cp's OEIS Frontend

A024327 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor( (n+1)/2 ), s = A023531, t = A014306.

%I A024327 #18 Feb 18 2022 02:01:36
%S A024327 0,0,1,1,0,1,1,1,2,2,1,2,2,1,2,2,3,2,3,3,2,3,3,2,3,3,4,3,4,4,4,4,3,4,
%T A024327 4,3,4,4,3,5,5,5,4,5,5,5,5,4,5,5,5,5,6,5,6,6,5,6,6,5,5,6,6,5,6,6,6,6,
%U A024327 5,7,7,7,7,7,6,7,7,7,7,7,7,6,7,7,6,7,8,7,8,7
%N A024327 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor( (n+1)/2 ), s = A023531, t = A014306.
%H A024327 G. C. Greubel, <a href="/A024327/b024327.txt">Table of n, a(n) for n = 1..1000</a>
%F A024327 a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A014306(n-k+1). - _G. C. Greubel_, Feb 17 2022
%t A024327 A014306:= With[{ms= Table[m(m+1)(m+2)/6, {m,0,20}]}, Table[If[MemberQ[ms, n], 0, 1], {n,0,150}]];
%t A024327 Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A014306[[n+1]]; n -= m++]; t, {n, 120}] (* _G. C. Greubel_, Feb 17 2022 *)
%o A024327 (Sage)
%o A024327 nmax=120
%o A024327 @CachedFunction
%o A024327 def b_list(N):
%o A024327     A = []
%o A024327     for m in range(ceil((6*N)^(1/3))):
%o A024327         A.extend([0]*(binomial(m+2, 3) - len(A)) + [1])
%o A024327     return A
%o A024327 A023533 = b_list(nmax+5)
%o A024327 def A014306(n): return 1 - A023533[n]
%o A024327 def b(n, j): return A014306(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
%o A024327 @CachedFunction
%o A024327 def A024327(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )
%o A024327 [A024327(n) for n in (1..nmax)] # _G. C. Greubel_, Feb 17 2022
%Y A024327 Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024328.
%Y A024327 Cf. A014306, A023531.
%K A024327 nonn
%O A024327 1,9
%A A024327 _Clark Kimberling_
%E A024327 Title corrected by _Sean A. Irvine_, Jun 30 2019