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

%I A256494 #32 May 20 2025 10:37:40
%S A256494 0,1,1,2,3,7,13,26,51,103,205,410,819,1639,3277,6554,13107,26215,
%T A256494 52429,104858,209715,419431,838861,1677722,3355443,6710887,13421773,
%U A256494 26843546,53687091,107374183,214748365,429496730,858993459,1717986919,3435973837,6871947674,13743895347,27487790695,54975581389,109951162778
%N A256494 Expansion of -x^2*(x^3+x-1) / ((x-1)*(x+1)*(2*x-1)*(x^2+1)).
%C A256494 Previous name was: Golden Book's Level Leap Sequence.
%C A256494 x-positions a(n) of transition from phase 1 (I I) to 2 (/\) for the Golden Book’s y-position n.
%H A256494 Colin Barker, <a href="/A256494/b256494.txt">Table of n, a(n) for n = 1..1000</a>
%H A256494 Armands Strazds, <a href="http://www.zime.lv/book/?oeis">The Golden Book</a>, 1990. [broken link]
%H A256494 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,1,-2).
%F A256494 a(n) = 2 * a(n - 1) + r((n - 1) % 4); r = array(1, -1, 0, -1).
%F A256494 From _Colin Barker_, Apr 09 2015: (Start)
%F A256494 a(n) = 2*a(n-1)+a(n-4)-2*a(n-5) for n>5.
%F A256494 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).
%F A256494 G.f.: -x^2*(x^3+x-1) / ((x-1)*(x+1)*(2*x-1)*(x^2+1)). (End)
%F A256494 E.g.f.: (5*cosh(x) + cosh(2*x) - cos(x) + sinh(2*x) - 2*sin(x) - 5)/10. - _Stefano Spezia_, May 18 2025
%t A256494 Join[{0}, LinearRecurrence[{2, 0, 0, 1, - 2}, {1, 1, 2, 3, 7}, 50]] (* _Vincenzo Librandi_, Dec 25 2015 *)
%o A256494 (PHP) $r = array(1, -1, 0, -1);
%o A256494 $a[0] = 0;
%o A256494 for ($n = 1; $n < 40; $n++) {
%o A256494 $a[$n] = 2 * $a[$n - 1] + $r[($n - 1) % 4];
%o A256494 }
%o A256494 echo(implode(", ", $a));
%o A256494 (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
%o A256494 (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
%Y A256494 Cf. A248646, A001045.
%K A256494 nonn,easy
%O A256494 1,4
%A A256494 _Armands Strazds_, Mar 30 2015
%E A256494 New name (using g.f. from Colin Barker) from _Joerg Arndt_, Dec 26 2015