A021006 Pisot sequence P(4,11), a(0)=4, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1). Evidently satisfies a(n) = 2*a(n-1)+2*a(n-2).

4, 11, 30, 82, 224, 612, 1672, 4568, 12480, 34096, 93152, 254496, 695296, 1899584, 5189760, 14178688, 38736896, 105831168, 289136128, 789934592, 2158141440, 5896152064, 16108587008, 44009478144, 120236130304, 328491216896, 897454694400, 2451891822592, 6698693033984, 18301169713152, 49999725494272, 136601790414848

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1,144, 9,180, 24,,.. - R. J. Mathar, Aug 10 2012

Inverse binomial transform of A001353 without its first two terms, and downshift. - Richard R. Forberg, Aug 24 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (2,2).
• Index entries for Pisot sequences

Programs

Magma

I:=[4,11]; [n le 2 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 26 2011

Mathematica

LinearRecurrence[{2,2},{4,11},30] (* Harvey P. Dale, Oct 25 2011 *)

Formula

G.f.: (4+3*x)/(1-2*x-2*x^2). [Philippe Deléham, Nov 19 2008]

a(n) = (sqrt(3)/24)*((1+sqrt(3))^(n+4) - (1-sqrt(3))^(n+4)). - Taras Goy, Jan 04 2025