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 A130578 #28 Jan 27 2025 06:42:56
%S A130578 0,0,1,3,6,10,16,26,43,71,116,188,304,492,797,1291,2090,3382,5472,
%T A130578 8854,14327,23183,37512,60696,98208,158904,257113,416019,673134,
%U A130578 1089154,1762288,2851442,4613731,7465175,12078908,19544084
%N A130578 Number of different possible rows (or columns) in an n X n crossword puzzle.
%C A130578 The number of linear arrangements of n black and white squares subject to the conditions that there must be at least one run of white squares and all runs of white squares must be of length at least three.
%C A130578 Crossword puzzles such as those in the New York Times do not include one-letter or two-letter words. Since the daily NYT puzzle is 15 X 15, there are a(15) = 797 different possible arrangements for each row.
%H A130578 Reinhard Zumkeller, <a href="/A130578/b130578.txt">Table of n, a(n) for n = 1..5000</a>
%H A130578 Fumio Hazama, <a href="http://dx.doi.org/10.1016/j.disc.2011.06.008">Spectra of graphs attached to the space of melodies</a>, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
%H A130578 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-1).
%F A130578 Recurrence: a(n+4) = 2*a(n+3) - a(n+2) + a(n) + 1, with a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 3.
%F A130578 Formula: a(n) = (30 - 30*sqrt(5) - 30*(1/2 - sqrt(5)/2)^n + 12*sqrt(5)*(1/2 - sqrt(5)/2)^n + 15*(1/2 + sqrt(5)/2)^n + 3*sqrt(5)*(1/2 + sqrt(5)/2)^n - 15*cos((n*Pi)/3) + 15*sqrt(5)*cos((n*Pi)/3) + 5*sqrt(3)*sin((n*Pi)/3) - 5*sqrt(15)*sin((n*Pi)/3))/(30*(-1 + sqrt(5))).
%F A130578 O.g.f.: x^3/((-1+x)*(x^2+x-1)*(x^2-x+1)). - _R. J. Mathar_, Nov 23 2007
%F A130578 a(n) = A005252(n+1) - 1. - _R. J. Mathar_, Nov 15 2011
%F A130578 G.f.: Q(0)*x^2/(2-2*x), where Q(k) = 1 + 1/(1 - x*( 4*k+2 -x +x^3)/( x*( 4*k+4 -x +x^3) +1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 07 2014
%e A130578 a(5) = 6 because using 0's for white squares and 1's for black, the possible rows are: 00011, 10001, 11000, 00001, 10000, 00000.
%t A130578 possiblerows = {}; For[n = 1, n <= 36, n++, table = Table[{n, k, Coefficient[(x^0 + Sum[x^i, {i, 3, n - k}])^(k + 1), x, n - k]}, {k, 0, n}]; total = Sum[table[[j, 3]], {j, 1, n}]; possiblerows = Append[possiblerows, total]; totalstable = Table[{t, possiblerows[[t]]}, {t, 1, Length[ possiblerows]}]]; TableForm[totalstable, TableHeadings -> {None, {" n = squares", "total number of permissible rows"}}]
%o A130578 (Haskell)
%o A130578 a130578 n = a130578_list !! (n-1)
%o A130578 a130578_list = 0 : 0 : 1 : 3 : zipWith (+)
%o A130578 (map (* 2) $ drop 3 a130578_list)
%o A130578 (zipWith (-) (map (+ 1) a130578_list) (drop 2 a130578_list))
%o A130578 -- _Reinhard Zumkeller_, May 23 2013
%o A130578 (PARI) a(n)=([0,1,0,0,0;0,0,1,0,0;0,0,0,1,0;0,0,0,0,1;-1,1,1,-3,3]^n*[0;0;0;1;3])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015
%K A130578 nonn,easy
%O A130578 1,4
%A A130578 Marc A. Brodie (mbrodie(AT)wju.edu), Aug 10 2007, Aug 24 2007