A259598 -id:A259598 - cp's OEIS frontend

A259556 Rectangular array, read by antidiagonals: T(h,k) = u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), and h >= 1, k >= 1.

3, 6, 5, 8, 8, 6, 11, 10, 9, 8, 14, 13, 11, 11, 10, 16, 16, 14, 13, 13, 11, 19, 18, 17, 16, 15, 14, 13, 21, 21, 19, 19, 18, 16, 16, 14, 24, 23, 22, 21, 21, 19, 18, 17, 16, 27, 26, 24, 24, 23, 22, 21, 19, 19, 18, 29, 29, 27, 26, 26, 24, 24, 22, 21, 21, 19, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21

Offset: 1

Clark Kimberling, Jul 22 2015

			Northwest corner:
3    6    8    11   14   16   19
5    8    10   13   16   18   21
6    9    11   14   17   19   22
8    11   13   16   19   22   24
10   13   15   18   21   23   26
11   14   16   19   22   24   27
T(2,3) = u(2) + v(3) = 3 + 7 = 10.

Clark Kimberling, Antidiagonals n=1..60, flattened

Cf. A259598, A259600, A259601.

r = GoldenRatio; z = 12;
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}]
TableForm[t] (* A259556 array *)
Table[s[n - k + 1, k], {n, z}, {k, n, 1, -1}] // Flatten (* A259556 sequence *)

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.

Original entry on oeis.org

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

Paul D. Hanna, Nov 30 2017

nonn

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

			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, ...].

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

Cf. A259598, A000201, A001950.

{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),", "))

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.

cp's OEIS Frontend

A259556 Rectangular array, read by antidiagonals: T(h,k) = u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), and h >= 1, k >= 1.

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