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A188541 a(n) = 2 * A079500(n) - A079500(n+1).

%I A188541 #50 Jan 04 2024 19:48:43
%S A188541 1,0,1,1,2,2,4,5,9,14,24,40,70,120,211,371,658,1172,2102,3786,6856,
%T A188541 12470,22782,41789,76947,142180,263578,490104,913858,1708386,3201290,
%U A188541 6011962,11313274,21329276,40282947,76202831,144370582,273906268,520359324,989804122,1884992934,3593832942,6859139352,13104584156,25061042050,47971076906,91906883496
%N A188541 a(n) = 2 * A079500(n) - A079500(n+1).
%C A188541 Arises in studying a conjecture related to lunar divisors in base 2.
%C A188541 a(n) is the number of compositions of n where the first part is even, say, 2*f and the other parts are <= f. - _Joerg Arndt_, Jan 04 2024
%H A188541 Alois P. Heinz, <a href="/A188541/b188541.txt">Table of n, a(n) for n = 0..3343</a>
%H A188541 D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="http://arxiv.org/abs/1107.1130">Dismal Arithmetic</a>, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
%F A188541 G.f.: Sum_{n>=0} x^(2*n)/(1 - Sum_{k=1..n} x^k). - _Joerg Arndt_, Jan 04 2024
%e A188541 From _Joerg Arndt_, Jan 04 2024: (Start)
%e A188541 There are a(10) = 24 compositions of 10 of the specified type:
%e A188541    1:  [ 2 1 1 1 1 1 1 1 1 ]
%e A188541    2:  [ 4 1 1 1 1 1 1 ]
%e A188541    3:  [ 4 1 1 1 1 2 ]
%e A188541    4:  [ 4 1 1 1 2 1 ]
%e A188541    5:  [ 4 1 1 2 1 1 ]
%e A188541    6:  [ 4 1 1 2 2 ]
%e A188541    7:  [ 4 1 2 1 1 1 ]
%e A188541    8:  [ 4 1 2 1 2 ]
%e A188541    9:  [ 4 1 2 2 1 ]
%e A188541   10:  [ 4 2 1 1 1 1 ]
%e A188541   11:  [ 4 2 1 1 2 ]
%e A188541   12:  [ 4 2 1 2 1 ]
%e A188541   13:  [ 4 2 2 1 1 ]
%e A188541   14:  [ 4 2 2 2 ]
%e A188541   15:  [ 6 1 1 1 1 ]
%e A188541   16:  [ 6 1 1 2 ]
%e A188541   17:  [ 6 1 2 1 ]
%e A188541   18:  [ 6 1 3 ]
%e A188541   19:  [ 6 2 1 1 ]
%e A188541   20:  [ 6 2 2 ]
%e A188541   21:  [ 6 3 1 ]
%e A188541   22:  [ 8 1 1 ]
%e A188541   23:  [ 8 2 ]
%e A188541   24:  [ 10 ]
%e A188541 (End)
%p A188541 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A188541       `if`(m=0, add(b(n-j, j), j=1..n),
%p A188541       add(b(n-j, min(n-j, m)), j=1..min(n, m))))
%p A188541     end:
%p A188541 a:= n-> 2*b(n, 0)-b(n+1, 0):
%p A188541 seq(a(n), n=0..46);  # _Alois P. Heinz_, Jan 04 2024
%t A188541 b[n_, m_] := b[n, m] = If[n == 0, 1, If[m == 0, Sum[b[n-j, j], {j, 1, n}], Sum[b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]];
%t A188541 a79500[n_] := b[n, 0];
%t A188541 a[n_] := -a79500[n+1] + 2 a79500[n];
%t A188541 Table[a[n], {n, 0, 48}] (* _Jean-François Alcover_, Sep 15 2018, after _Alois P. Heinz_ in A079500 *)
%o A188541 (SageMath)
%o A188541 def C(n): return sum(Compositions(n, max_part=k, inner=[k]).cardinality()
%o A188541                  for k in (0..n))
%o A188541 def a(n): return 2*C(n) - C(n+1) if n > 0 else 1
%o A188541 print([a(n) for n in (0..18)])  # _Peter Luschny_, Jan 04 2024
%Y A188541 Cf. A079500.
%K A188541 nonn
%O A188541 0,5
%A A188541 _N. J. A. Sloane_, Apr 03 2011
%E A188541 Offset changed to 0 by _N. J. A. Sloane_, Jan 04 2024 at the suggestion of _Joerg Arndt_.