OFFSET
0,2
COMMENTS
A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).
Floretion Algebra Multiplication Program, FAMP Code:
1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.
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 * A001109(n+1) - (-1)^n.
G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013
From G. C. Greubel, Aug 21 2022: (Start)
a(n) = A000129(2*n+2) - (-1)^n.
E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)
MATHEMATICA
CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x, 0, 30}], x] (* Robert G. Wilson v, Apr 06 2005 *)
LinearRecurrence[{5, 5, -1}, {1, 13, 69}, 30] (* Harvey P. Dale, Jun 03 2017 *)
PROG
(Magma) [Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // G. C. Greubel, Aug 21 2022
(SageMath) [lucas_number1(2*n+2, 2, -1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Apr 04 2005
STATUS
approved