A085601 a(n) = 2 * (4^n + 2^n) + 1.

5, 13, 41, 145, 545, 2113, 8321, 33025, 131585, 525313, 2099201, 8392705, 33562625, 134234113, 536903681, 2147549185, 8590065665, 34360000513, 137439477761, 549756862465, 2199025352705, 8796097216513, 35184380477441

Offset: 0

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jul 07 2003

Keywords

• nonn
• easy

Comments

1. Begin with a square tile.

2. Place square tiles on each edge to form a diamond shape.

3. Count the tiles: a(0) = 5.

4. Add tiles to fill the enclosing square.

5. Go to step 2.

Links

• Iain Fox, Table of n, a(n) for n = 0..1660
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8)

Crossrefs

Cf. A028403, A046142, A056640, A064412.

Cf. A343175 (essentially the same).

Programs

Mathematica

Table[2(4^n+2^n)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{5,13,41},30] (* Harvey P. Dale, Dec 30 2017 *)

PARI

first(n) = Vec((5 - 22*x + 20*x^2)/(1 - 7*x + 14*x^2 - 8*x^3) + O(x^n)) \\ Iain Fox, Dec 30 2017

Formula

From R. J. Mathar, Apr 20 2009: (Start)

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).

G.f.: -(5 - 22*x + 20*x^2)/((x - 1)*(2*x - 1)*(4*x - 1)).

(End)

E.g.f.: e^x + 2*(e^(2*x) + e^(4*x)). - Iain Fox, Dec 30 2017

Extensions

Edited by Franklin T. Adams-Watters and Don Reble, Aug 15 2006