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

1,1

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

Cf. A001045, A000975.

Sequence in context: A147878 A179902 A140992 * A093053 A192580 A075712

Adjacent sequences: A248643 A248644 A248645 * A248647 A248648 A248649

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