A003188 -id:A003188 - cp's OEIS frontend

A265385 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1) + a(n-2)), with gray(m) = A003188(m).

1, 1, 3, 6, 13, 26, 52, 105, 211, 418, 847, 1673, 3380, 6755, 13404, 27104, 53538, 108163, 216183, 428935, 867329, 1713228, 3461227, 6917868, 13725948, 27754524, 54823316, 110759272, 221371778, 439230367, 888144817, 1754346232, 3544296957, 7083888783

Offset: 1

Stanislav Sykora, Dec 07 2015

nonn

This recurrence is reminiscent of Fibonacci's, except that the result of each step is passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.

Conjecture: The mean growth rate r(n) = (a(2n)/a(n))^(1/n) appears to converge exactly to 2, with the consecutive-terms ratio s(n) = a(n)/a(n-1) exhibiting relatively small (~1%) but persistent fluctuations around the mean value. This contrasts what one might first expect, that sequence's growth rate were similar to that of the Fibonacci sequence, i.e., the golden ratio, since gray(m) just permutes every block of numbers ranging from 2^k to 2^l-1, for any 0

			r(10) = 2.000421531046..., r(1000) = 1.999999999903...
s(100) = 1.9841292..., s(101) = 2.0220518..., s(102) = 1.9752921...
s(10000) = 1.9841299..., s(10001) = 2.0220478..., s(10002) = 1.9752929...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Fibonacci number
Wikipedia, Gray code

Cf. A000045, A003188, A265386, A265387.

gray(m)=bitxor(m,m>>1);
a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(a[n-1]+a[n-2]));a

A265386 Sequence defined by a(1)=a(2)=1 and a(n) = gray(gray(a(n-1)) + gray(a(n-2))), with gray(m) = A003188(m).

Original entry on oeis.org

1, 1, 3, 2, 7, 4, 15, 9, 31, 19, 63, 39, 126, 79, 253, 158, 510, 315, 1012, 622, 2004, 1116, 4072, 2505, 8173, 5100, 16175, 10171, 32657, 20192, 64797, 39858, 128257, 71450, 260628, 160367, 523085, 326498, 1035105, 651126, 2090065, 1292517, 4146840

Offset: 1

Stanislav Sykora, Dec 07 2015

nonn

This recurrence is reminiscent of Fibonacci's, except that in each step the arguments as well as the result are passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.

Conjecture: the mean growth rate r(n) = (a(2n)/a(n))^(1/n) appears to converge to sqrt(2), with the consecutive-terms ratio s(n) = a(n)/a(n-1) exhibiting large and persistent fluctuations around the mean value.

			r(10) = 1.417436..., r(1000) = 1.414393...
s(100) = 0.629..., s(101) = 3.210..., s(102) = 0.618...
s(10000) = 0.631..., s(10001) = 3.183..., s(10002) = 0.608...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Fibonacci number
Wikipedia, Gray code

Cf. A000045, A003188, A265385, A265387.

gray(m)=bitxor(m,m>>1);
a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(gray(a[n-1])+gray(a[n-2])));a

A265387 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1)) + gray(a(n-2)), with gray(m) = A003188(m).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 39, 78, 157, 316, 629, 1265, 2520, 5053, 10135, 20159, 40508, 80642, 161701, 324346, 645118, 1296264, 2580557, 5174455, 10379095, 20643816, 41480472, 82577840, 165582588, 332131050, 660602145, 1327375184, 2642491049, 5298643189

Offset: 1

Stanislav Sykora, Dec 07 2015

nonn

This recurrence is reminiscent of Fibonacci's, except that in each step the arguments are passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.

			r(10) = 2.000470476732..., r(1000) = 2.000000000203...
s(100) = 2.0058315..., s(101) = 1.9889791..., s(102) = 2.0093437...
s(10000) = 2.0058331..., s(10001) = 1.9889803..., s(10002) = 2.0093413...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Fibonacci number
Wikipedia, Gray code

Cf. A000045, A003188, A265385, A265386.

gray(m)=bitxor(m,m>>1);
a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(a[n-1])+gray(a[n-2]));a

A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 3, 2, 7, 6, 4, 12, 15, 13, 9, 5, 4, 13, 12, 11, 10, 6, 7, 4, 5, 14, 15, 12, 7, 5, 12, 4, 10, 14, 13, 15, 8, 24, 27, 25, 29, 31, 26, 30, 17, 9, 8, 25, 24, 31, 30, 27, 26, 19, 18, 10, 11, 8, 9, 26, 27, 24, 25, 22, 23, 20, 11, 9, 24, 8, 30, 26, 25, 27, 18, 22, 21, 23, 12, 13, 14, 15, 8, 9, 10, 11, 28, 29, 30, 31, 24

Offset: 0

Antti Karttunen, Feb 13 2016

			The first fifteen rows of the triangle:
                             0
                           1   3
                         2   6   5
                       3   2   7   6
                     4  12  15  13   9
                   5   4  13  12  11  10
                 6   7   4   5  14  15  12
               7   5  12   4  10  14  13  15
             8  24  27  25  29  31  26  30  17
           9   8  25  24  31  30  27  26  19  18
        10  11   8   9  26  27  24  25  22  23  20
      11   9  24   8  30  26  25  27  18  22  21  23
    12  13  14  15   8   9  10  11  28  29  30  31  24
  13  15  10  14  24   8  11   9  20  28  31  29  25  27
14  10   9  11  27  25   8  24  23  21  28  20  26  30  29

Antti Karttunen, Table of n, a(n) for n = 0..15050; rows 0 .. 172 of the triangular table

Cf. A003188, A006068.
Cf. A002262, A003056, A268715.
Cf. A001477 (left edge), A001969 (right edge).
Cf. A268720 (row sums).

a88[n_] := BitXor[n, Floor[n/2]];
a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];
a68[0] = 0;
T[n_, k_] := a88[a68[n] + a68[k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268719 n) (A268715bi (A003056 n) (A002262 n)))

T(n,k) = A003188(A006068(n) + A006068(k)).

a(n) = A268715(A003056(n), A002262(n)). [As a linear sequence.]

A268720 Row sums of A268719: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

Original entry on oeis.org

0, 4, 13, 18, 53, 55, 63, 80, 217, 217, 205, 244, 234, 264, 305, 328, 881, 877, 841, 916, 790, 864, 977, 988, 900, 956, 1021, 1070, 1197, 1235, 1267, 1344, 3553, 3541, 3457, 3604, 3310, 3456, 3681, 3684, 3100, 3244, 3453, 3478, 3917, 3931, 3883, 4048, 3528, 3636, 3757, 3850, 4021, 4111, 4199, 4320, 4745, 4817

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Row sums of triangle A268719.

Cf. A003188, A006068, A268715.

(define (A268720 n) (add (lambda (k) (A268715bi n k)) 0 n)) ;; Code for A268715bi given in A268715.
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

Original entry on oeis.org

0, 2, 7, 18, 17, 48, 56, 80, 67, 122, 136, 194, 204, 268, 281, 328, 291, 378, 396, 498, 510, 640, 675, 792, 790, 886, 965, 1098, 1093, 1208, 1248, 1344, 1227, 1378, 1356, 1530, 1538, 1792, 1815, 2016, 2008, 2218, 2339, 2602, 2619, 2892, 2970, 3208, 3150, 3294, 3385, 3586, 3691, 4012, 4174, 4440, 4367, 4554, 4644

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A003188, A006068.

Cf. also A268720, A268836.

(define (A268837 n) (add (lambda (k) (A003188 (+ (A006068 k) (A006068 (- n k))))) 0 n))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 8, 4, 2, 11, 1, 13, 14, 7, 16, 8, 4, 19, 2, 21, 22, 11, 1, 25, 26, 13, 28, 14, 7, 31, 32, 16, 8, 35, 4, 37, 38, 19, 2, 41, 42, 21, 44, 22, 11, 47, 1, 49, 50, 25, 52, 26, 13, 55, 56, 28, 14, 59, 7, 61, 62, 31, 64, 32, 16, 67, 8, 69, 70, 35, 4, 73, 74, 37, 76, 38, 19, 79, 2, 81, 82, 41, 84, 42, 21, 87, 88, 44, 22, 91, 11, 93, 94, 47, 1, 97, 98, 49, 100

Offset: 1

Antti Karttunen, Nov 03 2016

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000069, A001969, A000265, A003188, A006068, A010060, A268669, A277813.
Cf. A277808 (gives the number of such iterations needed to reach a(n) from n).
Cf. A003945 (the positions of 1's in this sequence).

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = n, otherwise a(n) = a(A003188(A006068(n)/2)).
Other identities:
a(n) = A000069(A277813(n)).
If A010060(n) = 0 [when n is one of the evil numbers, A001969], then a(n)= a(A000265(n)) [the trailing zeros in binary expansion of n do not affect the result].
For all n >= 1, a(A000069(n)) = A000069(n). [By definition].
For all n > 1, a(A001969(n)) < A001969(n).

A286548 a(n) = A003188(n) - n.

Original entry on oeis.org

0, 0, 1, -1, 2, 2, -1, -3, 4, 4, 5, 3, -2, -2, -5, -7, 8, 8, 9, 7, 10, 10, 7, 5, -4, -4, -3, -5, -10, -10, -13, -15, 16, 16, 17, 15, 18, 18, 15, 13, 20, 20, 21, 19, 14, 14, 11, 9, -8, -8, -7, -9, -6, -6, -9, -11, -20, -20, -19, -21, -26, -26, -29, -31, 32, 32, 33, 31, 34, 34, 31, 29, 36, 36, 37, 35, 30, 30, 27, 25, 40, 40

Offset: 0

Antti Karttunen, May 18 2017

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A286546, A286549.

Scheme
```
(define (A286548 n) (- (A003188 n) n))
```

a(n) = A003188(n) - n.

a(n) = -A286546(A003188(n)). - Antti Karttunen, Oct 02 2017

A286549 Restricted growth sequence of A286548 (A003188(n) - n).

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 3, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 14, 7, 16, 16, 5, 10, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 22, 24, 25, 25, 26, 27, 28, 28, 29, 13, 30, 30, 11, 31, 32, 32, 31, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 43, 42, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 54, 54, 53, 46, 55, 55, 44, 49, 56, 56, 27

Offset: 0

Antti Karttunen, May 18 2017

nonn

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

Cf. A003188, A286547, A286548.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003188(n) = bitxor(n, n>>1);
A286548(n) = (A003188(n)-n);
write_to_bfile(0,rgs_transform(vector(65536,n,A286548(n-1))),"b286549.txt");

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cp's OEIS Frontend

A234613 Self-inverse permutation of nonnegative integers, "gray-blue" code: a(n) = A193231(A003188(n)).

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A265385 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1) + a(n-2)), with gray(m) = A003188(m).

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A265386 Sequence defined by a(1)=a(2)=1 and a(n) = gray(gray(a(n-1)) + gray(a(n-2))), with gray(m) = A003188(m).

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A265387 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1)) + gray(a(n-2)), with gray(m) = A003188(m).

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A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

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A268720 Row sums of A268719: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

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A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

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A286548 a(n) = A003188(n) - n.

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