A295540 Number of ways of writing n as the sum of a lower Wythoff number (A000201) and an upper Wythoff number (A001950), when zero is included in both sequences.

1, 1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 5, 5, 2, 7, 3, 5, 8, 1, 9, 5, 5, 10, 2, 9, 9, 3, 12, 5, 8, 13, 1, 13, 10, 5, 15, 5, 11, 15, 2, 17, 9, 9, 18, 3, 16, 15, 5, 20, 8, 13, 21, 1, 22, 13, 10, 23, 5, 19, 20, 5, 25, 11, 15, 26, 2, 25, 19, 9, 28, 9, 20, 27, 3, 30, 16, 15, 31, 5, 27, 25, 8, 33, 13, 21, 34, 1, 34, 23, 13, 36, 10, 27, 33, 5, 38, 19, 20, 39, 5, 35, 30, 11, 41, 15, 27, 41, 2, 43, 25, 19, 44, 9, 36, 37, 9, 46, 20, 27

Offset: 0

Author

Paul D. Hanna, Nov 30 2017

Keywords

• nonn

Comments

Note that floor(n*phi) and floor(n*phi^2), for n>=1, form Beatty sequences.

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 3*x^10 + 5*x^11 + x^12 + 5*x^13 + 5*x^14 + 2*x^15 + 7*x^16 + 3*x^17 + 5*x^18 + 8*x^19 + x^20 + 9*x^21 + 5*x^22 + 5*x^23 + 10*x^24 + 2*x^25 + 9*x^26 + 9*x^27 + 3*x^28 + 12*x^29 + 5*x^30 + 8*x^31 + 13*x^32 + x^33 + 13*x^34 + 10*x^35 + 5*x^36 + 15*x^37 + 5*x^38 + 11*x^39 + 15*x^40 + 2*x^41 + 17*x^42 + 9*x^43 + 9*x^44 + 18*x^45 + 3*x^46 + 16*x^47 + 15*x^48 + 5*x^49 + 20*x^50 +...+ a(n)*x^n +...
such that A(x) = WL(x) * WU(x) where
WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 + x^14 + x^16 + x^17 + x^19 + x^21 + x^22 + x^24 + x^25 + x^27 + x^29 + x^30 +...+ x^A000201(n) +...
WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 + x^20 + x^23 + x^26 + x^28 + x^31 + x^34 + x^36 + x^39 + x^41 + x^44 + x^47 + x^49 +...+ x^A001950(n) +...
Terms equal 1 only at positions:
[0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, ..., Fibonacci(n+1)-1, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A259598, A000201, A001950.

Programs

PARI

{a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
WL = sum(m=0,floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
WU = sum(m=0,floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
polcoeff(WL*WU,n)}
for(n=0,120, print1(a(n),", "))

Formula

G.f.: [ Sum_{n>=0} x^floor(n*phi) ] * [ Sum_{n>=0} x^floor(n*phi^2) ], where phi = (1+sqrt(5))/2.

G.f.: [1 + Sum_{n>=1} x^A000201(n) ] * [1 + Sum_{n>=1} x^A001950(n) ], where A000201 and A001950 are the lower and upper Wythoff sequences, respectively.

a(Fibonacci(n+1)-1) = 1 for n>=1.

a(Fibonacci(n+2)-2) = Fibonacci(n) for n>=1.