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Convolve Fibonacci and Pell numbers.
6

%I #50 Sep 08 2022 08:44:35

%S 0,0,1,3,9,24,62,156,387,951,2323,5652,13716,33228,80405,194415,

%T 469845,1135092,2741626,6620928,15987663,38603019,93204647,225030024,

%U 543293352,1311663096,3166694569,7645173627,18457238241,44559967920,107577688310,259716176580

%N Convolve Fibonacci and Pell numbers.

%C Define a triangle T(r,c) by T(n,0) = T(n,n) = A000045(n) and T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The sum of the terms in the first n rows is 2*a(n+1). - _J. M. Bergot_, Apr 07 2013

%H T. D. Noe, <a href="/A006684/b006684.txt">Table of n, a(n) for n = 0..500</a>

%H A. Özkoç, <a href="https://doi.org/10.1186/s13662-015-0486-7">Some algebraic identities on quadra Fibona-Pell integer sequence</a>, Advances in Difference Equations, 2015:148 (2015).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,-1).

%F a(n) = Pell(n) - Fibonacci(n).

%F G.f.: x^2/( (1-x-x^2)*(1-2*x-x^2) ). - _Joerg Arndt_, Apr 17 2013

%F a(n) = 3*a(n-1) - 3*a(n-3) - a(n-4) with a(0) = a(1) = 0, a(2) = 1, a(3) = 3. - _Taras Goy_, Mar 12 2019

%p with(combinat):seq(fibonacci(i,2)-fibonacci(i, 1),i=0..27); # _Zerinvary Lajos_, Mar 20 2008

%t LinearRecurrence[{3,0,-3,-1}, {0,0,1,3}, 50] (* _T. D. Noe_, Apr 16 2013 *)

%t Table[Fibonacci[n, 2] - Fibonacci[n], {n, 0, 30}] (* _Vladimir Reshetnikov_, Sep 27 2016 *)

%o (Magma)

%o Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;

%o [Pell(n) - Fibonacci(n): n in [0..30]]; // _G. C. Greubel_, Aug 05 2021

%o (Sage) [lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # _G. C. Greubel_, Aug 05 2021

%Y Cf. A000045, A000129, A106515 (first differences).

%K nonn

%O 0,4

%A _N. J. A. Sloane_