A059906 -id:A059906 - cp's OEIS frontend

A072637 Inverse permutation to A072636.

0, 1, 2, 3, 6, 4, 5, 14, 15, 7, 16, 8, 19, 42, 43, 51, 52, 11, 9, 39, 37, 10, 28, 38, 112, 123, 121, 151, 149, 122, 376, 150, 466, 20, 53, 17, 44, 154, 155, 126, 127, 18, 47, 54, 156, 135, 136, 480, 481, 477, 475, 387, 385, 476, 1531, 386, 1234, 415, 413, 1542, 1540

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A029837(A014486(A072637(n))+1)/2 [A029837(n+1) gives the binary width of n].

A072637(n) = A057163(A072635(n)). Cf. also A014486, A059905, A059906, A071654, A072646.

A083935 Inverse function of N -> N injection A083934.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 13 2003

nonn

a(0)=0 because A083934(0)=0, but a(n) = 0 also for those n which do not occur as values of A083934. All positive natural numbers occur here once.

a(A080934(n)) = n for all n. Cf. A083925-A083929, A014486, A080300, A059905, A059906.

A163326 Pick digits at the odd distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 3, 3, 3, 4, 4, 4, 5, 5, 5, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 6, 6, 6, 7, 7, 7, 8, 8, 8, 6, 6, 6, 7, 7, 7, 8, 8, 8, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0

Offset: 0

Antti Karttunen, Jul 29 2009

			42 in ternary base (A007089) is written as '1120' (1*27 + 1*9 + 2*3 + 0), from which we pick the first and 3rd digits from the right (zero-based!), giving '12' = 1*3 + 2 = 5, thus a(42) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..728
Kevin Ryde, Plot2 of X=A163325,Y=A163326, illustrating the ternary Z-order curve.
Index entries for sequences related to coordinates of 2D curves

A059906 is an analogous sequence for binary. Note that A037314(A163325(n)) + 3*A037314(A163326(n)) = n for all n. Cf. A007089, A163327-A163329.

a(n) = fromdigits(digits(n,9)\3,3); \\ Kevin Ryde, May 15 2020

a(n) = A163325(floor(n/3))

a(n) = Sum_{k>=0} A030341(n,k)*b(k) with (b) = (0,1,0,3,0,9,0,27,0,81,0,243,0,...): powers of 3 alternating with zeros. - Philippe Deléham, Oct 22 2011

Edited by Charles R Greathouse IV, Nov 01 2009

A163900 Squared distance between n's location in A054238 array and A163357 array.

Original entry on oeis.org

0, 0, 1, 1, 8, 18, 5, 5, 4, 2, 9, 5, 2, 2, 9, 9, 0, 2, 1, 1, 0, 0, 1, 1, 4, 4, 9, 1, 2, 4, 5, 9, 16, 10, 25, 17, 16, 16, 25, 9, 36, 36, 49, 25, 10, 4, 5, 1, 10, 18, 5, 5, 10, 16, 17, 25, 10, 20, 25, 29, 36, 36, 25, 49, 128, 162, 113, 113, 128, 128, 113, 145, 100, 100, 89, 113, 162

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A163901. See also A163898, A163899.

a(n) = A000290(abs(A059906(n)-A059252(n))) + A000290(abs(A059905(n)-A059253(n))).

A072635 Inverse permutation to A072634.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 19, 16, 5, 15, 4, 14, 52, 43, 51, 42, 20, 22, 53, 60, 21, 61, 56, 179, 155, 178, 154, 177, 164, 557, 163, 556, 11, 39, 13, 41, 151, 123, 153, 125, 12, 40, 33, 117, 152, 124, 471, 381, 477, 553, 479, 555, 505, 1797, 507, 1799, 478, 554, 1536

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

A. Karttunen, Gatomorphisms
Index entries for sequences that are permutations of the natural numbers

A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A029837(A014486(A072635(n))+1)/2 [A029837(n+1) gives the binary width of n].

A072635(n) = A057163(A072637(n)). Cf. also A014486, A059905, A059906, A071652, A072647, A072637.

A292372 A binary encoding of 2-digits in base-4 representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 3, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 3, 2, 0, 0, 1, 0, 4, 4, 5, 4, 4, 4, 5, 4, 6, 6, 7, 6, 4, 4, 5, 4, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 3, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 3, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 3, 2, 0, 0, 1, 0, 4, 4, 5, 4, 4, 4, 5, 4, 6, 6, 7, 6, 4, 4, 5, 4, 0, 0, 1, 0, 0, 0, 1, 0, 2

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        1          2           1
   3        0          3           0
   4        0         10           0
   5        0         11           0
   6        1         12           1
   7        0         13           0
   8        2         20          10
   9        2         21          10
  10        3         22          11
  11        2         23          10
  12        0         30           0
  13        0         31           0
  14        1         32           1
  15        0         33           0
  16        0        100           0
  17        0        101           0
  18        1        102           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004198, A007088, A007090, A048724, A059906, A160382, A292370, A292371, A292373.

Cf. A289814 (analogous sequence for base-3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 2, 1, 0], 2], {n, 0, 120}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==2 else '0' for i in k), 2)
print([a(n) for n in range(121)]) # Indranil Ghosh, Sep 21 2017

def A292372(n): return 0 if (m:=n&~(n<<1)) < 2 else int(bin(m)[-2:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059906(n AND A048724(n)), where AND is a bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160382(n).

A292373 A binary encoding of 3-digits in base-4 representation of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 6, 6, 6, 7, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 4, 4, 5, 4

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        0          2           0
   3        1          3           1
   4        0         10           0
   5        0         11           0
   6        0         12           0
   7        1         13           1
   8        0         20           0
   9        0         21           0
  10        0         22           0
  11        1         23           1
  12        2         30          10
  13        2         31          10
  14        2         32          10
  15        3         33          11
  16        0        100           0
  17        0        101           0
  18        0        102           0
  19        1        103           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A007088, A007090, A048735, A059905, A059906, A160383, A213370, A292370, A292371, A292372.

def A292373(n): return int(bin(n&n>>1)[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A048735(n)) = A059906(A213370(n)).

For all n >= 0, A000120(a(n)) = A160383(n).

A082857 Inverse function of N -> N injection A082856.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 6, 0, 0, 0, 4, 0, 0, 0, 14, 0, 0, 0, 0, 0, 7, 0, 16, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 5, 0, 0, 0, 15, 0, 0, 0, 11, 0, 0, 0, 39, 0, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 123, 0, 0, 0, 0, 0, 8, 0, 19, 0, 0, 0, 0, 0, 0, 0, 51, 0, 0, 0, 0, 0, 20, 0, 53, 0, 0, 0, 0, 0, 0, 0, 154, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 0, 0, 151, 0, 0, 0, 0, 0, 0, 0, 155

Offset: 0

Antti Karttunen, May 06 2003

nonn

a(n) = 0 for those n which do not occur as the values of A082856. All positive natural numbers occur here once.

Antti Karttunen, Alternative Catalan Orderings (with the complete Scheme source)

Cf. A082856, A072634-A072637, A075173-A075174.

a(A082856(n)) = n for all n.

A152754 "Upper positive integers": n is in the sequence iff in the representation n=A000695(k)+2*A000695(l) satisfies inequality A000695(k) < A000695(l).

Original entry on oeis.org

2, 8, 9, 10, 11, 14, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 56, 57, 58, 59, 62, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160

Offset: 1

Vladimir Shevelev, Dec 12 2008

nonn

In the mapping: every integer m corresponds to a unique pair (k,l) with m=A000695(k)+2*A000695(l) (k=A059905(m), l=A059906(m)), the numbers a(n) are mapped into the lattice points lying upper the diagonal l=k. If the binary expansion of N is Sum b_j*2^j, then N is in the sequence iff Sum b_(2j)*2^jA139370. This explains, somewhat, why many terms of the sequence are in A139370 as well.

Vincenzo Librandi, Table of n, a(n) for n = 1..8128

Cf. A000695, A139370.

fh[n_,h_] := If[h==1, Mod[n,2], If[Mod[n,4]>=2,1,0]]; half[n_, h_ ] := Module[{t=1, s=0, m=n}, While[m>0, s += fh[m,h]*t; m=Quotient[m,4]; t *= 2]; s]; mb[n_] := FromDigits[Riffle[IntegerDigits[n, 2], 0], 2]; aQ[n_] := mb[half[n,1]] < mb[half[n, 2]]; Select[Range[160], aQ] (* Amiram Eldar, Dec 16 2018 from the PARI code *)

a000695(n) = fromdigits(binary(n), 4);
half1(n) = { my(t=1, s=0); while(n>0, s += (n%2)*t; n \= 4; t *= 2); (s); }; \\ A059905
half2(n) = { my(t=1, s=0); while(n>0, s += ((n%4)>=2)*t; n \= 4; t *= 2); (s); }; \\ A059906
isok(n) = a000695(half1(n)) < a000695(half2(n)); \\ Michel Marcus, Dec 15 2018

Missing 9 and more terms from Michel Marcus, Dec 15 2018

A163234 Inverse permutation to A163233.

Original entry on oeis.org

0, 1, 2, 4, 6, 3, 11, 7, 9, 13, 5, 8, 24, 18, 17, 12, 28, 21, 37, 29, 10, 15, 16, 22, 58, 48, 47, 38, 31, 39, 23, 30, 35, 43, 27, 34, 62, 52, 51, 42, 14, 19, 20, 26, 32, 25, 41, 33, 112, 98, 97, 84, 73, 85, 61, 72, 70, 59, 83, 71, 40, 49, 50, 60, 120, 105, 137, 121, 78

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A163233. a(n) = A163236(A057300(n)). Cf. A163236.

def A(x, y): return (((x + y)**2) + x + 3*y)//2
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a059905(n): return sum([(n>>2*i&1)<Indranil Ghosh, Jun 25 2017

Scheme

(define (A163234 n) (A001477bi (A006068 (A059905 n)) (A006068 (A059906 n)))) (define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

Formula

a(n) = A001477bi(A006068(A059905(n)),A006068(A059906(n))), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

cp's OEIS Frontend

A072637 Inverse permutation to A072636.

Views

Author

Keywords

Links

Crossrefs

A083935 Inverse function of N -> N injection A083934.

Views

Author

Keywords

Comments

Crossrefs

A163326 Pick digits at the odd distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A163900 Squared distance between n's location in A054238 array and A163357 array.

Views

Author

Keywords

Links

Crossrefs

Formula

A072635 Inverse permutation to A072634.

Views

Author

Keywords

Links

Crossrefs

A292372 A binary encoding of 2-digits in base-4 representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

A292373 A binary encoding of 3-digits in base-4 representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula

A082857 Inverse function of N -> N injection A082856.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A152754 "Upper positive integers": n is in the sequence iff in the representation n=A000695(k)+2*A000695(l) satisfies inequality A000695(k) < A000695(l).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A163234 Inverse permutation to A163233.

Views