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A248646 Expansion of x*(5+x+x^2)/(1-2*x). 8

%I #55 Sep 29 2020 04:19:34

%S 2,5,11,23,46,92,184,368,736,1472,2944,5888,11776,23552,47104,94208,

%T 188416,376832,753664,1507328,3014656,6029312,12058624,24117248,

%U 48234496,96468992,192937984,385875968,771751936,1543503872,3087007744,6174015488,12348030976

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

%C Previous name was: The Golden Book sequence.

%C Golden Book is a weighted binary pattern, which instead of 0 and 1 uses distance elements, namely 2 and 3 units long. All the horizontal junction points between the elements (2 and 2, 2 and 3, 3 and 2, or 3 and 3) are connected by a straight line on adjacent levels if the vertical distance between those points is sqrt(2) or less. The weighted binary pattern is:

%C L(0): 2, 3, 2, 3, 2, 3, 2, 3, ...

%C L(1): 2, 2, 3, 3, 2, 2, 3, 3, ...

%C L(2): 2, 2, 2, 2, 3, 3, 3, 3, ...

%C ...

%C Starting from the level 2 all single levels of the Golden Book have always these 5 phases: |||, /\ |, / /, | \/, | |. A combination of any 2 adjacent levels (2..n) have 11 phases, etc.

%H G. C. Greubel, <a href="/A248646/b248646.txt">Table of n, a(n) for n = 1..1000</a>

%H Andris Dzenītis, <a href="http://www.lmic.lv/core.php/core.php?pageId=726?pageId=726&amp;id=9227&amp;&amp;subPageId=759&amp;action=showSubPage">Writer of the Golden Book</a>, Interview with Armands Strazds (in Latvian) in the music journal, Mūzikas Saule, April/May 2006. [broken link]

%H A. Strazds, <a href="http://www.zime.lv/book/?oeis">The Golden Book</a> [broken link]

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).

%F Full cycle length: 2 + 3*A001045(0)..A001045(L-1) + (1/2)*(-1^L + 1 + 3*2^(L-1)) + A001045(0)..A001045(L); L, level (0..n).

%F From _Colin Barker_, Oct 11 2014: (Start)

%F a(n) = 23*2^(n-3) for n > 2.

%F a(n) = 2*a(n-1) for n > 3.

%F G.f.: -x*(x^2 + x + 5) / (2*x-1). (End)

%o (PHP)

%o $a = array(0 => 2);

%o $m = array(1 => 1, 2 => 0, 3 => 0, 4 => 0);

%o for ($n = 1; $n < 20; $n++) { $a[$n] = 2 * $a[$n - 1] + ($m[pow(2, $n) % 5]++ ? 0 : 1); }

%o print_r($a); /* _Armands Strazds_, Oct 30 2014 */

%o (Python) print([int(23*2**(n-4)) for n in range(1, 34)]) # _Karl V. Keller, Jr._, Sep 28 2020

%Y Cf. A001045, A000975.

%K nonn,easy

%O 1,1

%A _Armands Strazds_, Oct 10 2014

%E More terms from _Vincenzo Librandi_, Oct 17 2014

%E New name using g.f. from _Joerg Arndt_, Sep 29 2020

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