OFFSET
1,1
COMMENTS
Previous name was: The Golden Book sequence.
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:
L(0): 2, 3, 2, 3, 2, 3, 2, 3, ...
L(1): 2, 2, 3, 3, 2, 2, 3, 3, ...
L(2): 2, 2, 2, 2, 3, 3, 3, 3, ...
...
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Andris Dzenītis, Writer of the Golden Book, Interview with Armands Strazds (in Latvian) in the music journal, Mūzikas Saule, April/May 2006. [broken link]
A. Strazds, The Golden Book [broken link]
Index entries for linear recurrences with constant coefficients, signature (2).
FORMULA
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).
From Colin Barker, Oct 11 2014: (Start)
a(n) = 23*2^(n-3) for n > 2.
a(n) = 2*a(n-1) for n > 3.
G.f.: -x*(x^2 + x + 5) / (2*x-1). (End)
PROG
(PHP)
$a = array(0 => 2);
$m = array(1 => 1, 2 => 0, 3 => 0, 4 => 0);
for ($n = 1; $n < 20; $n++) { $a[$n] = 2 * $a[$n - 1] + ($m[pow(2, $n) % 5]++ ? 0 : 1); }
print_r($a); /* Armands Strazds, Oct 30 2014 */
(Python) print([int(23*2**(n-4)) for n in range(1, 34)]) # Karl V. Keller, Jr., Sep 28 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Armands Strazds, Oct 10 2014
EXTENSIONS
More terms from Vincenzo Librandi, Oct 17 2014
New name using g.f. from Joerg Arndt, Sep 29 2020
STATUS
approved