A136290 a(0)=1, a(1)=3, a(2)=9, a(3)=12, a(4)=15; thereafter a(n) = a(n-1) + a(n-3) - a(n-4).

1, 3, 9, 12, 15, 21, 24, 27, 33, 36, 39, 45, 48, 51, 57, 60, 63, 69, 72, 75, 81, 84, 87, 93, 96, 99, 105, 108, 111, 117, 120, 123, 129, 132, 135, 141, 144, 147, 153, 156, 159, 165, 168, 171, 177, 180, 183, 189, 192, 195, 201, 204, 207, 213, 216, 219, 225, 228, 231, 237

Offset

0,2

Comments

This is the coordination sequence for Marjorie Rice's tiling of the plane shown in Fig. 15 of Schattschneider (1981), with respect to the central vertex. The Schattschneider illustration below shows that the first differences of the coordination sequence are 2, 6, 3, 3, 6, 3, 3, 6, 3, 3, ..., and so the sequence itself satisfies the recurrence in the definition. The tiling has symmetry group D_6 (the dihedral group of order 6).

Continuing from the arrangement of pennies described in A136289, we also wish to place dimes over the holes in the array, where the n-th generation of dimes can be placed only when all three of its supporting pennies are in place already; then a(n-1) is the number of dimes in generation n for >= 1. - Colin Mallows, Apr 13 2008

References

Doris Schattschneider, In Praise of Amateurs, pp. 140-166 in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Colin Mallows, Analysis of coordination sequence for Marjorie Rice tiling

Doris Schattschneider, Illustration of terms a(0) to a(12) of the coordination sequence for the Rice tiling

N. J. A. Sloane, Illustration of terms a(0) to a(9) of the coordination sequence for the Rice tiling [Annotated copy of Fig. 15 of Schattschneider (1981)]

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

G.f.: (1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1+x+x^2)). - Colin Barker, Jul 12 2014

a(n) = [n=0] + 4*n - ChebyshevU(n-1, -1/2). - G. C. Greubel, Apr 13 2021

For n > 0, a(n) = 3*A042965(n+1). - Jon E. Schoenfield, Jun 03 2022

Maple

1, seq(4*n -simplify(ChebyshevU(n-1, -1/2)), n = 1..20); # G. C. Greubel, Apr 13 2021

Mathematica

{1}~Join~LinearRecurrence[{1, 0, 1, -1}, {3, 9, 12, 15}, 59] (* Jean-François Alcover, Oct 23 2019 *)

Prog

(Magma) a:=[1, 3, 9, 12, 15]; [n le 5 select a[n] else Self(n-1)+Self(n-3)-Self(n-4):n in [1..60]]; // Marius A. Burtea, Oct 23 2019
(Sage) [1]+[4*n-chebyshev_U(n-1, -1/2) for n in (1..60)] # G. C. Greubel, Apr 13 2021

Crossrefs

Cf. A042965, A136289.

Sequence in context: A255686 A138921 A194412 * A244147 A103531 A333441

Adjacent sequences: A136287 A136288 A136289 * A136291 A136292 A136293

Keyword

nonn

Author

Colin Mallows, Apr 13 2008

Extensions

Entry revised by N. J. A. Sloane, Apr 06 2019, replacing the old definition with Colin Barker's recurrence.

Status

approved