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A098212 Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)). 1
5, 25, 149, 865, 5045, 29401, 171365, 998785, 5821349, 33929305, 197754485, 1152597601, 6717831125, 39154389145, 228208503749, 1330096633345, 7752371296325, 45184131144601, 263352415571285, 1534930362283105, 8946229758127349, 52142448186480985 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (5,5,-1)

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

a(n) = (1/2)*(-1)^n -(3/2)*sqrt(2)*((3-2*sqrt(2))^n-(3+2*sqrt(2))^n) +(9/4)*((3+2*sqrt(2))^n+(3-2*sqrt(2))^n), with n>=0. - Paolo P. Lava, Nov 28 2008

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, 20}] (* Robert G. Wilson v, Nov 05 2004 *)

CoefficientList[Series[(5-x^2)/((1+x)(1-6x+x^2)), {x, 0, 20}], x] (* or *) LinearRecurrence[{5, 5, -1}, {5, 25, 149}, 20] (* Harvey P. Dale, Jun 09 2011 *)

PROG

(Floretion Algebra Multiplication Program, FAMP)

1vesseq[(j' + k' + 'ii')*('j + 'k + 'ii')] - Creighton Dement, Aug 16 2005

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

(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

CROSSREFS

Cf. A079291, A090390, A000129, A001333.

Sequence in context: A049427 A121639 A098349 * A002050 A047782 A106565

Adjacent sequences:  A098209 A098210 A098211 * A098213 A098214 A098215

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Oct 25 2004

EXTENSIONS

More terms from Robert G. Wilson v, Nov 05 2004

a(20)-a(21) from Vincenzo Librandi, Jul 26 2015

STATUS

approved

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Last modified March 26 04:55 EDT 2017. Contains 284111 sequences.