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A177787 Number of paths from (0,0) to (n+2,n) using only up and right steps and avoiding two or more consecutive moves up or three or more consecutive moves right.

%I A177787 #46 Oct 15 2022 08:07:04
%S A177787 2,5,10,18,30,47,70,100,138,185,242,310,390,483,590,712,850,1005,1178,
%T A177787 1370,1582,1815,2070,2348,2650,2977,3330,3710,4118,4555,5022,5520,
%U A177787 6050,6613,7210,7842,8510,9215,9958,10740,11562,12425,13330,14278,15270
%N A177787 Number of paths from (0,0) to (n+2,n) using only up and right steps and avoiding two or more consecutive moves up or three or more consecutive moves right.
%C A177787 Strings of length 2n+2 over the alphabet {U, R} with n Rs and avoiding UU or RRR as substrings.
%C A177787 Also number of binary words with 3 1's and n 0's that do not contain the substring 101. a(2) = 5: 00111, 10011, 11001, 11100, 01110. - _Alois P. Heinz_, Jul 18 2013
%C A177787 Let (b(n)) be the p-INVERT of A010892 using p(S) = 1 - S^2; then b(n) = a(n+1) for n >= 0. See A292301. - _Clark Kimberling_, Sep 30 2017
%C A177787 From _Gus Wiseman_, Oct 13 2022: (Start)
%C A177787 Also the number of integer compositions of n+3 with half-alternating sum n-1, where we define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ... For example, the a(1) = 2 through a(4) = 10 compositions are:
%C A177787   (112)   (122)    (132)
%C A177787   (1111)  (212)    (222)
%C A177787           (1211)   (312)
%C A177787           (2111)   (1311)
%C A177787           (11111)  (2211)
%C A177787                    (3111)
%C A177787                    (11112)
%C A177787                    (12111)
%C A177787                    (21111)
%C A177787                    (111111)
%C A177787 A001700/A138364 = compositions with alternating sum 0, ranked by A344619.
%C A177787 A357621 = half-alternating sum of standard compositions, reverse A357622.
%C A177787 A357641 = compositions with half-alternating sum 0, ranked by A357625.
%C A177787 Cf. A357136, A357182, A357626, A357631, A357639, A357642, A357706.
%C A177787 (End)
%H A177787 Vincenzo Librandi, <a href="/A177787/b177787.txt">Table of n, a(n) for n = 1..1000</a>
%H A177787 Ângela Mestre, José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H A177787 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1). - _R. J. Mathar_, May 22 2010
%F A177787 a(n) = 1/6 * n (11 + n^2).
%F A177787 From _R. J. Mathar_, May 22 2010: (Start)
%F A177787 a(n) = A140226(n)/2.
%F A177787 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A177787 G.f.: x*(2-3*x+2*x^2)/(x-1)^4. (End)
%p A177787 a:= n-> n/6*(11+n^2): seq(a(n), n=1..40);
%t A177787 CoefficientList[Series[(2-3*x+2*x^2)/(x-1)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o A177787 (Magma) I:=[2, 5, 10, 18]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jul 04 2012
%Y A177787 First differences of A227161. - _Alois P. Heinz_, Jul 18 2013
%Y A177787 Cf. A035363, A088218.
%K A177787 nonn,easy
%O A177787 1,1
%A A177787 _Shanzhen Gao_, May 13 2010
%E A177787 More terms from _R. J. Mathar_, May 22 2010