OFFSET
0,4
COMMENTS
The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).
FORMULA
G.f.: x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)). Colin Barker, Nov 13 2012
From Peter Bala, Mar 26 2015: (Start)
The following remarks assume the o.g.f. for this sequence is x*(1 + x^2)/((1 + x - x^2)*(1 - 2*x - x^2)).
This sequence is a fourth-order linear divisibility sequence. It is the case P1 = 1, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.
exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)
From G. C. Greubel, Jul 12 2023: (Start)
a(n) = Sum_{j=0..n} T(n, j)*A001045(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
EXAMPLE
MATHEMATICA
(See A192423.)
LinearRecurrence[{1, 4, -1, -1}, {0, 1, 1, 6}, 40] (* G. C. Greubel, Jul 12 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023
(SageMath)
@CachedFunction
def a(n): # a = A192425
if (n<4): return (0, 1, 1, 6)[n]
else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 12 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved