login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A326367 Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly two unit "lozenges" or "diamonds" (also of side length 1). 7
0, 0, 24, 126, 387, 915, 1845, 3339, 5586, 8802, 13230, 19140, 26829, 36621, 48867, 63945, 82260, 104244, 130356, 161082, 196935, 238455, 286209, 340791, 402822, 472950, 551850, 640224, 738801, 848337, 969615, 1103445, 1250664, 1412136, 1588752, 1781430, 1991115 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Richard J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = (3/8)*(n-2)*(n-1)*(3*n^2 + 3*n - 4) (conjectured by R. J. Mathar, proved by Greg Dresden and E. Sijaric).

From Colin Barker, Jul 01 2019: (Start)

G.f.: 3*x^3*(4 - x)*(2 + x) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

E.g.f.: (3/8)*exp(x)*x^2*(32 + 24*x + 3*x^2). - Stefano Spezia, Jul 01 2019

EXAMPLE

We can represent a unit triangle this way:

       o

      / \

     o - o

and a unit "lozenge" or "diamond" has these three orientations:

     o

    / \          o - o            o - o

   o   o  and   /   /   and also   \   \

    \ /        o - o                o - o

     o

and for n=3, here is one of the 24 different tiling of the triangle of side length 3 with exactly two lozenges:

          o

         / \

        o   o

       / \ / \

      o - o - o

     /   / \ / \

    o - o - o - o

MATHEMATICA

Rest@ CoefficientList[Series[3 x^3*(4 - x) (2 + x)/(1 - x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Jul 04 2019 *)

PROG

(PARI) concat([0, 0], Vec(3*x^3*(4 - x)*(2 + x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Jul 01 2019

CROSSREFS

Cf. A273464, A326368, A326369.

Sequence in context: A044356 A044737 A305159 * A182186 A188304 A167981

Adjacent sequences:  A326364 A326365 A326366 * A326368 A326369 A326370

KEYWORD

nonn,easy

AUTHOR

Greg Dresden, Jul 01 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)