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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments. 3
1, 0, 3, 7, 34, 123, 495, 1912, 7501, 29253, 114342, 446545, 1744489, 6814224, 26618619, 103979239, 406172770, 1586623227, 6197795703, 24210320296, 94572284197, 369425778645, 1443080391558, 5637075481729, 22019992977457 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = 2*x*p(n-1,x) + (x^2)*p(n-1,x).  See A192872.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

FORMULA

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).

G.f.: (1+x)*(1-3*x-x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - R. J. Mathar, May 07 2014

a(n) = Fibonacci(n-1)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - G. C. Greubel, Jul 29 2019

MATHEMATICA

(* First program *)

q = x^2; s = x + 1; z = 25;

p[0, x_]:= 1; p[1, x_]:= x;

p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^2;

Table[Expand[p[n, x]], {n, 0, 7}]

reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192880 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)

FindLinearRecurrence[u1]

FindLinearRecurrence[u2]

(* Additional programs *)

LinearRecurrence[{2, 7, 2, -1}, {1, 0, 3, 7}, 30] (* G. C. Greubel, Jan 08 2019 *)

Table[Fibonacci[n-1]*LucasL[n, 2]/2, {n, 0, 30}] (* G. C. Greubel, Jul 29 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)) \\ G. C. Greubel, Jan 08 2019

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019

(Sage) ((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

(sage) [fibonacci(n-1)*lucas_number2(n, 2, -1)/2 for n in (0..30)] # G. C. Greubel, Jul 29 2019

(GAP) a:=[1, 0, 3, 7];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jan 08 2019

CROSSREFS

Cf. A000045, A002203, A192872, A192882.

Sequence in context: A208989 A024496 A081890 * A318444 A334314 A179115

Adjacent sequences:  A192877 A192878 A192879 * A192881 A192882 A192883

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 11 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 June 21 16:47 EDT 2021. Contains 345365 sequences. (Running on oeis4.)