A025120 - OEIS

%I #6 Sep 14 2022 20:40:30

%S 0,0,1,3,4,0,0,0,1,3,4,6,8,9,11,12,14,0,1,3,4,6,8,9,11,12,14,16,17,19,

%T 21,22,24,26,30,33,36,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,

%U 33,36,40,42,46,50,52,56,58,62,66,68,72,76,22,24,25,27,29,30,32,33,35,37,38,40,42,43,45

%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000201 (lower Wythoff sequence), t = A023533.

%H G. C. Greubel, <a href="/A025120/b025120.txt">Table of n, a(n) for n = 1..5000</a>

%t b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];

%t A025120[n_]:= A025120[n]= Sum[Floor[(n-j+2)*GoldenRatio]*b[j], {j, Floor[(n+4)/2], n+1}];

%t Table[A025120[n], {n,100}] (* _G. C. Greubel_, Sep 14 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o A025120:= func< n | (&+[Floor(k*(1+Sqrt(5))/2)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;

%o [A025120(n): n in [1..100]]; // _G. C. Greubel_, Sep 14 2022

%o (SageMath)

%o @CachedFunction

%o def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..13))

%o @CachedFunction

%o def A025120(n): return sum(floor((n-j+2)*golden_ratio)*b(j) for j in (((n+4)//2)..n+1))

%o [A025120(n) for n in (1..100)] # _G. C. Greubel_, Sep 14 2022

%Y Cf. A000201, A023533.

%K nonn

%O 1,4

%A _Clark Kimberling_