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 A380176 #10 Feb 05 2025 22:21:14
%S A380176 0,0,1,2,6,12,26,56,124,266,563,1204,2573,5468,11559,24370,51281,
%T A380176 107720,225867,472660,987378,2059180,4287932,8916624,18517398,
%U A380176 38406486,79563118,164636582,340308519,702713844,1449664783,2987870476,6152930738,12660419370,26030245642
%N A380176 Number of pairs of adjacent equal parts in all gap-free compositions of n.
%C A380176 An integer composition is gap-free if its set of parts covers an interval.
%F A380176 G.f.: B(x) = d/dz Sum_{j>0} Sum_{k>=j} C({j..k},x,z)|_{z=1} where C({s},x,z) = Sum_{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.
%e A380176 The gap-free compositions of n = 4 are: (4), (2,2), (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 6 pairs of equal adjacent parts giving a(4) = 6.
%o A380176 (PARI)
%o A380176 C_xz(s,N) = {my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_xz(s[^i],N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1,#s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
%o A380176 B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, for(k=0,N, h += C_xz([(1+k)..(j+k)], N+1)); j+=1); h}
%o A380176 P_xz(N) = Pol(B_xz(N), {x})
%o A380176 B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
%o A380176 B_x(20)
%Y A380176 Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.
%K A380176 nonn
%O A380176 0,4
%A A380176 _John Tyler Rascoe_, Jan 14 2025