A122946 a(0)=a(1)=0, a(2)=2; for n >= 3, a(n) = a(n-1) + 4*a(n-3).

0, 0, 2, 2, 2, 10, 18, 26, 66, 138, 242, 506, 1058, 2026, 4050, 8282, 16386, 32586, 65714, 131258, 261602, 524458, 1049490, 2095898, 4193730, 8391690, 16775282, 33550202, 67116962, 134218090, 268418898, 536886746, 1073759106, 2147434698, 4294981682, 8590018106

Offset: 0

Author

Benoit Cloitre, Oct 24 2006

Keywords

• nonn
• easy

Comments

See lemma 5.2 of Reznick's preprint.

Conjecture: count of even Markov numbers in generation n (with generations 0, 1 and 2 labeled as {5}, {13, 29} and {34, 194, 433, 169}. (Checked up to generation 20.) - Wouter Meeussen, Jan 16 2024

Wouter Meeussen's conjecture is true. Proof: label the Markov tree with Markov triples according to the scheme described at A368546. Mod 2, the triples are: row 0: (1,1,0); row 1: (1,1,1), (1,1,0); row 2: (1,0,1), (1,0,1), (1,1,1), (1,1,0); row 3: (1,1,0), (0,1,1), (1,1,0), (0,1,1), (1,0,1), (1,0,1), (1,1,1), (1,1,0); etc. Note that the Markov number labels of the tree (the center numbers of the triples) in rows 0 and 1 include no even numbers, while those in row 2 include two even numbers. Observing that the second triple in row 1 and the first four triples in row 3 are the same or the reverse of the root triple, and noting that every vertex in row 3 and beyond is in a subtree with one of these triples as root, the recurrence follows. - William P. Orrick, Mar 05 2024

Links

• A.H.M. Smeets, Table of n, a(n) for n = 0..300
• Bruce Reznick, Regularity properties of the Stern enumeration of the rationals, arXiv:math/0610601 [math.NT], 2006.
• Index entries for linear recurrences with constant coefficients, signature (1,0,4).

Crossrefs

Cf. A002487, A002559, A089977, A110512, A368546.

Programs

Mathematica

LinearRecurrence[{1,0,4},{0,0,2},36] (* James C. McMahon, Jan 16 2024 *)

PARI

a0=a1=0;a2=2;for(n=3,50,a3=a2+4*a0;a0=a1;a1=a2;a2=a3;print1(a3,","))

Formula

a(n) = (1/7)*2^(-2 + n/2)*(7*2^(n/2) - 7*cos(n*(Pi - arctan(sqrt(7)))) + 5*sqrt(7)*sin(n*(-Pi + arctan(sqrt(7))))). - Zak Seidov, Oct 26 2006

G.f.: 2*x^2 / ((1-2*x)*(2*x^2+x+1)). - Colin Barker, Jun 20 2013

a(n) = 2 * A089977(n-2) for n >= 2. - Alois P. Heinz, Jan 16 2024

From A.H.M. Smeets, Jan 16 2024: (Start)

Limit_{n -> oo} a(n)/a(n-1) = 2.

a(n) = 2^(n-2) + A110512(n-2), for n >= 2. (End)

Extensions

Entries checked by Zak Seidov, Oct 26 2006