A285361 The number of tight 3 X n pavings.

0, 1, 11, 64, 282, 1071, 3729, 12310, 39296, 122773, 378279, 1154988, 3505542, 10598107, 31957661, 96200098, 289255020, 869075073, 2609845875, 7834779640, 23514823730, 70565441671, 211738266921, 635298685614, 1906063827672, 5718527025901, 17156252164799, 51470098670020

Offset: 0

Author

Don Knuth, Apr 17 2017

Keywords

• nonn
• easy

Comments

Also, zero together with the third row of the square array A285357.

Examples

			For n=2 the 11 solutions are 12|32|44, 12|13|44, 12|33|44, 11|22|34, 11|23|43, 12|13|43, 12|32|42, 12|13|14, 12|32|34, 11|23|24, 11|23|44.
(Use the "interactive illustration" link in A285357 (with n=3!) for a graphic display.)
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2092
• D. E. Knuth (Proposer), Problem 12005, Amer. Math. Monthly 124 (No. 8, Oct. 2017), page 755. For solution see op. cit., 126 (No. 7, 2019), 660-664.
• Roberto Tauraso, Problem 12005, Proposed solution.
• Index entries for linear recurrences with constant coefficients, signature (8,-24,34,-23,6).

Crossrefs

Cf. A285357, A298362, A336732, A336734.

Programs

Magma

[(1/4)*(3^(n+3)-5*2^(n+4)+4*n^2+26*n+53): n in [0..30]]; // Vincenzo Librandi, Mar 16 2018

Maple

seq((1/4) * (3^(n+3) - 5*2^(n+4) + 4*n^2 + 26*n + 53),n=0..50); # Robert Israel, Mar 15 2018

Mathematica

LinearRecurrence[{8, -24, 34, -23, 6}, {0, 1, 11, 64, 282}, 30] (* Vincenzo Librandi, Mar 16 2018 *)

Formula

a(n) = (1/4) * (3^(n+3) - 5*2^(n+4) + 4*n^2 + 26*n + 53). - Hugo Pfoertner, Mar 14 2018

G.f.: (x+3*x^2)/((1-x)^3*(1-2*x)*(1-3*x)). - Robert Israel, Mar 15 2018

Extensions

a(10) from Hugo Pfoertner, Jan 17 2018

More terms from M. F. Hasler, Jan 21 2018