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 A114901 #34 Jan 22 2025 05:19:12
%S A114901 1,0,1,1,2,1,5,3,10,10,21,22,49,51,105,126,233,292,529,678,1181,1585,
%T A114901 2654,3654,6016,8416,13606,19395,30840,44517,70087,102070,159304,
%U A114901 233941,362429,535520,825358,1225117,1880220,2801749,4285086,6404354,9769782,14634907
%N A114901 Number of compositions of n such that each part is adjacent to an equal part.
%H A114901 Alois P. Heinz, <a href="/A114901/b114901.txt">Table of n, a(n) for n = 0..4000</a>
%H A114901 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A114901 INVERT(iMOEBIUS(iINVERT(A000012 shifted right 2 places)))
%F A114901 G.f.: A(x,1) is the k = 1 case of A(x,k) = 1/(1 - Sum_{i>0} ( (Sum_{j>k} x^(i*j))/(1 + Sum_{j>k} x^(i*j)) )) where A(x,k) is the g.f. for compositions of n with all run-lengths > k. - _John Tyler Rascoe_, May 16 2024
%e A114901 The 5 compositions of 6 are 3+3, 2+2+2, 2+2+1+1, 1+1+2+2, 1+1+1+1+1+1.
%e A114901 From _Gus Wiseman_, Nov 25 2019: (Start)
%e A114901 The a(2) = 1 through a(9) = 10 compositions:
%e A114901 (11) (111) (22) (11111) (33) (11122) (44) (333)
%e A114901 (1111) (222) (22111) (1133) (11133)
%e A114901 (1122) (1111111) (2222) (33111)
%e A114901 (2211) (3311) (111222)
%e A114901 (111111) (11222) (222111)
%e A114901 (22211) (1111122)
%e A114901 (111122) (1112211)
%e A114901 (112211) (1122111)
%e A114901 (221111) (2211111)
%e A114901 (11111111) (111111111)
%e A114901 (End)
%p A114901 g:= proc(n, i) option remember; add(b(n-i*j, i), j=2..n/i) end:
%p A114901 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A114901 add(`if`(i=l, 0, g(n,i)), i=1..n/2))
%p A114901 end:
%p A114901 a:= n-> b(n, 0):
%p A114901 seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 29 2019
%t A114901 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Min@@Length/@Split[#]>1&]],{n,0,10}] (* _Gus Wiseman_, Nov 25 2019 *)
%t A114901 g[n_, i_] := g[n, i] = Sum[b[n - i*j, i], {j, 2, n/i}] ;
%t A114901 b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[i==l, 0, g[n, i]], {i, 1, n/2}]];
%t A114901 a[n_] := b[n, 0];
%t A114901 a /@ Range[0, 50] (* _Jean-François Alcover_, May 23 2021, after _Alois P. Heinz_ *)
%o A114901 (PARI)
%o A114901 A_x(N,k) = { my(x='x+O('x^N), g=1/(1-sum(i=1,N,sum(j=k+1,N, x^(i*j))/(1+ sum(j=k+1,N, x^(i*j)))))); Vec(g)}
%o A114901 A_x(50,1) \\ _John Tyler Rascoe_, May 17 2024
%Y A114901 The case of partitions is A007690.
%Y A114901 Compositions with no adjacent parts equal are A003242.
%Y A114901 Compositions with all multiplicities > 1 are A240085.
%Y A114901 Compositions with minimum multiplicity 1 are A244164.
%Y A114901 Compositions with at least two adjacent parts equal are A261983.
%Y A114901 Cf. A178470, A238130, A274174, A329863.
%K A114901 nonn
%O A114901 0,5
%A A114901 _Christian G. Bower_, Jan 05 2006