A001950 -id:A001950 - cp's OEIS frontend

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.

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

A283767 Numbers k such that U(k) is even, where U = A001950 = upper Wythoff sequence.

Original entry on oeis.org

1, 4, 7, 8, 10, 11, 13, 14, 17, 20, 21, 23, 24, 26, 27, 30, 33, 36, 37, 39, 40, 43, 46, 49, 50, 52, 53, 56, 59, 62, 63, 65, 66, 68, 69, 72, 75, 76, 78, 79, 81, 82, 85, 88, 91, 92, 94, 95, 98, 101, 104, 105, 107, 108, 111, 114, 117, 118, 120, 121, 123, 124

Offset: 1

Clark Kimberling, Mar 17 2017

Complement of A283768.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A001950, A001622, A283765, A283768.

r = GoldenRatio^2; z = 250; t = Table[Floor[n*r], {n, 1, z}]; u = Mod[t, 2];
Flatten[Position[u, 0]]  (* A283767 *)
Flatten[Position[u, 1]]  (* A283768 *)

a(n+1) - a(n) is in {1,2,3} for every n.

A283768 Numbers k such that U(k) is odd, where U = A001950 = upper Wythoff sequence.

Original entry on oeis.org

2, 3, 5, 6, 9, 12, 15, 16, 18, 19, 22, 25, 28, 29, 31, 32, 34, 35, 38, 41, 42, 44, 45, 47, 48, 51, 54, 55, 57, 58, 60, 61, 64, 67, 70, 71, 73, 74, 77, 80, 83, 84, 86, 87, 89, 90, 93, 96, 97, 99, 100, 102, 103, 106, 109, 110, 112, 113, 115, 116, 119, 122, 125

Offset: 1

Clark Kimberling, Mar 17 2017

Complement of A283767.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A001950, A001622, A283765, A283767.

r = GoldenRatio^2; z = 250; t = Table[Floor[n*r], {n, 1, z}]; u = Mod[t, 2];
Flatten[Position[u, 0]]  (* A283767 *)
Flatten[Position[u, 1]]  (* A283768 *)

a(n+1) - a(n) is in {1,2,3} for every n.

A283772 Numbers k such that U(k) = 0 mod 3, where U = A001950 = upper Wythoff sequence.

Original entry on oeis.org

6, 7, 14, 15, 21, 22, 23, 29, 30, 31, 37, 38, 39, 45, 46, 47, 53, 54, 61, 62, 69, 70, 76, 77, 78, 84, 85, 86, 92, 93, 94, 100, 101, 102, 108, 109, 116, 117, 124, 125, 131, 132, 133, 139, 140, 141, 147, 148, 149, 155, 156, 157, 163, 164, 171, 172, 179, 180

Offset: 1

Clark Kimberling, Mar 18 2017

The sequences A283772, A283773, A283774 partition the positive integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000201, A001622, A283773, A283774.

r = GoldenRatio^2; z = 350; t = Table[Floor[n*r], {n, 1, z}]; u = Mod[t, 3];
Flatten[Position[u, 0]]  (* A283772 *)
Flatten[Position[u, 1]]  (* A283773 *)
Flatten[Position[u, 2]]  (* A283774 *)

r = (3 + sqrt(5))/2;
for(n=1, 351, if(floor(n*r)%3==0, print1(n,", "))) \\ Indranil Ghosh, Mar 19 2017

import math
from sympy import sqrt
r = (3 + sqrt(5))/2
[n for n in range(1, 351) if int(math.floor(n*r))%3==0] # Indranil Ghosh, Mar 19 2017

a(n+1) - a(n) is in {1,6,7} for every n.

A283773 Numbers k such that U(k) = 1 mod 3, where U = A001950 = upper Wythoff sequence.

Original entry on oeis.org

3, 4, 5, 11, 12, 13, 19, 20, 27, 28, 35, 36, 42, 43, 44, 50, 51, 52, 58, 59, 60, 66, 67, 68, 74, 75, 82, 83, 90, 91, 97, 98, 99, 105, 106, 107, 113, 114, 115, 121, 122, 123, 129, 130, 137, 138, 144, 145, 146, 152, 153, 154, 160, 161, 162, 168, 169, 170, 176

Offset: 1

Clark Kimberling, Mar 18 2017

The sequences A283772, A283773, A283774 partition the positive integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000201, A001622, A283772, A283774.

r = GoldenRatio^2; z = 350; t = Table[Floor[n*r], {n, 1, z}]; u = Mod[t, 3];
Flatten[Position[u, 0]]  (* A283772 *)
Flatten[Position[u, 1]]  (* A283773 *)
Flatten[Position[u, 2]]  (* A283774 *)

r = (3 + sqrt(5))/2;
for(n=1, 351, if(floor(n*r)%3==1, print1(n,", "))) \\ Indranil Ghosh, Mar 19 2017

import math
from sympy import sqrt
r = (3 + sqrt(5))/2
[n for n in range(1, 351) if int(math.floor(n*r))%3==1] # Indranil Ghosh, Mar 19 2017

a(n+1) - a(n) is in {1,6,7} for every n.

A283774 Numbers k such that U(k) == 2 mod 3, where U = A001950 = upper Wythoff sequence.

Original entry on oeis.org

1, 2, 8, 9, 10, 16, 17, 18, 24, 25, 26, 32, 33, 34, 40, 41, 48, 49, 55, 56, 57, 63, 64, 65, 71, 72, 73, 79, 80, 81, 87, 88, 89, 95, 96, 103, 104, 110, 111, 112, 118, 119, 120, 126, 127, 128, 134, 135, 136, 142, 143, 150, 151, 158, 159, 165, 166, 167, 173

Offset: 1

Clark Kimberling, Mar 19 2017

The sequences A283772, A283773, A283774 partition the positive integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000201, A001622, A283772, A283774.

r = GoldenRatio^2; z = 350; t = Table[Floor[n*r], {n, 1, z}]; u = Mod[t, 3];
Flatten[Position[u, 0]]  (* A283772 *)
Flatten[Position[u, 1]]  (* A283773 *)
Flatten[Position[u, 2]]  (* A283774 *)

r = (3 + sqrt(5))/2;
for(n=1, 351, if(floor(n*r)%3==2, print1(n, ", "))) \\ Indranil Ghosh, Mar 21 2017

import math
from sympy import sqrt
r = (3 + sqrt(5))/2
[n for n in range(1, 351) if int(math.floor(n*r))%3==2] # Indranil Ghosh, Mar 21 2017

a(n+1) - a(n) is in {1,6,7} for every n.

A289926 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S - S^2.

Original entry on oeis.org

2, 13, 71, 376, 1991, 10564, 56051, 297384, 1577797, 8371133, 44413759, 235640987, 1250213362, 6633113651, 35192550325, 186717077925, 990643385291, 5255942989944, 27885853904294, 147950776760552, 784965467407868, 4164701250741605, 22096177765889378

Offset: 0

Clark Kimberling, Aug 14 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Cf. A001950, A289925.

z = 60; r = 1 + GoldenRatio; s = Sum[Floor[k*r] x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A001950 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289926 *)

A356106 a(n) = A001950(A022839(n)).

Original entry on oeis.org

5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, 75, 81, 86, 91, 99, 104, 109, 115, 120, 128, 133, 138, 143, 151, 157, 162, 167, 175, 180, 185, 191, 198, 204, 209, 214, 219, 227, 233, 238, 243, 251, 256, 261, 267, 274, 280, 285, 290, 298, 303, 308, 314, 319

Offset: 1

Clark Kimberling, Sep 08 2022

This is the third of four sequences that partition the positive integers. See A356104.

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = this sequence
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356105, A356107, A351415 (intersections), A356217 (reverse composites).

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* this sequence *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

Definition corrected by Georg Fischer, May 24 2024

A192185 Number of partitions of n into upper Wythoff numbers (A001950).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 2, 3, 2, 4, 3, 5, 6, 5, 8, 7, 9, 13, 10, 16, 14, 18, 22, 21, 28, 29, 31, 42, 37, 50, 51, 57, 70, 69, 83, 91, 95, 120, 118, 139, 153, 161, 193, 200, 224, 254, 262, 312, 324, 360, 404, 427, 485, 525, 561, 640, 668, 758, 817, 878, 982, 1046, 1150, 1265, 1340, 1499, 1597, 1745, 1911, 2036, 2241, 2420, 2602, 2866, 3041, 3332, 3597, 3864, 4221, 4518

Offset: 0

Paul D. Hanna, Jun 25 2011

nonn

This sequence is motivated by the identity:
Product_{n>=1} (1 - x^[n*phi])*(1 - x^[n*phi^2]) / (1 - x^n) = 1, where [.] denotes floor(.).
Therefore, the product of the g.f. of this sequence with the g.f. of A192184 yields the g.f. of the partition numbers (A000041).

			G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 3*x^10 +...
where the g.f. may be expressed by the product:
A(x) = 1/((1-x^2)*(1-x^5)*(1-x^7)*(1-x^10)*(1-x^13)*...)
in which the exponents of x are the upper Wythoff numbers (A001950):
[2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,...].
a(12) counts these partitions: [10,2], [7,5], [5,5,2], [2,2,2,2,2,2]. _Clark Kimberling_, Mar 09 2014

Paul D. Hanna, Table of n, a(n) for n = 0..5000

Cf. A192184, A001950, A000041.

t = Table[Floor[n+n*GoldenRatio], {n, 1, 200}]; p[n_] := IntegerPartitions[n, All, t]; Table[ p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)

{a(n)=local(phi=(sqrt(5)+1)/2,PWU=1/prod(m=1,ceil(n/phi),1-x^floor(m*phi^2)+x*O(x^n)));polcoeff(PWU,n)}

G.f.: Product_{n>=1} 1/(1 - x^floor(n*phi^2)), where phi = (sqrt(5)+1)/2.

G.f.: Product_{n>=1} 1/(1 - x^A001950(n)), where A001950 is the upper Wythoff sequence.

A203237 (n-1)-st elementary symmetric function of the first n terms of the upper Wythoff sequence, A001950.

Original entry on oeis.org

1, 7, 59, 660, 9280, 148300, 2805900, 58575000, 1396365000, 37435710000, 1077585600000, 34227953760000, 1189257232800000, 43680491749440000, 1734759507499200000, 72342732649037760000, 3233001543062054400000

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A203236.

f[k_] := k + Floor[GoldenRatio*k]
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 18}]     (* A203237 *)
t[16]  (* A001950 *)

cp's OEIS Frontend

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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A283767 Numbers k such that U(k) is even, where U = A001950 = upper Wythoff sequence.

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A283768 Numbers k such that U(k) is odd, where U = A001950 = upper Wythoff sequence.

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A283772 Numbers k such that U(k) = 0 mod 3, where U = A001950 = upper Wythoff sequence.

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A283773 Numbers k such that U(k) = 1 mod 3, where U = A001950 = upper Wythoff sequence.

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A283774 Numbers k such that U(k) == 2 mod 3, where U = A001950 = upper Wythoff sequence.

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A289926 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S - S^2.

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A356106 a(n) = A001950(A022839(n)).

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