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A192422 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments. 2
0, 1, 1, 5, 7, 20, 35, 83, 161, 355, 720, 1541, 3185, 6733, 14027, 29500, 61663, 129403, 270865, 567911, 1189440, 2492905, 5222449, 10943813, 22928815, 48044900, 100665083, 210927155, 441948689, 926020171, 1940274000, 4065458669, 8518311809 (list; graph; refs; listen; history; text; internal format)
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+1, see A192232.
Assuming the o.g.f. given below, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 1, P2 = -1, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A100047. - Peter Bala, Aug 28 2019
LINKS
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
FORMULA
From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
G.f.: x*(1 + x^2)/(1 - x - 3*x^2 + x^3 + x^4). (End)
From Vladimir Kruchinin, Mar 20 2016: (Start)
G.f.: ((1+x^2)/(1-x^2)) * F(x/(1-x^2)), where F(x) is g.f. of Fibonacci numbers (A000045).
a(n) = n*Sum_{i=0..floor((n-1)/2)} (binomial(n-i-1,i)/(n-2*i))*Fibonacci(n-2*i). (End)
a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023
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 -> 3 + x
p(3,x) = 3*x + x^3 -> 1 + 5*x
p(4,x) = 2 + 4*x^2 + x^4 -> 8 + 7*x.
From these, read A192421 = (2, 0, 3, 1, 8, ...) and a(n) = (0, 1, 1, 5, 7, ...).
MATHEMATICA
(See A192421.)
LinearRecurrence[{1, 3, -1, -1}, {0, 1, 1, 5}, 40] (* G. C. Greubel, Jul 11 2023 *)
PROG
(Maxima)
a(n):=n*sum((binomial(n-i-1, i))/(n-2*i)*fib(n-2*i), i, 0, (n-1)/2); /* Vladimir Kruchinin, Mar 20 2016 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023
(SageMath)
@CachedFunction
def a(n): # a = A192422
if (n<4): return (0, 1, 1, 5)[n]
else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023
CROSSREFS
Sequence in context: A279957 A249047 A258282 * A120035 A198302 A091154
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved

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Last modified June 20 19:46 EDT 2024. Contains 373532 sequences. (Running on oeis4.)