cp's OEIS Frontend

A025123 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.

%I A025123 #6 Sep 14 2022 20:40:57
%S A025123 0,0,2,5,7,0,0,0,2,5,7,10,13,15,18,20,23,0,2,5,7,10,13,15,18,20,23,26,
%T A025123 28,31,34,36,39,43,49,54,59,13,15,18,20,23,26,28,31,34,36,39,41,44,47,
%U A025123 49,52,54,59,65,69,75,81,85,91,95,101,107,111,117,123,36,39,41,44,47,49,52,54,57,60,62
%N A025123 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.
%H A025123 G. C. Greubel, <a href="/A025123/b025123.txt">Table of n, a(n) for n = 1..5000</a>
%t A025123 b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
%t A025123 A025123[n_]:= A025123[n]= Sum[Floor[(n-j+2)*GoldenRatio^2]*b[j], {j, Floor[(n+4)/2], n+1}];
%t A025123 Table[A025123[n], {n,100}] (* _G. C. Greubel_, Sep 14 2022 *)
%o A025123 (Magma)
%o A025123 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A025123 A025123:= func< n | (&+[Floor(k*(3+Sqrt(5))/2)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
%o A025123 [A025123(n): n in [1..100]]; // _G. C. Greubel_, Sep 14 2022
%o A025123 (SageMath)
%o A025123 @CachedFunction
%o A025123 def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..13))
%o A025123 @CachedFunction
%o A025123 def A025123(n): return sum(floor((n-j+2)*golden_ratio^2)*b(j) for j in (((n+4)//2)..n+1))
%o A025123 [A025123(n) for n in (1..100)] # _G. C. Greubel_, Sep 14 2022
%Y A025123 Cf. A001950, A023533.
%K A025123 nonn
%O A025123 1,3
%A A025123 _Clark Kimberling_