cp's OEIS Frontend

A024324 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 = A000201 (lower Wythoff sequence).

%I A024324 #20 Feb 06 2022 02:16:41
%S A024324 0,0,3,4,6,8,9,11,20,23,27,29,33,37,39,43,60,65,70,74,80,84,89,94,98,
%T A024324 104,131,137,143,150,157,163,169,176,183,189,195,202,241,248,256,265,
%U A024324 272,281,289,296,306,313,321,329,337,346,397,406,416,425,436,445,454,466,474,484
%N A024324 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 = A000201 (lower Wythoff sequence).
%H A024324 G. C. Greubel, <a href="/A024324/b024324.txt">Table of n, a(n) for n = 1..1000</a>
%F A024324 a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000201(n-j+1).
%t A024324 Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Floor[n*GoldenRatio^2]; n -= m++]; t, {n, 120}] (* _G. C. Greubel_, Jan 28 2022 *)
%o A024324 (Magma)
%o A024324 b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;
%o A024324 A024324:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;
%o A024324 [A024324(n) : n in [1..80]]; // _G. C. Greubel_, Jan 28 2022
%o A024324 (Sage)
%o A024324 def b(n,j): return floor( (n+1-j)*(1+sqrt(5))/2 ) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
%o A024324 def A024324(n): return sum( b(n,k) for k in (1..((n+1)//2)) )
%o A024324 [A024324(n) for n in (1..80)] # _G. C. Greubel_, Jan 28 2022
%o A024324 (PARI) my(phi=quadgen(5)); a(n) = my(L=n>>1,m=2,ret=0); n--; while(n>L, ret += floor(n*phi); n-=(m++)); ret; \\ _Kevin Ryde_, Feb 03 2022
%Y A024324 Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024325, A024326, A024327.
%Y A024324 Cf. A000201, A023531.
%K A024324 nonn
%O A024324 1,3
%A A024324 _Clark Kimberling_
%E A024324 a(62) corrected by _Sean A. Irvine_, Jun 27 2019