A259556 -id:A259556 - cp's OEIS frontend

A259600 Triangular array: sums of two distinct lower Wythoff numbers.

4, 5, 7, 7, 9, 10, 9, 11, 12, 14, 10, 12, 13, 15, 17, 12, 14, 15, 17, 19, 20, 13, 15, 16, 18, 20, 21, 23, 15, 17, 18, 20, 22, 23, 25, 26, 17, 19, 20, 22, 24, 25, 27, 28, 30, 18, 20, 21, 23, 25, 26, 28, 29, 31, 33, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36

Offset: 2

Clark Kimberling, Jul 22 2015

Row n shows the numbers u(m) + u(n), where u = A000201 (lower Wythoff sequence), for m=1..n-1, for n >= 2. (The offset is 2, so that the top row is counted as row 2.)

			10 = 4 + 6 = u(3) + u(4), so that 10 appears as the final term in row 4. (The offset is 2, so that the top row is counted as row 2.) Rows 2 to 9:
4
5    7
7    9    10
9    11   12   14
10   12   13   15   17
12   14   15   17   19   20
13   15   16   18   20   21   23
15   17   18   20   22   23   25   26

Cf. A259556, A259601.

r = GoldenRatio; z = 20; u[n_] := u[n] = Floor[n*r];
s[m_, n_] := u[m] + u[n]; t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}];
TableForm[t]  (* A259600 array *)
Flatten[t]  (* A259600 sequence *)

tabl(nn) = {r=(sqrt(5)+1)/2; for (n=2, nn, for (k=1, n-1, print1(floor(n*r) + floor(k*r), ", ");); print(););} \\ Michel Marcus, Jul 30 2015

A259601 Triangular array: sums of two distinct upper Wythoff numbers.

Original entry on oeis.org

7, 9, 12, 12, 15, 17, 15, 18, 20, 23, 17, 20, 22, 25, 28, 20, 23, 25, 28, 31, 33, 22, 25, 27, 30, 33, 35, 38, 25, 28, 30, 33, 36, 38, 41, 43, 28, 31, 33, 36, 39, 41, 44, 46, 49, 30, 33, 35, 38, 41, 43, 46, 48, 51, 54, 33, 36, 38, 41, 44, 46, 49, 51, 54, 57

Offset: 2

Clark Kimberling, Jul 22 2015

Row n shows the numbers v(m) + v(n), where v = A001950 (upper Wythoff sequence), for m=1..n-1, for n >= 2. (The offset is 2, so that the top row is counted as row 2.)

			17 = 7 + 10 = v(3) + v(4), so that 17 appears as the final term in row 4. (The offset is 2, so that the top row is counted as row 2.) Rows 2 to 9:
7
9    12
12   15   17
15   18   20   23
17   20   22   25   28
20   23   25   28   31   33
22   25   27   30   33   35   38
25   28   30   33   36   38   41   43

Cf. A000045, A001950, A259556, A259600.

r = GoldenRatio; z = 13; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := v[m] + v[n]; t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]
TableForm[t] (* A259601 array *)
Flatten[t]   (* A259601 sequence *)

tabl(nn) = {r=(sqrt(5)+1)/2; for (n=2, nn, for (k=1, n-1, print1(floor(n*r^2) + floor(k*r^2), ", ");); print(););} \\ Michel Marcus, Jul 30 2015

A259598 Number of representations of n as u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), h>=1, k>=1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 3, 1, 2, 4, 0, 4, 4, 1, 6, 2, 4, 7, 0, 8, 4, 4, 9, 1, 8, 8, 2, 11, 4, 7, 12, 0, 12, 9, 4, 14, 4, 10, 14, 1, 16, 8, 8, 17, 2, 15, 14, 4, 19, 7, 12, 20, 0, 21, 12, 9, 22, 4, 18, 19, 4, 24, 10, 14, 25, 1, 24, 18, 8, 27, 8, 19, 26, 2, 29, 15

Offset: 1

Clark Kimberling, Jul 22 2015

Three conjectures. The numbers that are not a sum u(h) + v(k) are (1,2,4,7,12, ...) = A000071 = -1 + Fibonacci numbers. The numbers that have exactly one such representation are (3, 5, 9, 15, 25, 41, ...) = A001595. The numbers that have exactly two such representations are (6, 10, 17, 28, 46, ...) = A001610.

Cf. A259556, A000071, A001595, A001610, A295540.

r = GoldenRatio; z = 500;
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}];
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z/5}]]  (* A259598 *)

{a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
WL = sum(m=1, floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
WU = sum(m=1, floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
polcoeff(WL*WU, n)}
for(n=1, 120, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 02 2017

G.f.: [Sum_{n>=1} x^floor(n*phi)] * [Sum_{n>=1} x^floor(n*phi^2)], where phi = (1+sqrt(5))/2. - Paul D. Hanna, Dec 02 2017

G.f.: [Sum_{n>=1} x^A000201(n)] * [Sum_{n>=1} x^A001950(n)], where A000201 and A001950 are the lower and upper Wythoff sequences, respectively. - Paul D. Hanna, Dec 02 2017

A260311 Difference sequence of A260317.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 5, 3, 2, 3, 5, 3, 2, 3, 5, 3, 5, 3, 2, 3, 5, 3, 5, 3, 2, 3, 5, 3, 5, 5, 3, 5, 3, 2, 3, 5, 3, 5, 5, 3, 5, 3, 2, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3, 2, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3

Offset: 1

Clark Kimberling, Jul 31 2015

Conjecture: a(n) is a Fibonacci number (A000045) for every n.
In fact, a(n) is in {1,2,3,5}; proved with the Walnut theorem-prover. - Jeffrey Shallit, Oct 12 2022
Comment from Jonathan F. P. Grube and Benjamin Mason, Oct 10 2024 [Corrected by Jonathan F. P. Grube, Nov 05 2024]: (Start)
With an offset of 41, the sequence is of the form 2[c_{1,1},...,c_{1,n_1}]2[c_{2,1},...,c_{2,n_2}]2[c_{3,1},...,c_{3,n_3}]2..., where [c_{i,1},...,c_{i,n_i}] is the word 35(355)^{c_{i,1}}35(355)^{c_{i,2}}35...35(355)^{c_{i,n_i}}353 over the alphabet {3,5} for some nonnegative integers c_{i,j}. Furthermore c_{i,j} is in {1,2}. Proved with Walnut, the theorem-prover.
Conjectures: n_{2k-1} = n_{2k} and c_{2k-1, j} = c_{2k, j} for all positive k and 0 0} is the Fibonacci sequence. (End)

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025.
The Walnut code at https://cs.uwaterloo.ca/~shallit/oeis-walnut.txt proves the conjecture. Walnut itself can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.

Cf. A259556, A259600, A259601, A260317, A000045.

r = GoldenRatio; z = 1060;
u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := v[m] + v[n];
t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
p0 = Flatten[Position[w, 0]]  (* A260317 *)
d = Differences[p0] (* A260311 *)

A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 14, 16, 19, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 50, 53, 55, 58, 63, 66, 68, 71, 76, 79, 84, 87, 89, 92, 97, 100, 105, 108, 110, 113, 118, 121, 126, 131, 134, 139, 142, 144, 147, 152, 155, 160, 165, 168, 173, 176, 178, 181

Offset: 1

Clark Kimberling, Jul 22 2015

It appears that the difference sequence consists entirely of Fibonacci numbers (A000045); see A260311.

In fact, the difference sequence consists only of the numbers 1,2,3,5. Proved with the Walnut theorem-prover. - Jeffrey Shallit, Oct 12 2022

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025. See pp. 2-3.

Cf. A000045, A259556, A259600, A259601, A260311.

r = GoldenRatio; z = 1060;
u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := v[m] + v[n];
t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
p0 = Flatten[Position[w, 0]]  (* A260317 *)
d = Differences[p0] (* A260311 *)

n <= a(n) < 5n, see Shallit comment. - Charles R Greathouse IV, Nov 22 2024

cp's OEIS Frontend

A259600 Triangular array: sums of two distinct lower Wythoff numbers.

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A259601 Triangular array: sums of two distinct upper Wythoff numbers.

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A259598 Number of representations of n as u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), h>=1, k>=1.

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A260311 Difference sequence of A260317.

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A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.

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