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A105058 G.f. (1+8x-x^2)/((x+1)(x^2-6x+1)). 0
1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..19.

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

FORMULA

FAMP result: (-1)^(n+1) = A046729(n) - a(n) + 2*A090390(n+1) - 2*A079291(n+1) A046729 = 4*A084158 (Pell triangles) A090390(n) = A001333(n)^2 (Squares of "Numerators of continued fraction convergents to sqrt(2)") A079291(n) = A000129(n)^2 (Squares of Pell numbers) (see FAMP code for identity used) SuperSeeker results: a(n) + a(n+1) = A077444(n+1) (Numbers n such that (n^2+4)/2 is a square. Offset at 1.) a(n) + a(n+1) = A082639(n+2) - A082639(n+1) (Numbers n such that 2*n*(n+2) is a square.) a(n+2) - a(n) = A077444(n+3) - A077444(n+2) (Numbers n such that (n^2+4)/2 is a square. Offset at 1.) a(n) + 2*a(n+1) + a(n+2) = A077445(n+3) - A077445(n+2) (Numbers n such that (n^2-8)/2 is a square. Offset at 1.)

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

MATHEMATICA

CoefficientList[ Series[(1 + 8x - x^2)/((x + 1)(x^2 - 6x + 1)), {x, 0, 19}], x] (* Robert G. Wilson v, Apr 06 2005 *)

LinearRecurrence[{5, 5, -1}, {1, 13, 69}, 30] (* Harvey P. Dale, Jun 03 2017 *)

PROG

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.

CROSSREFS

Cf. A046729, A090390, A079291, A077444, A077445.

Equals 2*A001109(n+1) + (-1)^n.

Sequence in context: A137188 A055338 A055880 * A146469 A146381 A085461

Adjacent sequences:  A105055 A105056 A105057 * A105059 A105060 A105061

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Apr 04 2005

STATUS

approved

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Last modified August 17 07:46 EDT 2017. Contains 290635 sequences.