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A359021 Number of inequivalent tilings of a 5 X n rectangle using integer-sided square tiles.

%I A359021 #25 Mar 18 2023 11:35:31
%S A359021 1,1,5,10,39,77,521,1985,8038,32097,130125,525676,2131557,8635656,
%T A359021 35017970,141968455,575692056,2334344849,9465939422,38384559168,
%U A359021 155652202456,631178976378,2559476952229,10378857744374,42087027204278,170665938023137,692062856184512
%N A359021 Number of inequivalent tilings of a 5 X n rectangle using integer-sided square tiles.
%H A359021 John Mason, <a href="/A359021/b359021.txt">Table of n, a(n) for n = 0..1000</a>
%H A359021 John Mason, <a href="/A359021/a359021.pdf">Counting tilings of width 5 rectangles</a>
%F A359021 For even n > 5:
%F A359021 a(n) = (A054857(n) + A079975(n) + 2*A054857(n/2) + 2* fixed_md(n/2) + 2*A054857((n-4)/2) + 4*A054857((n-2)/2) + 2* (A054857((n/2)-1) + fixed_md((n/2)-1)))/4.
%F A359021 For odd n > 5:
%F A359021 a(n) = (A054857(n) + A079975(n) + 2*A054857((n-1)/2) + 4*A054857((n-3)/2) + 2*fixed_md((n-3)/2) + 2*A054857((n-5)/2) + 2*fixed_md((n-1)/2))/4.
%F A359021 where
%F A359021 fixed_md(1)=1, fixed_md(2)=3, fixed_md(3)=15 and for n > 3, fixed_md(n) = A054857(n-1) + A054857(n-2) + fixed_md(n-2)+ fixed_md(n-1) + 2*A054857(n-3) + fixed_md(n-3).
%e A359021 a(2) is 5 because of:
%e A359021   +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
%e A359021   | | | |   | |   | |   | |   |
%e A359021   +-+-+ +-+-+ +   + +   + +-+-+
%e A359021   | | | |   | |   | |   | |   |
%e A359021   +-+-+ +   + +-+-+ +-+-+ +   +
%e A359021   | | | |   | |   | | | | |   |
%e A359021   +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
%e A359021   | | | |   | |   | | | | | | |
%e A359021   +-+-+ +   + +   + +-+-+ +-+-+
%e A359021   | | | |   | |   | | | | | | |
%e A359021   +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
%Y A359021 Column k = 5 of A227690.
%Y A359021 Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
%Y A359021 Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
%Y A359021 Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
%Y A359021 Cf. A079975.
%K A359021 nonn
%O A359021 0,3
%A A359021 _John Mason_, Dec 12 2022