A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Author

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

Keywords

• nonn
• easy

Comments

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.

It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.

This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
• W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
• Willibald Limbrunner, Family of sequences for k
• M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Crossrefs

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.

Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.

Cf. A083098, A083099, A083100, A084057, A158780.

Programs

Magma

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

Mathematica

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

SageMath

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

Formula

a(n) = 2*a(n-2) + 2*a(n-4).

a(2*n+1) = A002605(n).

a(2*n) = A026150(n-1).

Extensions

Edited by R. J. Mathar, May 14 2009