A266505 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = -1, a(1) = 1, a(2) = 3, a(3) = 5.

-1, 1, 3, 5, 5, 11, 13, 27, 31, 65, 75, 157, 181, 379, 437, 915, 1055, 2209, 2547, 5333, 6149, 12875, 14845, 31083, 35839, 75041, 86523, 181165, 208885, 437371, 504293, 1055907, 1217471, 2549185, 2939235, 6154277, 7095941, 14857739, 17131117, 35869755, 41358175, 86597249, 99847467

Offset

0,3

Comments

a(n)/A266504(n) converges to sqrt(2).

Alternatively, bisection of A266506.

Alternatively, A135532(n) and A048655(n) interlaced.

Alternatively, A255236(n-1), A054490(n), A038762(n) and A101386(n) interlaced.

Let b(n) = (a(n) - (a(n) mod 2))/2, that is b(n) = {-1, 0, 1, 2, 2, 5, 6, 13, 15, 32, 37, 78, 90, ...}. Then:

A006451(n) = {b(4n+0) U b(4n+1)} gives n in N such that triangular(n) + 1 is square;

A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that triangular(n) follows form n^2 + n + 1 (twice a triangular number + 1).

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Formula

G.f.: (-1 + 3*x)*(1 + x)^2/(1 - 2 x^2 - x^4).

a(n) = (-(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.

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

a(n) = sqrt(2*A266504(n))^2 - 7(-1)^A266504(n))*sgn(2*n -1), where A266504 gives all x in N such that 2*x^2 - 7(-1)^x = y^2. This sequence gives associated y values.

a(2n) = (-(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n - (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n) = A135532(n).

a(2n) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) - (((1+sqrt(2))^(n-1)-(1-sqrt(2))^(n-1))/sqrt(8)) = A135532(n).

a(2n+1) = (+(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n + (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n + 1) = A048655(n).

a(2n+1) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) + (((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/sqrt(8)) = A048655(n).

a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A255236(n-1).

a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A054490(n).

a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038762(n).

a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A101386(n).

(sqrt(2*(a(2n + 1) )^2 + 14*(-1)^floor(n/2)))/2 = A266504(n).

(a(2n + 1) + a(2n))/8 = A000129(n), where A000129 gives the Pell numbers.

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

(a(2n + 2) + a(2n + 1))/2 = A000129(n+2).

(a(2n + 2) - a(2n + 1))/2 = A000129(n-1).

Maple

a:=proc(n) option remember; if n=0 then -1 elif n=1 then 1 elif n=2 then 3 elif n=3 then 5 else 2*a(n-2)+a(n-4); fi; end:  seq(a(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016

Mathematica

LinearRecurrence[{0, 2, 0, 1}, {-1, 1, 3, 5}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[(-1 + 3 x) (1 + x)^2/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 42}] (* Michael De Vlieger, Dec 31 2015 *)

Prog

(Magma) I:=[-1, 1, 3, 5]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
(PARI) x='x+O('x^30); Vec((-1+3*x)*(1+x)^2/(1-2*x^2-x^4)) \\ G. C. Greubel, Jul 26 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, A266507.

Sequence in context: A093572 A317650 A240731 * A118132 A089167 A188345

Adjacent sequences: A266502 A266503 A266504 * A266506 A266507 A266508

Keyword

sign,easy

Author

Raphie Frank, Dec 30 2015

Status

approved