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A114619 a(n) = 2*A079291(n) (twice squares of Pell numbers). 2
0, 2, 8, 50, 288, 1682, 9800, 57122, 332928, 1940450, 11309768, 65918162, 384199200, 2239277042, 13051463048, 76069501250, 443365544448, 2584123765442, 15061377048200, 87784138523762, 511643454094368 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Cross-referenced sequences A116484, A001109, A108475, A090390 are also generated by A*B given in the program code.

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).

FORMULA

a(n) = 2*A000129(n)^2.

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

a(n) = ( ( (1-sqrt(2))^n + (1+sqrt(2))^n) /2)^2 + (-1)^(n+1). The formula (1-sqrt( 2))^n + (1+sqrt(2))^n) /2) is the that given by Reinhard Zumkeller in A001333. Furthermore, ( ( A001333(n) )^2 + ( a(n) )^2 - 1)^(1/2) = A001542(n) and ( 2*A001542(n) + 1)^(1/2) = A001333(2n) - Antonio Pane (apane1(AT)spc.edu), Dec 15 2007

MATHEMATICA

2*Fibonacci[Range[0, 30], 2]^2 (* G. C. Greubel, Aug 18 2022 *)

PROG

Floretion Algebra Multiplication Program, FAMP Code:

1jbasejseq[A*B] with

A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and

B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ; apart from initial term.

(Magma) [n le 3 select 2*(n-1)^2 else 5*Self(n-1) +5*Self(n-2) -Self(n-3): n in [1..31]]; // G. C. Greubel, Aug 18 2022

(SageMath) [2*lucas_number1(n, 2, -1)^2 for n in (0..30)] # G. C. Greubel, Aug 18 2022

CROSSREFS

Cf. A000129, A001109, A001333, A001542, A079291.

Cf. A090390, A108475, A114619, A116484.

Sequence in context: A136226 A046165 A227264 * A027047 A034491 A231352

Adjacent sequences:  A114616 A114617 A114618 * A114620 A114621 A114622

KEYWORD

easy,less,nonn

AUTHOR

Creighton Dement, Feb 17 2006

STATUS

approved

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Last modified September 26 12:00 EDT 2022. Contains 356997 sequences. (Running on oeis4.)