A116483 Expansion of (1 + x) / (5*x^2 - 2*x + 1).

1, 3, 1, -13, -31, 3, 161, 307, -191, -1917, -2879, 3827, 22049, 24963, -60319, -245453, -189311, 848643, 2643841, 1044467, -11130271, -27482877, 685601, 138785587, 274143169, -145641597, -1661999039, -2595790093, 3118415009, 19215780483

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, ...).

Inverse binonomial transform of (1, 4, 8, 0, -64, -256, -512, 0, 4096, 16384, 32768, 0, -262144, -1048576, -2097152, 0, ...).

G.f.*(1-x)/(1+x) (i.e, convolution with 1,-2,2,-2,2,-2, ... ) yields A006495.

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseforseq[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' ;

Links

• Colin Barker, 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, A116484.

Programs

PARI

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])+imag(v[k]));}
/* cf. A138749 */ /* Joerg Arndt, Jul 06 2011 */

PARI

Vec((1 + x) / (5*x^2 - 2*x + 1) + O(x^50)) \\ Colin Barker, Aug 25 2017

Formula

a(n) = 2*a(n-1) -5*a(n-2). - Paul Curtz, Apr 18 2011

a(n) = (1/2 + i/2)*((1 - 2*i)^n - i*(1 + 2*i)^n) where i=sqrt(-1). - Colin Barker, Aug 25 2017