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

2, 5, 11, 23, 46, 92, 184, 368, 736, 1472, 2944, 5888, 11776, 23552, 47104, 94208, 188416, 376832, 753664, 1507328, 3014656, 6029312, 12058624, 24117248, 48234496, 96468992, 192937984, 385875968, 771751936, 1543503872, 3087007744, 6174015488, 12348030976

Offset: 1

Author

Armands Strazds, Oct 10 2014

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001045, A000975.

Programs

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

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)

Extensions

More terms from Vincenzo Librandi, Oct 17 2014

New name using g.f. from Joerg Arndt, Sep 29 2020