A003231 -id:A003231 - cp's OEIS frontend

A003234 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)) - 1.

3, 8, 11, 16, 19, 21, 24, 29, 32, 37, 42, 45, 50, 53, 55, 58, 63, 66, 71, 74, 76, 79, 84, 87, 92, 97, 100, 105, 108, 110, 113, 118, 121, 126, 129, 131, 134, 139, 142, 144, 147, 152, 155, 160, 163, 165, 168, 173, 176, 181, 186, 189, 194, 197, 199, 202, 207

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 s 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 n = a003234_list !! (n-1)
a003234_list = [x | x <- [1..],
                    a003231 (a001950 x) == a001950 (a003231 x) - 1]
-- Reinhard Zumkeller, Oct 03 2014

A003234 := proc(n)
    option remember;
    if n =1 then
        3;
    else
        for a from procname(n-1)+1 do
            if A003231(A001950(a)) = A001950(A003231(a))-1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003234(n),n=1..80) ; # R. J. Mathar, Jul 16 2024

a3[n_] := Floor[n (Sqrt[5] + 3)/2];
a5[n_] := Floor[n (Sqrt[5] + 5)/2];
Select[Range[300], a5[a3[#]] == a3[a5[#]]-1&] (* Jean-François Alcover, Jul 31 2018 *)

A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
isok(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1; \\ Michel Marcus, Feb 02 2014

from math import isqrt
from itertools import count, islice
def A003234_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)+1==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k,count(max(1,startvalue)))
A003234_list = list(islice(A003234_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

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

A249115 Floor(r*n), where r = (5 - sqrt(5))/2; the Beatty complement of A003231.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 89, 91

Offset: 1

Clark Kimberling, Oct 21 2014

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 A249115(n) is the position of n*(tau - 1) in the ordered union of R and S.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.

Cf. A003231, A001622.

[Floor(n*(5-Sqrt(5))/2): n in [1..100]]; // Vincenzo Librandi, Oct 25 2014

Table[Floor[(5 - Sqrt[5])/2*n], {n, 1, 200}]

A276867 First differences of the Beatty sequence A003231 for 2 + tau, where tau = golden ratio = (1 + sqrt(5))/2.

Original entry on oeis.org

3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4

Offset: 1

Clark Kimberling, Sep 24 2016

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

Cf. A003231, A276886.

z = 500; r = 2+GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A003231 *)
Differences[b] (* A276867 *)

a(n) = floor(n*r) - floor(n*r - r), where r = 2 + tau, n >= 1.

A003258 The number m such that c'(m) = A005206(A003231(n)), where c'(m) = A249115(m) is the m-th positive integer not in A003231.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 105

Offset: 1

N. J. A. Sloane

nonn

This is the function named phi in the Carlitz-Scoville-Vaughan link. - 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.

Conjecture: a(n) = A078489(n) + n - 1. - Ralf Stephan, Feb 24 2004

More terms and a definition from Eric M. Schmidt, Aug 17 2014

Definition edited by Eric M. Schmidt, Aug 07 2015

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

A003252 The number m such that A003251(m) = A003231(n).

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 61, 64, 67, 70, 73, 76, 79, 82, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 122, 125, 128, 131, 134, 137, 140, 143, 145, 148, 151, 154, 157, 159, 162, 165, 168, 171

Offset: 1

N. J. A. Sloane

nonn

This is the function named lambda 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. A003253.

a(n) = 3n - j(n), where j(n) is the maximum number such that j(n) <= A003249(n). [Carlitz, Theorem 7.15.] - Eric M. Schmidt, Aug 17 2014

Sequence corrected and extended by, and definition from Eric M. Schmidt, Aug 17 2014

A247431 The largest integer m such that A001950(m) < A003231(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 90, 92

Offset: 1

Eric M. Schmidt, Sep 17 2014

nonn

This is the function named K in [Carlitz].

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

Cf. A001950, A003231.

a31(n) = (5*n+sqrtint(5*n^2))\2; \\ A003231
a50(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
a(n) = my(m=1, N=a31(n)); while(a50(m) < N, m++); m-1; \\ Michel Marcus, Nov 14 2023

A276871 Sums-complement of the Beatty sequence for sqrt(5).

Original entry on oeis.org

1, 10, 19, 28, 37, 48, 57, 66, 75, 86, 95, 104, 113, 124, 133, 142, 151, 162, 171, 180, 189, 198, 209, 218, 227, 236, 247, 256, 265, 274, 285, 294, 303, 312, 323, 332, 341, 350, 359, 370, 379, 388, 397, 408, 417, 426, 435, 446, 455, 464, 473, 484, 493, 502

Offset: 1

Clark Kimberling, Sep 24 2016

The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k.  If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order.  In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
  r                  B(r)        D(r)       SC(r)
  ----------------------------------------------------
  sqrt(5)            A022839     A081427    A276871
  sqrt(6)            A022840     A276856    A276872
  sqrt(7)            A022841     A276857    A276873
  sqrt(8)            A022842     A276858    A276874
  e                  A022843     A276859    A276875
  2*e                A276853     A276860    A276876
  Pi                 A022844     A063438    A276877
  2*Pi               A028130     A276861    A276878
  1+sqrt(2)          A003151     A276862    A276879
  1+sqrt(3)          A054088     A007538    A276880
  1+sqrt(5)          A276854     A276863    A276881
  2+sqrt(2)          A001952     A276864    A276882
  2+sqrt(3)          A003512     A276865    A276883
  2+sqrt(5)          A004976     A276866    A276884
  1+tau              A001950   2 + A003849  A276885
  2+tau              A003231     A276867    A276886
  3+tau              A276855     A276868    A276887
  2+sqrt(1/2)        A182769     A276869    A276888
  sqrt(2)+sqrt(3)    A110117     A276870    A276889
From Jeffrey Shallit, Aug 15 2023: (Start)
Simpler description:  this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf).  It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)

			The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 16.
Jeffrey Shallit, Fibonacci automaton for A276871
Index entries for sequences related to Beatty sequences

Cf. A022839, A081427.

z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
t = Differences[b]; (* A081427 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276871 *)

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 *)

cp's OEIS Frontend

A003234 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)) - 1.

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

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A249115 Floor(r*n), where r = (5 - sqrt(5))/2; the Beatty complement of A003231.

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A276867 First differences of the Beatty sequence A003231 for 2 + tau, where tau = golden ratio = (1 + sqrt(5))/2.

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A003258 The number m such that c'(m) = A005206(A003231(n)), where c'(m) = A249115(m) is the m-th positive integer not in A003231.

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

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A003252 The number m such that A003251(m) = A003231(n).

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A247431 The largest integer m such that A001950(m) < A003231(n).

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