OFFSET
0,4
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
A. Özkoç, Some algebraic identities on quadra Fibona-Pell integer sequence, Advances in Difference Equations, 2015:148 (2015).
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,-1).
FORMULA
a(n) = Pell(n) - Fibonacci(n).
G.f.: x^2/( (1-x-x^2)*(1-2*x-x^2) ). - Joerg Arndt, Apr 17 2013
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
MAPLE
with(combinat):seq(fibonacci(i, 2)-fibonacci(i, 1), i=0..27); # Zerinvary Lajos, Mar 20 2008
MATHEMATICA
LinearRecurrence[{3, 0, -3, -1}, {0, 0, 1, 3}, 50] (* T. D. Noe, Apr 16 2013 *)
Table[Fibonacci[n, 2] - Fibonacci[n], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
PROG
(Magma)
Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[Pell(n) - Fibonacci(n): n in [0..30]]; // G. C. Greubel, Aug 05 2021
(SageMath) [lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # G. C. Greubel, Aug 05 2021
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, -3, 0, 3]^n*[0; 0; 1; 3])[1, 1] \\ Charles R Greathouse IV, May 15 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved