A259598 - OEIS

%I #14 Dec 02 2017 07:51:27

%S 0,0,1,0,1,2,0,3,1,2,4,0,4,4,1,6,2,4,7,0,8,4,4,9,1,8,8,2,11,4,7,12,0,

%T 12,9,4,14,4,10,14,1,16,8,8,17,2,15,14,4,19,7,12,20,0,21,12,9,22,4,18,

%U 19,4,24,10,14,25,1,24,18,8,27,8,19,26,2,29,15

%N Number of representations of n as u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), h>=1, k>=1.

%C Three conjectures. The numbers that are not a sum u(h) + v(k) are (1,2,4,7,12, ...) = A000071 = -1 + Fibonacci numbers. The numbers that have exactly one such representation are (3, 5, 9, 15, 25, 41, ...) = A001595. The numbers that have exactly two such representations are (6, 10, 17, 28, 46, ...) = A001610.

%F G.f.: [Sum_{n>=1} x^floor(n*phi)] * [Sum_{n>=1} x^floor(n*phi^2)], where phi = (1+sqrt(5))/2. - _Paul D. Hanna_, Dec 02 2017

%F G.f.: [Sum_{n>=1} x^A000201(n)] * [Sum_{n>=1} x^A001950(n)], where A000201 and A001950 are the lower and upper Wythoff sequences, respectively. - _Paul D. Hanna_, Dec 02 2017

%t r = GoldenRatio; z = 500;

%t u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];

%t s[m_, n_] := s[m, n] = u[m] + v[n]; t = Table[s[m, n], {m, 1, z}, {n, 1, z}];

%t w = Flatten[Table[Count[Flatten[t], n], {n, 1, z/5}]]  (* A259598 *)

%o (PARI) {a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);

%o WL = sum(m=1, floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));

%o WU = sum(m=1, floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));

%o polcoeff(WL*WU, n)}

%o for(n=1, 120, print1(a(n), ", ")) \\ _Paul D. Hanna_, Dec 02 2017

%Y Cf. A259556, A000071, A001595, A001610, A295540.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Jul 22 2015