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A264038
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Convolution of Lucas and Jacobsthal numbers.
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1
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0, 2, 3, 10, 20, 47, 98, 210, 435, 902, 1848, 3775, 7670, 15542, 31403, 63330, 127500, 256367, 514938, 1033450, 2072675, 4154702, 8324528, 16673535, 33386670, 66837422, 133778523, 267724810, 535721060, 1071881327, 2144473298, 4290096450, 8582053395, 17167117142, 34339105128, 68686091455, 137384934950, 274790503142, 549614391563, 1099282801650
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OFFSET
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0,2
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COMMENTS
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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.
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."
I would suggest another open question: Are there convolutions of other linear homogeneous recurrences with constant coefficients which are equivalent to very simple forms?
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LINKS
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FORMULA
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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.
a(n) = 2*a(n-1)+2*a(n-2)-3*a(n-3)-2*a(n-4) for n > 3.
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)
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EXAMPLE
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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.
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MATHEMATICA
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LinearRecurrence[{1, 2}, {1, 5}, 40]-LinearRecurrence[{1, 1}, {1, 3}, 40]
LinearRecurrence[{2, 2, -3, -2}, {0, 2, 3, 10}, 50] (* Harvey P. Dale, Dec 11 2016 *)
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PROG
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(PARI)
/* Prints first 40 terms of sequence a(n) */
Lucas(n)={if(n==0, 2, if(n==1, 1, Lucas(n-1)+Lucas(n-2))); }
j(n)={if(n==0, 2, if(n==1, 1, j(n-1)+2*j(n-2))); } /*Lucas-Jacobsthal*/
a(n)=j(n+1)-Lucas(n+1);
for(n=0, 40, print(a(n)));
(PARI) concat(0, Vec(-x*(x-2)/((x+1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Nov 02 2015
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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