A275439 Sum of the asymmetry degrees of all compositions of n with parts in {1,2}.
0, 0, 0, 2, 2, 6, 12, 22, 42, 78, 140, 252, 448, 788, 1380, 2402, 4158, 7170, 12316, 21082, 35982, 61246, 103992, 176184, 297888, 502728, 846984, 1424738, 2393114, 4014270, 6725196, 11253694, 18810930, 31410894, 52400132, 87335604, 145438624, 242001692
Offset: 0
Examples
a(5) = 6 because the compositions of 5 with parts in {1,2} are 122, 212, 221, 1112, 1121, 1211, 2111, and 11111 and the sum of their asymmetry degrees is 1 + 0 + 1 + 1 + 1 + 1 + 1 + 0 = 6.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,2,1,-2,-3,-1).
Programs
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Maple
f := n -> combinat:-fibonacci(n): a := n -> (n+1)/2-(3/2)*floor((n+2)/3)+(3/5)*(n+1)*f(n)-(1/10)*(2*n+5)*f(n+1): seq(a(n), n = 0..40); # alternative program: g := 2*z^3/((1+z+z^2)*(1-z-z^2)^2): gser := series(g, z=0, 45): seq(coeff(gser, z, n), n = 0..40);
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Mathematica
Join[{0}, Table[Total@ Map[Total, Map[BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, _} /; a > 2]], 1]]], {n, 30}]] (* Michael De Vlieger, Aug 17 2016 *) -
PARI
concat(vector(3), Vec(2*x^3/((1+x+x^2)*(1-x-x^2)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
Formula
G.f.: g(z) = 2*z^3/((1+z+z^2)*(1-z-z^2)^2). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (F(z)^2-F(z^2))/((1+F(z))*(1-F(z))^2).
a(n) = (n+1)/2-(3/2)*floor((n+2)/3)+(3/5)*(n+1)*f(n)-(1/10)*(2*n+5)*f(n+1), where f(j) = A000045(j) are the Fibonacci numbers.
a(n) = Sum_{k>=0} k*A275438(n,k).
a(n) = 2*A291408(n-4) for n>=4. - Alois P. Heinz, May 20 2025
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