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A114901
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Number of compositions of n such that each part is adjacent to an equal part.
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46
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1, 0, 1, 1, 2, 1, 5, 3, 10, 10, 21, 22, 49, 51, 105, 126, 233, 292, 529, 678, 1181, 1585, 2654, 3654, 6016, 8416, 13606, 19395, 30840, 44517, 70087, 102070, 159304, 233941, 362429, 535520, 825358, 1225117, 1880220, 2801749, 4285086, 6404354, 9769782, 14634907
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OFFSET
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0,5
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LINKS
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FORMULA
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INVERT(iMOEBIUS(iINVERT(A000012 shifted right 2 places)))
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EXAMPLE
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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.
The a(2) = 1 through a(9) = 10 compositions:
(11) (111) (22) (11111) (33) (11122) (44) (333)
(1111) (222) (22111) (1133) (11133)
(1122) (1111111) (2222) (33111)
(2211) (3311) (111222)
(111111) (11222) (222111)
(22211) (1111122)
(111122) (1112211)
(112211) (1122111)
(221111) (2211111)
(11111111) (111111111)
(End)
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MAPLE
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g:= proc(n, i) option remember; add(b(n-i*j, i), j=2..n/i) end:
b:= proc(n, l) option remember; `if`(n=0, 1,
add(`if`(i=l, 0, g(n, i)), i=1..n/2))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Min@@Length/@Split[#]>1&]], {n, 0, 10}] (* Gus Wiseman, Nov 25 2019 *)
g[n_, i_] := g[n, i] = Sum[b[n - i*j, i], {j, 2, n/i}] ;
b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[i==l, 0, g[n, i]], {i, 1, n/2}]];
a[n_] := b[n, 0];
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CROSSREFS
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Compositions with no adjacent parts equal are A003242.
Compositions with all multiplicities > 1 are A240085.
Compositions with minimum multiplicity 1 are A244164.
Compositions with at least two adjacent parts equal are A261983.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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