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A266506 a(2n) =  a(2n - 4) + a(2n - 3) and a(2n + 1) = 2*a(2n - 4) + a(2n - 3), with a(0) = 2, a(1) = -1, a(2) = 2, a(3) = 1. Alternatively, interleave denominators (A266504) and numerators (A266505) of convergents to sqrt(2). 4
2, -1, 2, 1, 1, 3, 3, 5, 4, 5, 8, 11, 9, 13, 19, 27, 22, 31, 46, 65, 53, 75, 111, 157, 128, 181, 268, 379, 309, 437, 647, 915, 746, 1055, 1562, 2209, 1801, 2547, 3771, 5333, 4348, 6149, 9104, 12875, 10497, 14845, 21979, 31083, 25342, 35839, 53062, 75041, 61181, 86523 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(2n) gives all x in N | 2*x^2 - 7(-1)^x = y^2. a(2n +1) gives associated y values.

Consequence of the above, a(2n + 1)/a(2n) converges to sqrt(2). For example, a(25)/a(24) = 181/128 = 1.4140625, accurate to 4 digits, while a(55)/a(54) = (181165/128103) = 1.41421356252..., accurate to 10 digits.

The intermediate convergents to sqrt(2), beginning with 4/3, 10/7, 24/17, 58/41, etc., as outlined in A052542, are given by (a(4n + 17) - a(4n + 15))/(a(4n + 16) - a(4n + 14)). For example, for n = 0, then 4/3 = a(17) - a(15))/(a(16) - a(14)) = (31 - 27)/(22 - 19), and for n = 2, then 24/17 = (a(25) - a(23))/(a(24) - a(22)) = (181 - 157)/(128 - 111).

Alternatively, A048654(n-1), A135532(n), A078343(n+1) and A048655(n) interlaced.

Alternatively, A100525(n-1), A255236(n-1), A266507(n), A054490(n), A038761(n), A038762(n), A253811(n), and A101386(n) interlaced.

Also see comments in A222786 regarding a curious relationship/coincidence between this sequence and laminated lattice kissing numbers with an average number of spheres/dimension equal to either a pronic number or twice a square.

Where A002203 gives the companion Pell numbers, or, in Lucas sequence notation, V_n(2, -1), then a(2n) = 1/4*(3*A002203(floor[n/2]) - A002203(n-(-1)^n)). Where A000129(n) gives the Pell numbers, or, in Lucas sequence notation, U_n(2, -1), then a(2n + 1) = (3*A000129(floor[n/2]) - A000129(n-(-1)^n)).

Subsequent to n = 7, every 4 values of this sequence increase in value, followed by a decrease in value equal to a Pell number (A000129).

Let b(n) = (a(n) - a(n)(mod 2))/2. That is, b(n) = {1, -1, 1, 0, 0, 1, 1, 2, 2, 2, 4, 5, 4, 6, 9, 13, 11, 15, 23, 32, 26, 37, 55, 78, 64, 90, ...}.

b(4n + 0) = A006452(2n) and A216134(2n) interlaced.

b(4n + 1) = A098790(n - 1) = A006451(2n - 1) and A124124(2n) interlaced.

b(4n + 2) = A006452(2n + 1) and A216134(2n + 1) interlaced.

b(4n + 3) = A098586(n - 1) =  A006451(2n) and A124124(2n + 1) interlaced.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = (1/sqrt(12+(-1)^n*4))*((3*((1+sqrt(2))^(floor(n/4))+(-1)^n*(1-sqrt(2))^floor(n/4)))-(-1)^(floor(n/2))*((1+sqrt(2))^(floor(n/4)-(-1)^(floor(n/2)))+(-1)^n*(1-sqrt(2))^(floor(n/4)-(-1)^(floor(n/2))))).

a(2n) = A266504(n).

a(2n) = a(2n - 3) + a(2n - 4).

a(2n) = (sqrt(2*(a(2n + 1))^2 + 14*(-1)^(floor(n/2))))/2 = (sqrt(2*(A266505(n))^2 + 14*(-1)^(floor(n/2))))/2.

a(2n) = 1/4*(3*A002203(floor[n/2]) - A002203(n-(-1)^n)), where A002203 gives the Companion Pell numbers. Therefore:

a(2n) = 1/4*((3*((1+sqrt(2))^(floor(n/2))+(1-sqrt(2))^floor(n/2))) - (-1)^n*((1+sqrt(2))^(floor(n/2)-(-1)^n)+(1-sqrt(2))^(floor(n/2)-(-1)^n))).

a(2n) = (+sqrt(2)*(1+sqrt(2))^((floor(n/2))-(-1)^n)*(-1)^n - 3*(1-sqrt(2))^((floor(n/2))-(-1)^n) + sqrt(2)*(1-sqrt(2))^((floor(n/2))-(-1)^n)*(-1)^n + 3*(1+sqrt(2))^((floor(n/2))-(-1)^n))/sqrt(8)).

a(2n + 1) = A266505(n).

a(2n + 1) = 2*a(2n - 3) + a(2n - 4).

a(2n + 1) = sqrt(2*(a(2n))^2 - 7*(-1)^(a(2n)))*sgn(2*n - 1) = sqrt(2*(A266504(n))^2 - 7(-1)^(A266504(n)))*sgn(2*n - 1).

a(2n + 1) =  (3*A000129(floor[n/2]) - A000129(n-(-1)^n)), where A000129 gives the Pell numbers. Therefore:

a(2n + 1) = 3*(((1+sqrt(2))^(floor(n/2))-(1-sqrt(2))^(floor(n/2)))/sqrt(8)) - (-1)^n*(((1+sqrt(2))^((floor(n/2))-(-1)^n)-(1-sqrt(2))^((floor(n/2))-(-1)^n))/sqrt(8)).

a(2n + 1) = (-(1+sqrt(2))^(floor(n/2))*(-1)^n - sqrt(8)*(1-sqrt(2))^(floor(n/2)) - (1-sqrt(2))^(floor(n/2))*(-1)^n + sqrt(8)*(1+sqrt(2))^(floor(n/2)))/2.

In general, a(4n + k) = 2*a(4n - (4-k)) + a(4n - (8-k)). Thus:

a(4n + 0) = 2*a(4n - 4) + a(4n - 8) = A048654(n-1).

a(4n + 1) = 2*a(4n - 3) + a(4n - 7) = A135532(n).

a(4n + 2) = 2*a(4n - 2) + a(4n - 6) = A078343(n+1).

a(4n + 3) = 2*a(4n - 1) + a(4n - 5) = A048655(n).

In general, a(8n + k) = 6*a(8n - (8-k)) - a(8n - (16-k)). Thus:

a(8n + 0) = 6*a(8n - 8) - a(4n - 16) = A100525(n-1).

a(8n + 1) = 6*a(8n - 7) - a(4n - 15) = A255236(n-1).

a(8n + 2) = 6*a(8n - 6) - a(4n - 14) = A266507(n).

a(8n + 3) = 6*a(8n - 5) - a(4n - 13) = A054490(n).

a(8n + 4) = 6*a(8n - 4) - a(4n - 12) = A038761(n).

a(8n + 5) = 6*a(8n - 3) - a(4n - 11) = A038762(n).

a(8n + 6) = 6*a(8n - 2) - a(4n - 10) = A253811(n).

a(8n + 7) = 6*a(8n - 1) - a(4n - 9)  = A101386(n).

a(4n + 1) - a(4n) = A048654(n - 2).

(a(4n + 2) - a(4n + 1))/(3) = A000129(n - 1), where A000129 gives the Pell Numbers.

a(4n + 3) - a(4n + 2) = A078343(n).

(a(4n + 4) - a(4n + 3))/(-1) = A000129(n).

((a(4n + 1) - a(4n)) + (a(4n + 3) - a(4n + 2)))/2 = A002203(n - 1), where A002203 gives the companion Pell numbers.

((a(4n + 1) - a(4n)) + (a(4n + 3) - a(4n + 2)))/4 = A001333(n - 1), where A001333 gives the half companion Pell Numbers.

(a(4n + 2) - a(4n + 1))/(3) + (a(4n + 4) - a(4n + 3))/(-1) = A001333(n).

(a(4n) + a(4n + 1) + a(4n + 2) + a(4n + 3))/4 = A001333(n + 1).

a(4n+2) + a(4n + 3) + a(4n + 4) + a(4n + 5) = A001333(n + 3).

(a(4n + 4) - a(4n + 3))/(-1) alternating with (a(4n + 2) - a(4n + 1))/(3) + (a(4n + 4) - a(4n + 3))/(-1) = A002965(n).

From Chai Wah Wu, Sep 17 2016: (Start)

a(n) = 2*a(n-4) + a(n-8) for n > 7.

G.f.: (-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1). (End)

MATHEMATICA

CoefficientList[Series[(-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1), {x, 0, 50}], x] (* G. C. Greubel, Jul 27 2018 *)

PROG

(PARI) x='x+O('x^50); Vec((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8 + 2*x^4-1)) \\ G. C. Greubel, Jul 27 2018

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8+2*x^4-1))); // G. C. Greubel, Jul 27 2018

CROSSREFS

Cf. A000129, A001333, A002203, A002965, A006451, A006452, A002965, A038761, A038762, A048654, A048655, A054490, A078343, A098586, A098790, A100525, A101386, A135532, A216134, A216162, A253811, A255236, A266504, A266505, A226507.

Sequence in context: A137278 A205810 A139368 * A134303 A078997 A024680

Adjacent sequences:  A266503 A266504 A266505 * A266507 A266508 A266509

KEYWORD

sign,easy

AUTHOR

Raphie Frank, Dec 30 2015

STATUS

approved

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Last modified August 20 19:51 EDT 2019. Contains 326155 sequences. (Running on oeis4.)