This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A089928 #36 Jul 18 2024 11:09:42
%S A089928 1,2,4,10,25,60,144,348,841,2030,4900,11830,28561,68952,166464,401880,
%T A089928 970225,2342330,5654884,13652098,32959081,79570260,192099600,
%U A089928 463769460,1119638521,2703046502,6525731524,15754509550,38034750625,91824010800
%N A089928 a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=4, a(4)=10.
%C A089928 a(n) is the number of tilings of an n-board (a board of size n X 1) using white squares, black squares, and white (1,1)-fences. A (1,1)-fence is a tile composed of two squares separated by a gap of width 1. - _Michael A. Allen_, Mar 12 2021
%C A089928 a(n) is the number of tilings of an n-board using white squares, black squares, white trominoes, black trominoes, and white tetrominoes. - _Michael A. Allen_, Mar 12 2021
%H A089928 G. C. Greubel, <a href="/A089928/b089928.txt">Table of n, a(n) for n = 0..1000</a>
%H A089928 Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.
%H A089928 Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H A089928 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A089928 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,1).
%F A089928 a(n) = ( (1+sqrt(2))^(n+2) + (1-sqrt(2))^(n+2) + 2*(-1)^floor(n/2) )/8.
%F A089928 a(n) = (-i)^n*Sum_{k=0..floor(n/2)} U(n-2*k, i) with i^2 = -1.
%F A089928 a(n) + a(n+2) = A000129(n+3). - _Alex Ratushnyak_, Aug 06 2012
%F A089928 G.f.: 1/ ( (1+2*x)*(1-2*x-x^2) ). - _R. J. Mathar_, Apr 26 2013
%F A089928 4*a(n) = A057077(n) + A001333(n+2). - _R. J. Mathar_, Apr 26 2013
%F A089928 a(2*n) = (A000129(n+1))^2 = A079291(n+1). - _Michael A. Allen_, Mar 12 2021
%F A089928 a(2*n+1) = A000129(n+1)*A000129(n+2) = A114620(n+1). - _Michael A. Allen_, Mar 12 2021
%t A089928 CoefficientList[Series[1/(1-2x-2x^3-x^4),{x,0,30}],x] (* _Michael A. Allen_, Mar 12 2021 *)
%t A089928 LinearRecurrence[{2,0,2,1}, {1,2,4,10}, 41] (* _G. C. Greubel_, Aug 18 2022 *)
%t A089928 nxt[{a_,b_,c_,d_}]:={b,c,d,2d+2b+a}; NestList[nxt,{1,2,4,10},30][[;;,1]] (* _Harvey P. Dale_, Jul 18 2024 *)
%o A089928 (Magma) [(Evaluate(DicksonFirst(n+2,-1), 2) + 2*(-1)^Binomial(n,2))/8: n in [0..40]]; // _G. C. Greubel_, Aug 18 2022
%o A089928 (SageMath) [(lucas_number2(n+2,2,-1) +2*(-1)^binomial(n,2))/8 for n in (0..40)] # _G. C. Greubel_, Aug 18 2022
%Y A089928 Cf. A000129, A001333, A006498, A057077, A079291, A114620.
%K A089928 easy,nonn
%O A089928 0,2
%A A089928 _Paul Barry_, Nov 15 2003
%E A089928 Formula corrected by _Max Alekseyev_, Aug 22 2013