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A248646 The Golden Book sequence. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

A. Strazds, The Golden Book

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 */

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

STATUS

approved

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Last modified November 20 12:37 EST 2018. Contains 317402 sequences. (Running on oeis4.)