login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A192424
a(n) = A192423(n)/2.
3
1, 0, 2, 1, 8, 10, 39, 70, 208, 439, 1162, 2640, 6641, 15600, 38362, 91481, 222688, 534650, 1295559, 3119990, 7544888, 18194639, 43958642, 106072320, 256167361, 618303360, 1492941842, 3603915601, 8701212248, 21005629450
OFFSET
0,3
FORMULA
From G. C. Greubel, Jul 11 2023: (Start)
a(n) = (1/2)*Sum_{j=0..n} T(n, j)*A078008(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = (1/3)*((-1)^n*A000032(n) + A000129(n+1) - A000129(n)).
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
G.f.: (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)). (End)
EXAMPLE
The first five polynomials p(n,x) and their reductions are as follows:
p(0,x) = 2 -> 2
p(1,x) = x -> x
p(2,x) = 2 + x^2 -> 4 + x
p(3,x) = 3*x + x^3 -> 2 + 6*x
p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
From these, read A192423(n) = 2*a(n) = (2, 0, 4, 2, 16, ...) and A192425 = (0, 1, 1, 6, 9, ...).
MATHEMATICA
(See A192423.)
LinearRecurrence[{1, 4, -1, -1}, {1, 0, 2, 1}, 40] (* G. C. Greubel, Jul 12 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023
(SageMath)
@CachedFunction
def a(n): # a = A192424
if (n<4): return (1, 0, 2, 1)[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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved