A002487 -id:A002487 - cp's OEIS frontend

A360413 Irregular table T(n, k), n >= 0, k = 1..A002487(n+1), read by rows; the n-th row lists the numbers k such that A065361(k) = n.

0, 1, 2, 3, 4, 5, 6, 9, 7, 10, 8, 11, 12, 13, 14, 15, 18, 27, 16, 19, 28, 17, 20, 21, 29, 30, 22, 31, 23, 24, 32, 33, 36, 25, 34, 37, 26, 35, 38, 39, 40, 41, 42, 45, 54, 81, 43, 46, 55, 82, 44, 47, 48, 56, 57, 83, 84, 49, 58, 85, 50, 51, 59, 60, 63, 86, 87, 90

Offset: 0

Rémy Sigrist, Feb 06 2023

As a flat sequence, this is a permutation of the nonnegative integers with inverse A360414.

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   0  0
   1  1
   2  2, 3
   3  4
   4  5, 6, 9
   5  7, 10
   6  8, 11, 12
   7  13
   8  14, 15, 18, 27
   9  16, 19, 28
  10  17, 20, 21, 29, 30
  11  22, 31
  12  23, 24, 32, 33, 36
.
Table T(n, k) begins (with terms given in base 3):
  n   n-th row in base 3
  --  -------------------------
   0  0
   1  1
   2  2, 10
   3  11
   4  12, 20, 100
   5  21, 101
   6  22, 102, 110
   7  111
   8  112, 120, 200, 1000
   9  121, 201, 1001
  10  122, 202, 210, 1002, 1010
  11  211, 1011
  12  212, 220, 1012, 1020, 1100

Rémy Sigrist, Table of n, a(n) for n = 0..9841
Rémy Sigrist, PARI program
Index entries for sequences related to Stern's sequences
Index entries for sequences that are permutations of the natural numbers

Cf. A002487, A005836, A032924, A065361, A360414 (inverse).

PARI
```
See Links section.
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T(n, 1) = A032924(n) for any n > 0.
T(n, A002487(n+1)) = A005836(n+1).
A065361(T(n, k)) = n.

A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = xyz, where f(m) = A002487(m)/A002487(m+1).

Original entry on oeis.org

0, 1, 2, 1, 6, 2, 2, 14, 4, 30, 12, 35, 2, 4, 9, 20, 4, 8, 126, 56, 32, 152, 52, 254, 61, 84, 40, 16, 4, 510, 368, 320, 212, 48, 396, 72, 583, 1022, 792, 368, 98, 188, 340, 80, 583, 339, 140, 32, 233, 2046, 480, 384, 583, 2062, 852, 188, 328

Offset: 1

M. F. Hasler, Sep 16 2024

nonn

A376241 uses the Stern-Brocot sequence s = A002487 to enumerate all (nonnegative) rational x = s(n)/s(n+1) and similarly y = s(m)/s(m+1), WLOG m <= n, which yield an integer x*y*z = x+y+z with (necessarily) z = (x+y)/(xy-1). The present sequence lists the m-values corresponding to the n-values listed in A376241.

			The terms correspond to the following solutions, with y = A002487(m)/A002487(m+1):
   m |  x  |  y  |  z  | xyz = x+y+z
-----+-----+-----+-----+------------
   0 |  0  |  0  |  0  |   0
   1 |  2  |  1  |  3  |   6
   2 | 3/2 | 1/2 | -8  |  -6
   1 |  3  |  1  |  2  |   6
   6 | 4/3 | 2/3 | -18 |  -16
   2 | 5/2 | 1/2 |  12 |   15
   2 |  4  | 1/2 | 9/2 |   9
  14 | 5/4 | 3/4 | -32 |  -30
  ...| ... | ... | ... |  ...

Cf. A002487 (Stern-Brocot sequence), A376241 (corresponding n values), A376243 (set of absolute values of corresponding xyz = x+y+z).

A376242(n, k=A376241(n))={my(p, q=1, x=A002487(k)/A002487(k+1)); for(m=2, k, my(y=(p=q)/q=A002487(m)); x*y != 1 && denominator(x+y+(x+y)/(x*y-1))==1 && return(m-1))} \\ Short of a function A376241(n), one can simply provide a term k = A376241(n) as second argument and omit the first argument n.

A071016 Stirling_2 transform of A002487.

Original entry on oeis.org

1, 3, 8, 24, 88, 381, 1822, 9254, 49295, 275219, 1614968, 9993871, 65442602, 454736731, 3356546167, 26272011685, 217027548533, 1879608087006, 16946483469822, 158067735588494, 1518042086052977, 14962182709519890, 151056371629601794, 1560637060858070869

Offset: 1

N. J. A. Sloane, May 24 2002

nonn

N. J. A. Sloane, Transforms

A071412 A002487 mod 3.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 2, 0, 1, 1, 0, 2, 2, 2, 0, 1, 1, 2, 1, 1, 0, 2, 2, 1, 2, 1, 2, 2, 0, 1, 1, 2, 1, 0, 2, 0, 1, 2, 1, 1, 0, 2, 2, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 1, 2, 2, 0, 1, 1, 2, 1, 0, 2, 0, 1, 1, 0, 2, 2, 2, 0, 1, 1, 0, 2, 0, 1, 2, 1, 1, 0, 2, 2, 1, 2, 0, 1, 0, 2, 2, 0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 1, 1, 0, 2, 2

Offset: 0

N. J. A. Sloane, Jul 15 2002

nonn

E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence is called fusc).

E. W. Dijkstra, An exercise for Dr. R. M. Burstall
E. W. Dijkstra, More about the function ``fusc''

from functools import reduce
def A071412(n): return sum(reduce(lambda x,y:(x[0],(x[0]+x[1])%3) if int(y) else ((x[0]+x[1])%3,x[1]),bin(n)[-1:2:-1],(1,0)))%3 if n else 0 # Chai Wah Wu, May 18 2023

A071883 A002487(n)*A002487(n+2).

Original entry on oeis.org

0, 2, 1, 6, 2, 9, 2, 12, 3, 20, 6, 25, 6, 20, 3, 20, 4, 35, 12, 56, 15, 56, 10, 49, 10, 56, 15, 56, 12, 35, 4, 30, 5, 54, 20, 99, 28, 110, 21, 110, 24, 143, 40, 156, 35, 108, 14, 81, 14, 108, 35, 156, 40, 143, 24, 110, 21, 110, 28, 99, 20

Offset: 0

N. J. A. Sloane, Jun 10 2002

nonn

Cf. A002487, A070871.

A071898 CONTINUANT transform of A002487: 1, 1, 2, 1, 3, 2, ...

Original entry on oeis.org

1, 2, 5, 7, 26, 59, 203, 262, 1251, 4015, 21326, 46667, 254661, 810650, 3497261, 4307911, 25036816, 104455175, 756223041, 2373124298, 19741217425, 101079211423, 727295697386, 1555670606195, 11616989940751, 59640620309950, 488741952420351, 1525866477571003

Offset: 1

N. J. A. Sloane, Jun 10 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..400
N. J. A. Sloane, Transforms

Cf. A002487.

b:= proc(n) option remember; `if`(n<2, n,
      `if`(irem(n, 2, 'r')=0, b(r), b(r) + b(r+1)))
    end:
a:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, b(n)*a(n-1)+a(n-2)))
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Aug 06 2013

b[n_] := b[n] = If[n < 2, n, r = Quotient[n, 2]; If[Mod[n, 2] == 0, b[r], b[r] + b[r + 1]]];
a[n_] := a[n] = If[n < 0, 0, If[n == 0, 1, b[n] a[n - 1] + a[n - 2]]];
Array[a, 35] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

A071970 List the positive rationals in the order in which they are produced by the Stern sequence A002487 and apply the Sagher map to turn them into integers.

Original entry on oeis.org

1, 2, 4, 3, 18, 12, 9, 8, 48, 45, 50, 20, 75, 72, 16, 5, 200, 112, 147, 288, 320, 175, 98, 28, 245, 800, 192, 63, 392, 80, 25, 6, 180, 675, 648, 176, 847, 490, 300, 99, 3872, 832, 845, 600, 1008, 1323, 162, 108, 567, 1176, 720, 325, 5408, 704, 363, 90, 700, 539

Offset: 1

N. J. A. Sloane, Jun 19 2002

The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.

			The first few rationals and their images are 1/1 -> 1, 1/2 -> 2, 2/1 -> 4, 1/3 -> 3, 3/2 -> 18, 2/3 -> 12, 3/1 -> 9, 1/4 -> 8, ...

Y. Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823.

Cf. A002487, A060837.

nmax = 58; s[0] = 0; s[1] = 1; s[n_?EvenQ] := s[n/2]; s[n_] := s[(n-1)/2] + s[(n+1)/2]; v = Table[ FactorInteger /@ {s[n] , s[n+1]}, {n, 1, nmax}]; a[n_] := Times @@ (#[[1]]^(2*#[[2]])&) /@ v[[n, 1]]*Times @@ (#[[1]]^(2*#[[2]]-1)&) /@ v[[n, 2]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Nov 25 2011, after Pari *)

s(n)=if(n<2,n>0,if(n%2,s((n+1)/2)+s((n-1)/2),s(n/2))) /* A002487(n) */

a(n)=local(v); if(n,v=factor(s(n)/s(n+1))~; prod(k=1,length(v),v[1,k]^if(v[2,k]<0,-1-2*v[2,k],2*v[2,k])),0)

More terms from Michael Somos, Jul 19 2002

A127971 a(n) = fusc(n+1) + (1-(-1)^n)/2, fusc = A002487.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 2, 4, 4, 5, 3, 5, 4, 4, 2, 5, 5, 7, 4, 8, 6, 7, 3, 7, 6, 8, 4, 7, 5, 5, 2, 6, 6, 9, 5, 11, 8, 10, 4, 11, 9, 13, 6, 12, 8, 9, 3, 9, 8, 12, 6, 13, 9, 11, 4, 10, 8, 11, 5, 9, 6, 6, 2, 7, 7, 11, 6, 14, 10, 13, 5, 15, 12, 18, 8, 17, 11, 13, 4

Offset: 0

Paul Barry, Feb 09 2007

Row sums of A127970.

G. C. Greubel, Table of n, a(n) for n = 0..10000

[(1-(-1)^n)/2 + (&+[Binomial(n-k,k) mod 2: k in [0..Floor(n/2)]]) : n in [0..50]]; // G. C. Greubel, May 04 2018

Table[Sum[Mod[Binomial[n-k,k],2], {k, 0, Floor[n/2]}] + (1-(-1)^n)/2, {n, 0, 50}] (* G. C. Greubel, May 04 2018 *)

for(n=0, 50, print1((1-(-1)^n)/2 + sum(k=0,floor(n/2), lift(Mod( binomial(n-k,k), 2))), ", ")) \\ G. C. Greubel, May 04 2018

from functools import reduce
def A127971(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n+1)[-1:2:-1],(1,0)))+(n&1) # Chai Wah Wu, May 18 2023

a(n) = (1-(-1)^n)/2 + Sum_{k=0..floor(n/2)} mod(C(n-k,k),2).

A240388 A sequence related to the Stern sequence s(n) (A002487), defined by w(n) = s(3n)/2.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 4, 1, 4, 2, 3, 2, 5, 4, 6, 1, 6, 4, 5, 2, 3, 3, 7, 2, 9, 5, 7, 4, 9, 6, 8, 1, 8, 6, 9, 4, 7, 5, 9, 2, 7, 3, 4, 3, 8, 7, 11, 2, 13, 9, 12, 5, 8, 7, 15, 4, 17, 9, 11, 6, 13, 8, 10, 1, 10, 8, 13, 6, 11, 9, 17, 4, 15, 7, 8, 5, 12, 9, 13, 2, 11, 7, 8, 3, 4, 4, 10, 3, 14, 8, 12, 7, 16, 11, 15, 2, 17, 13, 20, 9, 16, 12, 22, 5, 18, 8, 10, 7, 18, 15, 23, 4, 25, 17, 22, 9, 14, 11, 23, 6, 25, 13, 15, 8, 17, 10, 12, 1

Offset: 0

Jennifer Lansing, Apr 04 2014

The even terms in the Stern sequence, divided by 2.

			w(7) = w(8-1) = w(3)+2w(1) = 2+2 = 4.
w(11) = w(8+3) = w(4+1)+w(2+1)-w(1)=w(5)+w(3)-w(1) = 2+2-1 = 3.
Comment from _N. J. A. Sloane_, Jul 01 2014: (Start)
May be arranged as a triangle:
  0
  1
  1
  2 1 2
  2 4 1 4 2
  3 2 5 4 6 1 6 4 5 2 3
  3 7 2 9 5 7 4 9 6 8 1 8 6 9 4 7 5 9 2 7 3
  ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16384
Jennifer Lansing, On the Stern sequence and a related sequence, Joint Mathematics Meetings, Baltimore, 2014.
Jennifer Lansing, Dissertation: On the Stern sequence and a related sequence, PhD dissertation, University of Illinois, 2014.
Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.

Cf. A002487.

A240388 := proc(n)
    option remember;
    local nloc;
    if n <=1  then
        n;
    elif n = 3 then
        2;
    elif type(n,'even') then
        procname(n/2) ;
    elif modp(n,8) = 1 then
        nloc := (n-1)/8 ;
        procname(4*nloc+1)+2*procname(nloc) ;
    elif modp(n,8) = 7 then
        nloc := (n+1)/8 ;
        procname(4*nloc-1)+2*procname(nloc) ;
    elif modp(n,8) = 3 then
        nloc := (n-3)/8 ;
        procname(4*nloc+1)+procname(2*nloc+1)-procname(nloc) ;
    else
        nloc := (n+3)/8 ;
        procname(4*nloc-1)+procname(2*nloc-1)-procname(nloc) ;
    end if;
end proc: # R. J. Mathar, Jul 05 2014
# second Maple program:
b:= proc(n) option remember; `if`(n<2, n,
      (q-> b(q)+(n-2*q)*b(n-q))(iquo(n, 2)))
    end:
a:= n-> b(3*n)/2:
seq(a(n), n=0..128);  # Alois P. Heinz, Jun 20 2022

Clear[s]; s[0] = 0; s[1] = 1; s[n_?EvenQ] := s[n] = s[n/2];
s[n_?OddQ] :=
s[n] = s[(n + 1)/2] + s[(n - 1)/2] (* For the Stern sequence *)
Clear[w]; w[n_] = 1/2 s[3 n]

a(n)=my(a=1, b=0); n*=3; while(n>0, if(n%2, b+=a, a+=b); n>>=1); b/2 \\ Charles R Greathouse IV, May 27 2014

from functools import reduce
def A240388(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(3*n)[-1:2:-1],(1,0)))//2 # Chai Wah Wu, Jun 20 2022

w(0)=0, w(1)=1, and w(3)=2. For n >= 1, w(n) satisfies the recurrences w(2n)=w(n), w(8n +/- 1)=w(4n +/- 1) + 2w(n), w(8n +/- 3)=w(4n +/- 1) + w(2n +/- 1) -w(n).

a(n) = A002487(3*n) / 2. - Joerg Arndt, Jun 20 2022

A293957 When A002487 is written as a triangle the n-th row has length 2^(n-1); a(n) is the maximal multiplicity of any entry in that row, considering the entries strictly between the initial 1 and the central 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 8, 12, 16, 22, 29, 36, 48, 67, 84, 118, 151, 203, 270, 362, 472, 636, 846, 1142, 1526, 2024, 2736, 3666, 4918, 6550, 8776, 11796, 15824

Offset: 0

N. J. A. Sloane, Nov 03 2017

The maximal entry is row n is Fibonacci(n+1), and the smallest missing number is A135510(n). The number of distinct numbers in each row is given by A293160.

It would be nice to have a formula for this sequence, or at least some bounds.

			Rows 0 through 6 of A002487 are:
0,
1,
1, 2,
1, 3, 2, 3,
1, 4, 3, 5, 2, 5, 3, 4,
1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5,
1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6,
To find a(5) we consider the entries 1, 5, 4, 7, 3, 8, 5, 7, 2 in row 5. Ignoring the initial 1 and the final 2, the maximal multiplicity is 2 (for example, 5 appears twice), so a(5) = 2.
From _Don Reble_, Nov 04 2017: (Start)
The initial values of a(n) for n >= 3 together with the terms that have the highest multiplicity are:
3    1 [3]
4    1 [3 4 5]
5    2 [5 7]
6    2 [5 7 9 11]
7    4 [11]
8    5 [13 17]
9    6 [19 23 31 41]
10    8 [23 37 43]
11   12 [71]
12   16 [71]
13   22 [127]
14   29 [109]
15   36 [199 251]
16   48 [263]
17   67 [433]
18   84 [701]
19  118 [839]
20  151 [1193]
21  203 [1801]
22  270 [2693]
23  362 [4229]
24  472 [4349]
25  636 [7759]
26  846 [11287]
27 1142 [14627]
28 1526 [20929]
29 2024 [37243]
30 2736 [43133]
31 3666 [67231]
32 4918 [90227]
33 6550 [127819]
34 8776 [181031]
35 11796 [251071]
36 15824 [394549]
(End)

Cf. A002487, A049456, A135510, A293160.

A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then procname(n/2); else procname((n-1)/2)+procname((n+1)/2); fi; end:
ans:=[];
for n from 3 to 18 do
b1:=2^(n-1); b2:=2^n-1; b3:=2^(n-2)-1; mx:=0;
ar:=Array(0..b1-1,0);
for k from 1 to b3 do
kk:=b1+k;
v:=A002487(kk);
ar[v]:=ar[v]+1;
od:
   for k from 0 to b1-1 do if ar[k]>mx then mx:=ar[k]; fi; od:
ans:=[op(ans),mx];
od:
ans;

from itertools import chain, product
from collections import Counter
from functools import reduce
def A293957(n): return 0 if n <= 2 else max(Counter(m for m in (sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if y else (x[0]+x[1],x[1]),chain(k,(1,)),(1,0))) for k in product((False,True),repeat=n-2)) if m != 1 and m != 2).values()) # Chai Wah Wu, Jun 20 2022

a(19)-a(36) from Don Reble, Nov 04 2017

cp's OEIS Frontend

A360413 Irregular table T(n, k), n >= 0, k = 1..A002487(n+1), read by rows; the n-th row lists the numbers k such that A065361(k) = n.

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A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = x*y*z, where f(m) = A002487(m)/A002487(m+1).

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PARI

A071016 Stirling_2 transform of A002487.

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A071412 A002487 mod 3.

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A071883 A002487(n)*A002487(n+2).

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A071898 CONTINUANT transform of A002487: 1, 1, 2, 1, 3, 2, ...

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Mathematica

A071970 List the positive rationals in the order in which they are produced by the Stern sequence A002487 and apply the Sagher map to turn them into integers.

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Mathematica

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A127971 a(n) = fusc(n+1) + (1-(-1)^n)/2, fusc = A002487.

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Magma

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A240388 A sequence related to the Stern sequence s(n) (A002487), defined by w(n) = s(3n)/2.

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A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = xyz, where f(m) = A002487(m)/A002487(m+1).