login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 3
1, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

LINKS

Table of n, a(n) for n=1..26.

FORMULA

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 2014

EXAMPLE

The first five polynomials p(n,x) and their reductions are as follows:

p(0,x)=1 -> 1

p(1,x)=2x -> 2x

p(2,x)=2+x+3x^2 -> 5+4x

p(3,x)=8x+4x^2+4x^3 -> 8+20x

p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.

From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).

MATHEMATICA

q[x_] := x + 1; d = Sqrt[x + 2];

p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)

Table[Expand[p[n, x]], {n, 1, 6}]

reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};

t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)

Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)

CROSSREFS

Cf. A192232, A192380, A192381.

Sequence in context: A256401 A109292 A275571 * A117474 A294665 A323139

Adjacent sequences:  A192376 A192377 A192378 * A192380 A192381 A192382

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 29 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 25 09:29 EST 2020. Contains 338623 sequences. (Running on oeis4.)