A001950 -id:A001950 - cp's OEIS frontend

A003233 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)).

1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91

Offset: 1

N. J. A. Sloane

nonn

See 3.3 p. 344 in Carlitz link. - Michel Marcus, Feb 02 2014

This is the function named r in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

Cf. A001950, A003231, A003234.

a003233 n = a003233_list !! (n-1)
a003233_list = [x | x <- [1..],
                    a003231 (a001950 x) == a001950 (a003231 x)]
-- Reinhard Zumkeller, Oct 03 2014

a3221[n_] := Floor[n(5 + Sqrt[5])/2];
a1950[n_] := Floor[n(1 + Sqrt[5])^2/4];
Select[Range[100], a3221[a1950[#]] == a1950[a3221[#]]&] (* Jean-François Alcover, Aug 04 2018 *)

A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
lista(nn) = { for(n=1, nn, if (A003231(A001950(n)) == A001950(A003231(n)), print1(n, ", ")));} \\ Michel Marcus, Feb 02 2014

from math import isqrt
from itertools import count, islice
def A003233_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:((m:=(n+isqrt(5*n**2)>>1)+n)+isqrt(5*m**2)>>1)+(m<<1)==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k,count(max(1,startvalue)))
A003233_list = list(islice(A003233_gen(),30)) # Chai Wah Wu, Sep 02 2022

More terms from Michel Marcus, Feb 02 2014

Definition from Michel Marcus moved from comment to name by Eric M. Schmidt, Aug 17 2014

A047924 a(n) = B_{A_n+1}+1, where A_n = floor(nphi) = A000201(n), B_n = floor(nphi^2) = A001950(n) and phi is the golden ratio.

Original entry on oeis.org

3, 6, 11, 14, 19, 24, 27, 32, 35, 40, 45, 48, 53, 58, 61, 66, 69, 74, 79, 82, 87, 90, 95, 100, 103, 108, 113, 116, 121, 124, 129, 134, 137, 142, 147, 150, 155, 158, 163, 168, 171, 176, 179, 184, 189, 192, 197, 202, 205, 210, 213, 218, 223, 226, 231, 234, 239

Offset: 0

N. J. A. Sloane

2nd column of array in A038150.

Apart from the first term also the second column of A126714; see also A223025. - Casey Mongoven, Mar 11 2013

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
Aviezri S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
Aviezri S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.
Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A007066.

A001950 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi^2) ;
end proc:
A000201 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi) ;
end proc:
A047924 := proc(n)
        1+A001950(1+A000201(n)) ;
end proc: # R. J. Mathar, Mar 20 2013

A[n_] := Floor[n*GoldenRatio]; B[n_] := Floor[n*GoldenRatio^2]; a[n_] := B[A[n]+1]+1; Table[a[n], {n, 0, 56}] (* Jean-François Alcover, Feb 11 2014 *)

from mpmath import *
mp.dps=100
import math
def A(n): return int(math.floor(n*phi))
def B(n): return int(math.floor(n*phi**2))
def a(n): return B(A(n) + 1) + 1 # Indranil Ghosh, Apr 25 2017

from math import isqrt
def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1 # Chai Wah Wu, Aug 25 2022

More terms from Naohiro Nomoto, Jun 08 2001

New description from Aviezri S. Fraenkel, Aug 03 2007

A179319 G.f.: WL(-x)*WU(x), where WL, WU are respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences.

Original entry on oeis.org

1, -1, 1, -2, 1, 0, 1, 1, 0, 0, 1, -1, 1, 1, 1, 2, -1, 1, 1, 0, 1, -1, 1, 1, 0, 0, 1, -1, 1, -2, 1, 0, 1, -1, 1, -2, 1, -3, 1, -1, 1, 0, 1, -1, 1, -2, 1, 0, 1, 1, 0, 0, 1, -1, 1, -2, 1, 0, 1, -1, 1, -2, 1, -3, 1, -1, 2, -2, 1, -3, 1, -4, 1, -2, 1, -1, 2

Offset: 0

N. J. A. Sloane, Jan 05 2011

sign

Mentioned in a posting by Paul D. Hanna to the Sequence Fans Mailing List, Dec 28 2010.

			WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 +...+ x^[n*phi] + ...
WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 +...+ x^[n*(phi+1)] + ...
G.f.: WL(-x)*WU(x) = 1 - x + x^2 - 2*x^3 + x^4 + x^6 + x^7 + x^10 - x^11 + x^12 + x^13 + x^14 + 2*x^15 - x^16 +...+ a(n)*x^n +...
Positions of records for positive coefficients (A183555) in WL(-x)*WU(x) begin:
1: 0
2: 15
3: 159
4: 303
5: 2887
6: 5471
7: 51839
8: 98207
9: 930247
10: 1762287
...
Positions of records for negative coefficients (A183556) in WL(-x)*WU(x) begin:
-1: 1
-2: 3
-3: 37
-4: 71
-5: 681
-6: 1291
-7: 12237
-8: 23183
-9: 219601
-10: 416019
...
Now compare the above positions to A059973:
[1,1, 2,4, 9,17, 38,72, 161,305, 682,1292, 2889,5473, 12238,23184, 51841,98209, 219602,416020, 930249,1762289, ...].

Cf. A000201, A001950; A183555, A183556, A183557; A059973.

 It appears that the records for positive integers occur at positions A059973(4n+1)-2 and A059973(4n+2)-2, while the records for negative integers occur at positions A059973(4n-1)-1 and A059973(4n)-1;
that is, the records seem to obey the following rule:
* a(A059973(4n+1)-2) = 2n-1 for n>1,
* a(A059973(4n+2)-2) = 2n for n>=1,
* a(A059973(4n-1)-1) = -(2n-1) for n>=1,
* a(A059973(4n)-1) = -(2n) for n>=1;
see A183555 and A183556.

Formula, examples, and program added by Paul D. Hanna, Jan 07 2011

A283234 2*A001950.

Original entry on oeis.org

4, 10, 14, 20, 26, 30, 36, 40, 46, 52, 56, 62, 68, 72, 78, 82, 88, 94, 98, 104, 108, 114, 120, 124, 130, 136, 140, 146, 150, 156, 162, 166, 172, 178, 182, 188, 192, 198, 204, 208, 214, 218, 224, 230, 234, 240, 246, 250, 256, 260, 266, 272, 276, 282, 286, 292

Offset: 1

Clark Kimberling, Mar 03 2017

This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where [ ]=floor.
Taking r=1, s=(-1+sqrt(5))/2, t=(1+sqrt(5))/2 gives
a=A283233, b=A283234, c=A005843.

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

Cf. A000201, A283233, A005843.

r = 1; s = (-1 + 5^(1/2))/2; t = (1 + 5^(1/2))/2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (* A283233 *)
Table[b[n], {n, 1, 120}]  (* A283234 *)
Table[c[n], {n, 1, 120}]  (* A005408 *)

from math import isqrt
def A283234(n): return ((n+isqrt(5*n**2))&-2)+(n<<1) # Chai Wah Wu, Aug 10 2022

a(n) = 2*floor(n*s), where r = (-1+sqrt(5))/2.

A356102 Intersection of A001950 and A022839.

Original entry on oeis.org

2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, 89, 91, 96, 102, 107, 109, 120, 125, 136, 138, 143, 149, 154, 167, 172, 178, 183, 185, 196, 201, 212, 214, 219, 225, 230, 243, 248, 259, 261, 272, 277, 290, 295, 301, 306, 308, 319, 324, 326, 328, 330, 333, 335

Offset: 1

Clark Kimberling, Sep 04 2022

This is the third of four sequences that partition the positive integers. See A351415.

			Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356103 Intersection of A001950 and A108598.

Original entry on oeis.org

5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, 57, 65, 68, 70, 75, 81, 83, 86, 94, 99, 104, 112, 115, 117, 123, 128, 130, 133, 141, 146, 151, 157, 159, 162, 164, 170, 175, 180, 188, 191, 193, 198, 204, 206, 209, 217, 222, 227, 233, 235, 238, 240, 246, 251

Offset: 1

Clark Kimberling, Sep 04 2022

This is the fourth of four sequences that partition the positive integers. See A351415.

			Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356102, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A003256 a(n) is the number m such that A242094(m) = A001950(n).

Original entry on oeis.org

2, 5, 7, 9, 12, 14, 17, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 47, 49, 51, 54, 56, 58, 61, 63, 66, 68, 70, 73, 75, 77, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 123, 125, 127, 129, 131, 134, 136, 138, 141, 143

Offset: 1

N. J. A. Sloane

nonn

This is the function named v in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

Ron Reble remarks that Carlitz has a typo on page 339: Carlitz writes "In particular since (b) is a proper subset of (a), there exists a function v such that b = av." It should be "(b) is a proper subset of (u), ... b = uv." - N. J. A. Sloane, Jan 20 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

Cf. A001950, A003234, A242094.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a003256 = (+ 1) . fromJust . (`elemIndex` a242094_list) . a001950
-- Reinhard Zumkeller, Oct 03 2014

a(n) = A001950(n) - j, where j is the largest integer such that A003234(j) < n. [Carlitz, Thm. 7.3]. - Eric M. Schmidt, Sep 16 2014

New definition by Eric M. Schmidt, Aug 17 2014

A095281 Upper Wythoff primes, i.e., primes in A001950.

Original entry on oeis.org

2, 5, 7, 13, 23, 31, 41, 47, 73, 83, 89, 107, 109, 149, 151, 157, 167, 191, 193, 227, 233, 251, 269, 277, 293, 311, 337, 353, 379, 397, 421, 431, 439, 463, 479, 523, 541, 547, 557, 599, 607, 617, 641, 659, 683, 691, 701, 709, 719, 727, 733, 743

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Contains all primes p whose Zeckendorf-expansion A014417(p) ends with an odd number of 0's.

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & A001950. Complement of A095280 in A000040. Cf. A095081, A095083, A095084, A095290.

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095281_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1)+n for n in count(1)))
A095281_list = list(islice(A095281_gen(),30)) # Chai Wah Wu, Aug 16 2022

A255671 Number of the column of the Wythoff array (A035513) that contains U(n), where U = A001950, the upper Wythoff sequence.

Original entry on oeis.org

2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 10, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4

Offset: 1

Clark Kimberling, Mar 03 2015

All the terms are even, and every even positive integer occurs infinitely many times.
From Michel Dekking, Dec 09 2024 and Ad van Loon: (Start)
This sequence has a self-similarity property:
      a(U(n)) = a(n) + 2 for all n.
Proof: it is known that the columns C_h of the Wythoff array are compound Wythoff sequences. For example: C_1 = L^2, C_2 = UL.
In general column C_h is equal to LU^{(h-1)/2} if h is odd, and to U^{h/2}L if h is even (see Theorem 10 in Kimberling’s 2008 paper in JIS).
Now if h is odd then the elements of column C_h are a subsequence of L, so no U(m) can occur in such a column.
If h is even then the elements of column C_h form a subsequence of U, and so many U(m) occur. Suppose that a(m) = h. Then U(U(m)) is an element of column UU^{h/2}L = U^{(h+2)/2}L. This implies a(U(m)) = a(m) +2. (End)

			Corner of the Wythoff array:
  1   2   3   5   8   13
  4   7   11  18  29  47
  6   10  16  26  42  68
  9   15  24  39  63  102
L = (1,3,4,6,8,9,11,...); U = (2,5,7,10,13,15,18,...), so that
A255670 = (1,3,1,1,5,...) and A255671 = (2,4,2,2,6,...).

Ad van Loon, The structure of the expansions, See Section 5.

Cf. A255670, A035612, A000201, A001950.

z = 13; r = GoldenRatio; f[1] = {1}; f[2] = {1, 2};
f[n_] := f[n] = Join[f[n - 1], Most[f[n - 2]], {n}]; f[z];
g[n_] := g[n] = f[z][[n]]; Table[g[n], {n, 1, 100}]  (* A035612 *)
Table[g[Floor[n*r]], {n, 1, (1/r) Length[f[z]]}]     (* A255670 *)
Table[g[Floor[n*r^2]], {n, 1, (1/r^2) Length[f[z]]}] (* A255671 *)

a(n) = 2 if and only if n = L(j) for some j; otherwise, n = U(k) for some k.

a(n) = A255670(n) + 1 = A035612(A001950(n)).

A258236 Number of steps from n to 0, where allowable steps are x -> [x/r] if x is in upper Wythoff sequence (A001950) and x -> [r*x] otherwise, where [ ] = floor and r = (3+sqrt(5))/2.

Original entry on oeis.org

0, 2, 1, 3, 5, 3, 5, 2, 4, 6, 4, 6, 8, 6, 8, 4, 6, 8, 6, 8, 3, 5, 7, 5, 7, 9, 7, 9, 5, 7, 9, 7, 9, 11, 9, 11, 7, 9, 11, 9, 11, 5, 7, 9, 7, 9, 11, 9, 11, 7, 9, 11, 9, 11, 4, 6, 8, 6, 8, 10, 8, 10, 6, 8, 10, 8, 10, 12, 10, 12, 8, 10, 12, 10, 12, 6, 8, 10, 8

Offset: 0

Clark Kimberling, Jun 05 2015

a(n) = number of edges from 0 to n in the tree at A258235.

			29->75->28->10->3->7->2->0, so that a(29) = 7.

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A258235, A258212, A001950.

r = GoldenRatio^2; w = Table[Floor[r*n], {n, 1, 1000}];
f[x_] := If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1+ Length[g[n]], {n, 0, 100}]

cp's OEIS Frontend

A003233 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)).

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A047924 a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.

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A179319 G.f.: WL(-x)*WU(x), where WL, WU are respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences.

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A283234 2*A001950.

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A356102 Intersection of A001950 and A022839.

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A356103 Intersection of A001950 and A108598.

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A003256 a(n) is the number m such that A242094(m) = A001950(n).

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A095281 Upper Wythoff primes, i.e., primes in A001950.

A047924 a(n) = B_{A_n+1}+1, where A_n = floor(nphi) = A000201(n), B_n = floor(nphi^2) = A001950(n) and phi is the golden ratio.