convolution of the Lucas numbers (A000032) and the Jacobsthal numbers (A001045), where, as usual, the m-th term of the convolution of sequences {b(n)}_{n>=0} and {c(n)}_{n>-0} equals Sum_{t+s=m} b(t)* c(s). (II) a(n) = A014551(n+1)-A000032(n+1), the difference of the Lucas-Jacobsthal numbers and the Lucas numbers with a shift of 1. The authors prove the equivalence of (I) and (II) using the generating function method.
%C A264038 Referring to the simplicity of definition (II), the authors formulate the following open question: "Since the convolution takes such a simple form, we ask whether it is possible to obtain a purely combinatorial proof of this result."
%C A264038 I would suggest another open question: Are there convolutions of other linear homogeneous recurrences with constant coefficients which are equivalent to very simple forms?
%H A264038 Colin Barker, <a href="/A264038/b264038.txt">Table of n, a(n) for n = 0..1000</a>
%H A264038 Martin Griffiths and Alex Bramham, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers/53-2/GriffithsBramham11172014.pdf">The Jacobsthal Numbers: Two Results and Two Questions</a>, Fibonacci Quarterly, Vol. 53, No. 2, May 2015, pp. 147-151.
%H A264038 Tamás Szakács, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_46_from205to216.pdf ">Convolution of second order linear recursive sequences I.</a>, Annales Mathematicae et Informaticae 46 (2016) pp. 205-216.
%H A264038 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3,-2).
%F A264038 a(n) = A014551(n+1)- A000032(n+1).
%F A264038 G.f.: 2/(1-2x)-1/(1+x)-alpha/(1-alpha*x)-beta/(1-beta*x) with alpha=(1+sqrt(5))/2 and beta=-1/alpha.
%F A264038 From _Colin Barker_, Nov 02 2015: (Start)
%F A264038 a(n) = 2*a(n-1)+2*a(n-2)-3*a(n-3)-2*a(n-4) for n > 3.
%F A264038 G.f.: 2/(1-2x)-1/(1+x)-alpha/(1-alpha*x)-beta/(1-beta*x)=-x*(x-2) / ((x+1)*(2*x-1)*(x^2+x-1)), with alpha = (sqrt(5)+1)/2, and beta=-1/alpha.(End)
%e A264038 Let L(n)=A000032(n), j(n)=A014551(n), and J(n)=A001045(n). Then using the convolution definition (I), a(3)=10 because a(3) = L(0)J(3) + L(1)J(2) + L(2)J(1) + L(3)J(0) = 2*3 + 1*1 + 3*1 + 4*0 = 10; similarly, using definition (II) we have a(3) = j(4) - L(4) = 17 - 7 = 10.
%t A264038 LinearRecurrence[{1,2},{1,5},40]-LinearRecurrence[{1,1},{1,3},40]
%t A264038 LinearRecurrence[{2,2,-3,-2},{0,2,3,10},50] (* _Harvey P. Dale_, Dec 11 2016 *)
%o A264038 (PARI)
%o A264038 /* Prints first 40 terms of sequence a(n) */
%o A264038 Lucas(n)={if(n==0,2,if(n==1,1,Lucas(n-1)+Lucas(n-2)));}
%o A264038 j(n)={if(n==0,2,if(n==1,1,j(n-1)+2*j(n-2)));} /*Lucas-Jacobsthal*/
%o A264038 a(n)=j(n+1)-Lucas(n+1);
%o A264038 for(n=0,40,print(a(n)));
%o A264038 (PARI) concat(0, Vec(-x*(x-2)/((x+1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ _Colin Barker_, Nov 02 2015
%Y A264038 Equals convolution of Lucas numbers (A000032) and Jacobsthal numbers (A001045); also equals difference of Lucas-Jacobsthal numbers (A014551) minus Lucas numbers (A000032) with a shift of 1.
%K A264038 nonn,easy
%O A264038 0,2
%A A264038 _Russell Jay Hendel_, Nov 01 2015