A256494 Expansion of -x^2*(x^3+x-1) / ((x-1)*(x+1)*(2*x-1)*(x^2+1)).

0, 1, 1, 2, 3, 7, 13, 26, 51, 103, 205, 410, 819, 1639, 3277, 6554, 13107, 26215, 52429, 104858, 209715, 419431, 838861, 1677722, 3355443, 6710887, 13421773, 26843546, 53687091, 107374183, 214748365, 429496730, 858993459, 1717986919, 3435973837, 6871947674, 13743895347, 27487790695, 54975581389, 109951162778

Offset: 1

Author

Armands Strazds, Mar 30 2015

Keywords

• nonn
• easy

Comments

Previous name was: Golden Book's Level Leap Sequence.

x-positions a(n) of transition from phase 1 (I I) to 2 (/\) for the Golden Book’s y-position n.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Armands Strazds, The Golden Book, 1990. [broken link]
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,-2).

Crossrefs

Cf. A248646, A001045.

Programs

Magma

I:=[0,1,1,2,3,7]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-4)-2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 25 2015

Mathematica

Join[{0}, LinearRecurrence[{2, 0, 0, 1, - 2}, {1, 1, 2, 3, 7}, 50]] (* Vincenzo Librandi, Dec 25 2015 *)

PARI

concat(0, Vec(-x^2*(x^3+x-1)/((x-1)*(x+1)*(2*x-1)*(x^2+1)) + O(x^100))) \\ Colin Barker, Apr 09 2015

PHP

$r = array(1, -1, 0, -1);
$a[0] = 0;
for ($n = 1; $n < 40; $n++) {
$a[$n] = 2 * $a[$n - 1] + $r[($n - 1) % 4];
}
echo(implode(", ", $a));

Formula

a(n) = 2 * a(n - 1) + r((n - 1) % 4); r = array(1, -1, 0, -1).

From Colin Barker, Apr 09 2015: (Start)

a(n) = 2*a(n-1)+a(n-4)-2*a(n-5) for n>5.

a(n) = (5+5*(-1)^n-(1+2*i)*(-i)^n-(1-i*2)*i^n+2^(1+n))/20 for n>0 where i=sqrt(-1).

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

E.g.f.: (5*cosh(x) + cosh(2*x) - cos(x) + sinh(2*x) - 2*sin(x) - 5)/10. - Stefano Spezia, May 18 2025

Extensions

New name (using g.f. from Colin Barker) from Joerg Arndt, Dec 26 2015