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 A077855 #50 May 06 2025 00:33:02
%S A077855 1,3,6,11,20,36,64,113,199,350,615,1080,1896,3328,5841,10251,17990,
%T A077855 31571,55404,97228,170624,299425,525455,922110,1618191,2839728,
%U A077855 4983376,8745216,15346785,26931731,47261894,82938843,145547524,255418100,448227520,786584465
%N A077855 Expansion of 1/((1-2*x+x^2-x^3)*(1-x)).
%C A077855 a(n) is the number of binary words of length n+2 such that there is at least one run of 0's and every run of 0's is of length >=2. a(1)=3 because we have: {0,0,0}, {0,0,1}, {1,0,0}. - _Geoffrey Critzer_, Jan 12 2013
%C A077855 INVERT transform of A099254: (1, 2, 1, -2, -4, -2, 3, 6, 3, ...). - _Gary W. Adamson_, Jan 11 2017
%C A077855 a(n) is the number of nonempty subsets A of {1, 2, ..., n+1} with the property that every element in A has at least one consecutive neighbor also in A. That is, for every element x in A, either x-1 is in A or x+1 is in A. - _MingKun Yue_, Mar 07 2025
%H A077855 Seiichi Manyama, <a href="/A077855/b077855.txt">Table of n, a(n) for n = 0..4092</a>
%H A077855 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-1).
%F A077855 G.f.: 1/((1-2*x+x^2-x^3)*(1-x)).
%F A077855 a(n) = A005251(n+4) - 1. a(n+1) - a(n) = A005314(n+2). - _R. J. Mathar_, Sep 19 2008
%F A077855 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4). - _Seiichi Manyama_, Nov 25 2016
%F A077855 a(n) = Sum_{i=1..(n+3)} binomial((n+3)-i, (n+3)-3*i). - _Wesley Ivan Hurt_, Jul 07 2020
%F A077855 a(n) ~ (48 - 11*r + 29*r^2) / (23 * r^n), where r = 0.569840290998... is the root of the equation r*(2 - r + r^2) = 1. - _Vaclav Kotesovec_, Apr 15 2024
%F A077855 From _MingKun Yue_, Mar 07 2025: (Start)
%F A077855 a(n) = 2*a(n-1) - a(n-2) + a(n-3) + 1.
%F A077855 a(n) = a(n-1) + Sum_{i=1..(n-3)} a(i) + n. (End)
%t A077855 nn=40; a=x^2/(1-x); Drop[CoefficientList[Series[(a+1)/(1-x a/(1-x))/(1-x)-1/(1-x), {x,0,nn}], x], 2] (* _Geoffrey Critzer_, Jan 12 2013 *)
%t A077855 LinearRecurrence[{3, -3, 2, -1}, {1, 3, 6, 11}, 36] (* or *)
%t A077855 CoefficientList[ Series[1/(x^4 - 2 x^3 + 3 x^2 - 3 x + 1), {x, 0, 35}], x] (* _Robert G. Wilson v_, Nov 25 2016 *)
%o A077855 (PARI) Vec((1-x)^(-1)/(1-2*x+x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012
%Y A077855 Cf. A018918, A099254, A005314 (first differences).
%K A077855 nonn,easy
%O A077855 0,2
%A A077855 _N. J. A. Sloane_, Nov 17 2002