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A146559 Expansion of (1-x)/(1 - 2*x + 2*x^2). 29
1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, 256, 0, -512, -1024, -1024, 0, 2048, 4096, 4096, 0, -8192, -16384, -16384, 0, 32768, 65536, 65536, 0, -131072, -262144, -262144, 0, 524288, 1048576, 1048576, 0, -2097152, -4194304 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Partial sums of this sequence give A099087. - Philippe Deléham, Dec 01 2008

From Philippe Deléham, Feb 13 2013, Feb 20 2013: (Start)

Terms of the sequence lie along the right edge of the triangle

  (1)

     (1)

   2    (0)

      2   (-2)

   4     0   (-4)

      4    -4   (-4)

   8     0    -8    (0)

      8    -8    -8    (8)

  16     0   -16     0   (16)

     16   -16   -16    16   (16)

  32     0   -32     0    32    (0)

     32   -32   -32    32    32  (-32)

  64     0   -64     0    64     0  (-64)

  ...

Row sums of triangle are in A104597.

(1+i)^n = a(n) + A009545(n)*i where i = sqrt(-1). (End)

From Tom Copeland, Nov 08 2014: (Start)

This array is a member of a Catalan family (A091867) related by compositions of C(x)= (1-sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, its inverse Cinv(x) = x(1-x), and the special linear fractional (Möbius) transformation P(x,t) = x / (1+t*x) with inverse P(x,-t) in x.

O.g.f.: G(x) = P[P[Cinv(x),-1],-1] = P[Cinv(x),-2] = x*(1-x)/[1-2x(1-x)]= x*A146599(x).

Ginv(x) = C[P(x,2)] = [1-sqrt(1-4x/(1+2x))]/2 = x*A126930(x).

G(-x) = -[x*(1+x) - 2*[x*(1+x)]^2 + 2^2*[x*(1+x)]^3 - ...], and so this array contains the -row sums of A030528 * Diag(1,(-2)^1,2^2,(-2)^3,...).

The inverse of -G(-x) is -C[-P(x,-2)]= (-1 + sqrt(1+4x/(1-2x)))/2, an o.g.f. for A210736 with a(0) set to zero there. (End)

{A146559, A009545} is the difference analog of {cos(x), sin(x)}. (Cf. the Shevelev link.) - Vladimir Shevelev, Jun 08 2017

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

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

FORMULA

a(0) = 1, a(1) = 1, a(n) = 2*a(n-1) - 2*a(n-2) for n>1.

a(n) = Sum_{k=0..n} A124182(n,k)*(-2)^(n-k).

a(n) = Sum_{k=0..n} A098158(n,k)*(-1)^(n-k). - Philippe Deléham, Nov 14 2008

a(n) = (1/2)*((1-I)^n + (1+I)^n), with n>=0 and I=sqrt(-1). - Paolo P. Lava, Nov 18 2008

a(n) = (-1)^n*A009116(n). - Philippe Deléham, Dec 01 2008

E.g.f.: exp(x)*cos(x). - Zerinvary Lajos, Apr 05 2009

E.g.f.: cos(x)*exp(x) = 1+x/(G(0)-x) where G(k)=4*k+1+x+(x^2)*(4*k+1)/((2*k+1)*(4*k+3)-(x^2)-x*(2*k+1)*(4*k+3)/( 2*k+2+x-x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011

a(n) = Re( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012

G.f.: 1 / (1 - x / (1 + x / (1 - 2*x))) = 1 + x / (1 + 2*x^2 / (1 - 2*x)). - Michael Somos, Jan 03 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

a(m+k) = a(m)*a(k) - A009545(m)*A009545(k). - Vladimir Shevelev, Jun 08 2017

EXAMPLE

G.f. = 1 + x - 2*x^3 - 4*x^4 - 4*x^5 + 8*x^7 + 16*x^8 + 16*x^9 - 32*x^11 - 64*x^12 - ...

MAPLE

G(x):=exp(x)*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..44 ); # Zerinvary Lajos, Apr 05 2009

MATHEMATICA

CoefficientList[Series[(1-x)/(1-2x+2x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, -2}, {1, 1}, 50] (* Harvey P. Dale, Oct 13 2011 *)

PROG

(PARI) Vec((1-x)/(1-2*x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

(Sage)

def A146559():

    x, y = -1, 0

    while true:

        yield -x

        x, y = x - y, x + y

a = A146559(); [a.next() for i in range(40)]  # Peter Luschny, Jul 11 2013

(MAGMA) I:=[1, 1, 0]; [n le 3 select I[n] else 2*Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

CROSSREFS

Cf. A000108, A009116, A009545, A030528, A091867, A126930, A210736.

Sequence in context: A195479 A112793 A009116 * A118434 A090132 A199051

Adjacent sequences:  A146556 A146557 A146558 * A146560 A146561 A146562

KEYWORD

sign,easy

AUTHOR

Philippe Deléham, Nov 01 2008

STATUS

approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)