|
| |
|
|
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.
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Note that floor(n*phi) and floor(n*phi^2), for n>=1, form Beatty sequences.
|
|
|
LINKS
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
