A047860 a(n) = T(3,n), array T given by A047858.

1, 5, 14, 34, 78, 174, 382, 830, 1790, 3838, 8190, 17406, 36862, 77822, 163838, 344062, 720894, 1507326, 3145726, 6553598, 13631486, 28311550, 58720254, 121634814, 251658238, 520093694, 1073741822, 2214592510, 4563402750

Offset

0,2

Comments

The Wikipedia article on L-system Example 2 is "Pythagoras Tree" given by the axiom: 0 and rules: 1 -> 11, 0 -> 1[0]0. The length of the n-th string of symbols is a(n). This interpretation leads to a matrix power formula for a(n). - Michael Somos, Jan 12 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Wikipedia, L-system Example 2: Pythagoras Tree

Formula

Main diagonal of the array defined by T(0, j)=j+1 j>=0, T(i, 0)=i+1 i>=0, T(i, j)=T(i-1, j-1)+T(i-1, j)+ 2; a(n)=2^(n-1)*(n+6)-2. - Benoit Cloitre, Jun 17 2003

a(0)=1, a(1)=5, a(2)=14, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Vincenzo Librandi, Sep 28 2011

Example

G.f. = 1 + 5*x + 14*x^2 + 34*x^3 + 78*x^4 + 174*x^5 + 382*x^6 + 830*x^7 + ...
Using the Pythagoras Tree L-system, a(0) = #0 = 1, a(1) = #1[0]0 = 5, a(2) = #11[1[0]0]1[0]0 = 14. - Michael Somos, Jan 12 2015

Mathematica

LinearRecurrence[{5, -8, 4}, {1, 5, 14}, 30] (* Harvey P. Dale, Sep 29 2012 *)

Prog

(Magma) [2^(n-1)*(n+6)-2: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011
(PARI) {a(n) = if( n<0, 0, [1, 1, 1, 1] * [2, 0, 0, 0; 1, 2, 0, 0; 1, 0, 1, 0; 1, 0, 0, 1]^n * [1, 0, 0, 0]~ )}; /* Michael Somos, Jan 12 2015 */
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 4, -8, 5]^n*[1; 5; 14])[1, 1] \\ Charles R Greathouse IV, Jul 19 2016

Crossrefs

n-th difference of a(n), a(n-1), ..., a(0) is (4, 5, 6, ...).

First differences of A027993.

Sequence in context: A192957 A094584 A023515 * A369887 A083332 A101015

Adjacent sequences: A047857 A047858 A047859 * A047861 A047862 A047863

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved