A116484 Expansion of (-1+3*x)/(5*x^2 + 1 - 2*x).

-1, 1, 7, 9, -17, -79, -73, 249, 863, 481, -3353, -9111, -1457, 42641, 92567, -28071, -518977, -897599, 799687, 6087369, 8176303, -14084239, -69049993, -67678791, 209892383, 758178721, 466895527, -2857102551, -8048682737, -1811852719

Offset: 0

Author

Creighton Dement, Feb 17 2006

Keywords

• easy
• sign

Comments

Binomial transform of signed powers of 2: (-1, 2, 4, -8, -16, 32, 64, -128, -256, 512, 1024). Inverse binomial transform of (-1, 0, 8, 32, 64, 0, -512, -2048, -4096, 0, 32768, 131072, 262144, 0, -2097152, -8388608). Compare with A116483.

Floretion Algebra Multiplication Program, FAMP Code: 2basekforseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ; 1vesforseq = A000004

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
• J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (2,-5).

Crossrefs

Cf. A000079, A006495, A116483.

Programs

Mathematica

CoefficientList[Series[(-1+3x)/(5x^2+1-2x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-5},{-1,1},40] (* Harvey P. Dale, Jun 24 2013 *)

Formula

a(n) = 3*A045873(n) - A045873(n+1). - R. J. Mathar, Apr 23 2009

E.g.f.: exp(x)*(sin(2*x) - cos(2*x)). - Arkadiusz Wesolowski, Aug 31 2012

a(0)=-1, a(1)=1, a(n) = 2*a(n-1) - 5*a(n-2). - Harvey P. Dale, Jun 24 2013

a(n) = (1/2)*((-1 - i)*(1 + 2*i)^n - (1 - i)*(1 - 2*i)^n), n >= 0, where i=sqrt(-1). - Taras Goy, Apr 20 2019