A015523 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469, 5477472, 22964761, 96281643, 403668734, 1692414417, 7095586921, 29748832848, 124724433149, 522917463687, 2192374556806, 9191710988853, 38537005750589

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

From Johannes W. Meijer, Aug 01 2010: (Start)

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

For n >= 1, the sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.

This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).

Inverse binomial transform of A052918 (with extra leading 0).

(End)

First differences in A197189. - Bruno Berselli, Oct 11 2011

Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12, ... - R. J. Mathar, Aug 10 2012

This is the Lucas U(P=3, Q=-5) sequence, and hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 3 + 5/(3 + 5/(3 + 5/(3 + ... + 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,5).

Crossrefs

Cf. A072263, A072264, A152187, A179606, A197189.

Programs

Magma

[ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Mathematica

Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* Alonso del Arte, Jan 16 2011 *)

PARI

x='x+O('x^30); concat([0], Vec(x/(1-3*x-5*x^2))) \\ G. C. Greubel, Jan 01 2018

Sage

[lucas_number1(n,3,-5) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = 3*a(n-1) + 5*a(n-2).

From Paul Barry, Jul 20 2004: (Start)

a(n) = ((3/2 + sqrt(29)/2)^n - (3/2 - sqrt(29)/2)^n)/sqrt(29).

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*5^k*3^(n-2*k-1). (End)

G.f.: x/(1 - 3*x - 5*x^2). - R. J. Mathar, Nov 16 2007

From Johannes W. Meijer, Aug 01 2010: (Start)

Limit_{k->oo} a(n+k)/a(k) = (A072263(n) + a(n)*sqrt(29))/2.

Limit_{n->oo} A072263(n)/a(n) = sqrt(29). (End)

G.f.: G(0)*x/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

E.g.f.: 2*exp(3*x/2)*sinh(sqrt(29)*x/2)/sqrt(29). - Stefano Spezia, Oct 06 2019