A116483 - OEIS

%I #34 Mar 06 2024 15:45:12

%S 1,3,1,-13,-31,3,161,307,-191,-1917,-2879,3827,22049,24963,-60319,

%T -245453,-189311,848643,2643841,1044467,-11130271,-27482877,685601,

%U 138785587,274143169,-145641597,-1661999039,-2595790093,3118415009,19215780483

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

%C Binomial transform of signed powers of 2: (1, 2, -4, -8, 16, 32, -64, -128, ...).

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

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

%C 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' ;

%H Colin Barker, <a href="/A116483/b116483.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Riordan, <a href="http://www.jstor.org/stable/2005477">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.

%H J. Riordan, <a href="/A003480/a003480.pdf">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-5).

%F a(n) = 2*a(n-1) -5*a(n-2). - _Paul Curtz_, Apr 18 2011

%F a(n) = (1/2 + i/2)*((1 - 2*i)^n - i*(1 + 2*i)^n) where i=sqrt(-1). - _Colin Barker_, Aug 25 2017

%o (PARI) a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])+imag(v[k]));}

%o /* cf. A138749 */ /* _Joerg Arndt_, Jul 06 2011 */

%o (PARI) Vec((1 + x) / (5*x^2 - 2*x + 1) + O(x^50)) \\ _Colin Barker_, Aug 25 2017

%Y Cf. A000079, A006495, A116484.

%K easy,sign

%O 0,2

%A _Creighton Dement_, Feb 17 2006