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 A379836 #43 Feb 05 2025 22:21:02
%S A379836 0,0,1,2,5,12,23,54,118,258,550,1178,2540,5394,11473,24174,51021,
%T A379836 107210,225099,471322,985202,2055542,4281847,8906676,18500425,
%U A379836 38379246,79516158,164561560,340179441,702506576,1449311429,2987297778,6151964642,12658841766,26027603925
%N A379836 Number of pairs of adjacent equal parts in all complete compositions of n.
%C A379836 An integer composition is complete if its set of parts covers an initial interval.
%F A379836 G.f.: B(x) = d/dz Sum_{k>0} C({1..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 A379836 The complete compositions of n = 4 are: (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 5 pairs of equal adjacent parts giving a(4) = 5.
%o A379836 (PARI)
%o A379836 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 A379836 B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += C_xz(vector(j, i, i), N+1); j+=1); h}
%o A379836 P_xz(N) = Pol(B_xz(N), {x})
%o A379836 B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
%o A379836 B_x(20)
%Y A379836 Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.
%K A379836 nonn
%O A379836 0,4
%A A379836 _John Tyler Rascoe_, Jan 14 2025