A098212 Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)).

5, 25, 149, 865, 5045, 29401, 171365, 998785, 5821349, 33929305, 197754485, 1152597601, 6717831125, 39154389145, 228208503749, 1330096633345, 7752371296325, 45184131144601, 263352415571285, 1534930362283105, 8946229758127349, 52142448186480985

Offset: 0

Author

Creighton Dement, Oct 25 2004

Keywords

• nonn
• easy

Comments

Old name was: Relates the squares of Pell numbers with the squares of the numerators of continued fraction convergents to sqrt(2).

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[(j' + k' + 'ii')*('j + 'k + 'ii')] - Creighton Dement, Aug 16 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Crossrefs

Cf. A000129, A001333, A001541, A079291, A075848, A090390.

Programs

Magma

I:=[5,25,149]; [n le 3 select I[n] else 5*Self(n-1)+5*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 26 2015

Mathematica

a[0]= 5; a[1]= 25; a[2]= 149; a[n_]:= a[n]= 5 a[n-1] + 5 a[n-2] - a[n-3]; Table[ a[n], {n,0,40}] (* Robert G. Wilson v, Nov 05 2004 *)
CoefficientList[Series[(5-x^2)/((1+x)(1-6x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{5,5,-1},{5,25,149},40] (* Harvey P. Dale, Jun 09 2011 *)

PARI

Vec((5-x^2)/((1+x)*(1-6*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

SageMath

def Pell(n): return lucas_number1(n,2,-1)
[4*Pell(n+1)^2 +(Pell(n+1) +Pell(n))^2  for n in (0..40)] # G. C. Greubel, Aug 20 2022

Formula

G.f.: (5-x^2)/((1+x)*(1-6*x+x^2)).

a(n) = 4*A079291(n+1) + A090390(n+1) = 4(A000129(n+1))^2 + (A001333(n+1))^2.

a(n) + a(n+1) = A075848(n+2) - A075848(n+1).

a(n) = A001541(n+1) + 2*A079291(n+1). - Creighton Dement, Oct 26 2004

a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3), a(0) = 5, a(1) = 25, a(2) = 149. - Robert G. Wilson v, Nov 05 2004

2*a(n) = (-1)^n + 3*A001541(n+1). - R. J. Mathar, Sep 11 2019