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

0,4

Comments

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

Links

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

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.

Example

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, ...].

Prog

(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), ", "))

Crossrefs

Cf. A259598, A000201, A001950.

Sequence in context: A194961 A195110 A167198 * A133299 A286537 A132163

Adjacent sequences: A295537 A295538 A295539 * A295541 A295542 A295543

Keyword

nonn

Author

Paul D. Hanna, Nov 30 2017

Status

approved