A259598 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.

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

Offset: 1

Author

Clark Kimberling, Jul 22 2015

Keywords

• nonn
• easy

Comments

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.

Crossrefs

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

Programs

Mathematica

r = GoldenRatio; z = 500;
u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := s[m, n] = u[m] + v[n]; t = Table[s[m, n], {m, 1, z}, {n, 1, z}];
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z/5}]]  (* A259598 *)

PARI

{a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
WL = sum(m=1, floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
WU = sum(m=1, floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
polcoeff(WL*WU, n)}
for(n=1, 120, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 02 2017

Formula

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

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