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 A329863 #20 Apr 12 2025 06:31:35
%S A329863 0,0,1,0,3,6,9,22,47,88,179,354,691,1344,2617,5042,9709,18632,35639,
%T A329863 68010,129556,246202,467188,885036,1674211,3163094,5969022,11251676,
%U A329863 21189382,39867970,74950464,140798302,264313039,495861874,929709687,1742193854,3263069271,6108762316
%N A329863 Number of compositions of n with cuts-resistance 2.
%C A329863 A composition of n is a finite sequence of positive integers summing to n.
%C A329863 For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.
%H A329863 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%F A329863 G.f.: -1 + (1 + Ca(x) * Sum_{m>0} x^(2*m) * (Ca(x)-1)/(1 + x^m * (2 + x^m * (1+Ca(x)))))/(1 - Ca(x) * Sum_{m>0} x^(2*m)/(1 + x^m * (2 + x^m * (1+Ca(x))))) where Ca(x) is the g.f. for A003242. - _John Tyler Rascoe_, Feb 20 2025
%e A329863 The a(2) = 1 through a(7) = 22 compositions (empty column not shown):
%e A329863 (1,1) (2,2) (1,1,3) (3,3) (1,1,5)
%e A329863 (1,1,2) (1,2,2) (1,1,4) (1,3,3)
%e A329863 (2,1,1) (2,2,1) (4,1,1) (2,2,3)
%e A329863 (3,1,1) (1,1,2,2) (3,2,2)
%e A329863 (1,1,2,1) (1,1,3,1) (3,3,1)
%e A329863 (1,2,1,1) (1,2,2,1) (5,1,1)
%e A329863 (1,3,1,1) (1,1,2,3)
%e A329863 (2,1,1,2) (1,1,3,2)
%e A329863 (2,2,1,1) (1,1,4,1)
%e A329863 (1,4,1,1)
%e A329863 (2,1,1,3)
%e A329863 (2,1,2,2)
%e A329863 (2,2,1,2)
%e A329863 (2,3,1,1)
%e A329863 (3,1,1,2)
%e A329863 (3,2,1,1)
%e A329863 (1,1,2,1,2)
%e A329863 (1,1,2,2,1)
%e A329863 (1,2,1,1,2)
%e A329863 (1,2,2,1,1)
%e A329863 (2,1,1,2,1)
%e A329863 (2,1,2,1,1)
%t A329863 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329863 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]==2&]],{n,0,10}]
%o A329863 (PARI)
%o A329863 Ca(N) = {1/(1-sum(k=1, N, x^k/(1+x^k)))}
%o A329863 A_x(N) = {my(x='x+O('x^N)); concat([0,0],Vec(-1+(1+sum(m=1,N, Ca(N)*x^(2*m)*(Ca(N)-1)/(1+x^m*(2+x^m*(1+Ca(N))))))/(1-sum(m=1,N, Ca(N)*x^(2*m)/(1+x^m*(2+x^m*(1+Ca(N))))))))}
%o A329863 A_x(38) \\ _John Tyler Rascoe_, Feb 20 2025
%Y A329863 Column k = 2 of A329861.
%Y A329863 Compositions with cuts-resistance 1 are A003242.
%Y A329863 Compositions with runs-resistance 2 are A329745.
%Y A329863 Numbers whose binary expansion has cuts-resistance 2 are A329862.
%Y A329863 Binary words with cuts-resistance 2 are conjectured to be A027383.
%Y A329863 Cuts-resistance of binary expansion is A319416.
%Y A329863 Binary words counted by cuts-resistance are A319421 and A329860.
%Y A329863 Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.
%K A329863 nonn
%O A329863 0,5
%A A329863 _Gus Wiseman_, Nov 23 2019
%E A329863 a(21) onwards from _John Tyler Rascoe_, Feb 20 2025