A003234 -id:A003234 - cp's OEIS frontend

A003249 a(n) = A001950(A003234(n)) + 1.

8, 21, 29, 42, 50, 55, 63, 76, 84, 97, 110, 118, 131, 139, 144, 152, 165, 173, 186, 194, 199, 207, 220, 228, 241, 254, 262, 275, 283, 288, 296, 309, 317, 330, 338, 343, 351, 364, 372, 377, 385, 398, 406, 419, 427, 432, 440, 453, 461, 474, 487, 495, 508, 516

Offset: 1

N. J. A. Sloane

nonn

This is the function named u' 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. A242094 (complement), A001950, A003234.

a003249 = (+ 1) . a001950 . a003234 -- Reinhard Zumkeller, Oct 03 2014

A003249 := proc(n)
    A001950(A003234(n)) +1 ;
end proc:
seq(A003249(n),n=1..80) ; # R. J. Mathar, Jul 16 2024

nmax = 80;
A001950[n_] := Floor[n*GoldenRatio^2];
A003231[n_] := 2*n + Floor[n*GoldenRatio];
A003234 = Select[Range[4*nmax],
   A003231[A001950[#]] == A001950[A003231[#]] - 1&];
a[n_] := A001950[A003234[[n]]] + 1;
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 21 2024 *)

Corrected and extended by, and a definition from Eric M. Schmidt, Aug 14 2014

A003248 a(n) = A000201(A003234(n)) + n.

Original entry on oeis.org

5, 14, 20, 29, 35, 39, 45, 54, 60, 69, 78, 84, 93, 99, 103, 109, 118, 124, 133, 139, 143, 149, 158, 164, 173, 182, 188, 197, 203, 207, 213, 222, 228, 237, 243, 247, 253, 262, 268, 272, 278, 287, 293, 302, 308, 312, 318, 327, 333, 342, 351, 357, 366, 372, 376

Offset: 1

N. J. A. Sloane

nonn

This is the function named t' 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. A000201, A003234.

More terms and a definition from Eric M. Schmidt, Aug 14 2014

A003254 The number m such that A003233(m) = A005206(A003234(n)).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 17, 19, 21, 23, 24, 25, 27, 29, 31, 33, 34, 35, 37, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 80, 82, 84, 86, 88, 89, 90, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106

Offset: 1

N. J. A. Sloane

nonn

This is the function named p 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.

More terms and a definition from Eric M. Schmidt, Aug 14 2014

A247421 The number m such that A242094(m) = A005206(A003234(n)).

Original entry on oeis.org

2, 5, 7, 9, 11, 12, 14, 17, 19, 21, 24, 26, 28, 30, 31, 33, 36, 38, 40, 42, 43, 45, 47, 49, 51, 54, 56, 58, 60, 61, 63, 66, 68, 70, 72, 73, 75, 77, 79, 80, 82, 85, 87, 89, 91, 92, 94, 97, 99, 101, 104, 106, 108, 110, 111, 113, 116, 118, 120, 122, 123, 125, 127

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named x in [Carlitz].

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

A003250 The number m such that A001950(m) = A003231(A003234(n)).

Original entry on oeis.org

4, 11, 15, 22, 26, 29, 33, 40, 44, 51, 58, 62, 69, 73, 76, 80, 87, 91, 98, 102, 105, 109, 116, 120, 127, 134, 138, 145, 149, 152, 156, 163, 167, 174, 178, 181, 185, 192, 196, 199, 203, 210, 214, 221, 225, 228, 232, 239, 243, 250, 257, 261, 268, 272, 275, 279

Offset: 1

N. J. A. Sloane

nonn

This is the function named z 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.

From Eric M. Schmidt, Aug 14 2014: (Start)
a(n) = ceiling((1/phi^2) * A003231(A003234(n))), where phi is the golden ratio.
a(n) = 5*k - 1 - A003231(k), where k = A003234(n). [Cf. Theorems 4.1(ii) and 5.9(vii) in Carlitz.]
Conjecture: a(n) = floor((3-phi)*A003234(n)).
(End)

More terms and a definition from Eric M. Schmidt, Aug 14 2014

A003231 a(n) = floor(n*(sqrt(5)+5)/2).

Original entry on oeis.org

3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 47, 50, 54, 57, 61, 65, 68, 72, 75, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 206, 209

Offset: 1

N. J. A. Sloane

Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A003231(n) is the position of n*tau in the ordered union of R and S. The position of n*(tau - 1) is A249115(n). - Clark Kimberling, Oct 21 2014
This is the function named c in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 06 2015
Natural numbers whose representation in base phi differs between the Bergmann representation and the "canonical" representation described by Dekking and van Loon. See proposition 3.3 in Dekking, van Loon (2021). - Hugo Pfoertner, May 26 2023

Dekking, Michel, and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." arXiv preprint arXiv:2111.07544 (2021); Fib. Quart. 61:2 (May 2023), 105-118.
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.
Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.
Index entries for sequences related to Beatty sequences

Cf. A000201, A003231, A003233, A003234, A249115.

Cf. A105424, A362917.

a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral
-- Reinhard Zumkeller, Oct 03 2014

[Floor(n*(Sqrt(5)+5)/2): n  in [1..100]]; // Vincenzo Librandi, Oct 23 2014

A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2015

With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)

PARI
```
a(n)=floor(n*(sqrt(5)+5)/2)
```

a(n)=(5*n+sqrtint(5*n^2))\2; \\ Michel Marcus, Nov 14 2023

from math import isqrt
def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 25 2022

a(n) = 2*n + A000201(n). - R. J. Mathar, Aug 22 2014

Better description and more terms from Michael Somos, Jun 07 2000

A242094 Complement of A003249.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Eric M. Schmidt, Aug 14 2014

nonn

This is the function named u in [Carlitz].

First differs from A187947 at a(46)=51.

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

Cf. A003249.

a242094 n = a242094_list !! (n-1)
a242094_list = c [1..] a003249_list where
   c (v:vs) ws'@(w:ws) = if v == w then c vs ws else v : c vs ws'
-- Reinhard Zumkeller, Oct 03 2014

nmax = 80;
A001950[n_] := Floor[n*GoldenRatio^2];
A003231[n_] := 2*n + Floor[n*GoldenRatio];
A003234 = Select[Range[4*nmax],
   A003231[A001950[#]] == A001950[A003231[#]] - 1 &];
A003249[n_] := A001950[A003234[[n]]] + 1;
Complement[Range[A003249[nmax]], Array[A003249, nmax]] (* Jean-François Alcover, Jul 21 2024 *)

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

Original entry on oeis.org

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

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

A247423 Complement of A247424.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 51, 52, 54, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 103, 104, 106, 108, 110

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named y in [Carlitz], which defines this sequence by the property A242094(A247419(a(n))) = A005206(A003234(n)).

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

cp's OEIS Frontend

A003249 a(n) = A001950(A003234(n)) + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A003248 a(n) = A000201(A003234(n)) + n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A003254 The number m such that A003233(m) = A005206(A003234(n)).

Views

Author

Keywords

Comments

References

Links

Extensions

A247421 The number m such that A242094(m) = A005206(A003234(n)).

Views

Author

Keywords

Comments

Links

A003250 The number m such that A001950(m) = A003231(A003234(n)).

Views

Author

Keywords

Comments

References

Links

Formula

Extensions

A003231 a(n) = floor(n*(sqrt(5)+5)/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A242094 Complement of A003249.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

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

Views

Author

Keywords

Comments

References