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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.

%I A295540 #18 Dec 02 2017 07:44:39
%S A295540 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,
%T A295540 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,
%U A295540 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
%N 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.
%C A295540 Note that floor(n*phi) and floor(n*phi^2), for n>=1, form Beatty sequences.
%H A295540 Paul D. Hanna, <a href="/A295540/b295540.txt">Table of n, a(n) for n = 0..10000</a>
%F A295540 G.f.: [ Sum_{n>=0} x^floor(n*phi) ] * [ Sum_{n>=0} x^floor(n*phi^2) ], where phi = (1+sqrt(5))/2.
%F A295540 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.
%F A295540 a(Fibonacci(n+1)-1) = 1 for n>=1.
%F A295540 a(Fibonacci(n+2)-2) = Fibonacci(n) for n>=1.
%e A295540 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 +...
%e A295540 such that A(x) = WL(x) * WU(x) where
%e A295540 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) +...
%e A295540 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) +...
%e A295540 Terms equal 1 only at positions:
%e A295540 [0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, ..., Fibonacci(n+1)-1, ...].
%o A295540 (PARI) {a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
%o A295540 WL = sum(m=0,floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
%o A295540 WU = sum(m=0,floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
%o A295540 polcoeff(WL*WU,n)}
%o A295540 for(n=0,120, print1(a(n),", "))
%Y A295540 Cf. A259598, A000201, A001950.
%K A295540 nonn
%O A295540 0,4
%A A295540 _Paul D. Hanna_, Nov 30 2017