A003714 -id:A003714 - cp's OEIS frontend

A379147 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the integers m such that A184617(abs(m)) = A003714(n).

0, -1, 1, -2, 2, -4, 4, -5, -3, 3, 5, -8, 8, -9, -7, 7, 9, -10, -6, 6, 10, -16, 16, -17, -15, 15, 17, -18, -14, 14, 18, -20, -12, 12, 20, -21, -19, -13, -11, 11, 13, 19, 21, -32, 32, -33, -31, 31, 33, -34, -30, 30, 34, -36, -28, 28, 36

Offset: 0

Rémy Sigrist, Dec 16 2024

A permutation of the integers (Z).
For any n >= 0:
- in the Zeckendorf expansion of n,
- replace each Fibonacci number, say A000045(2+i) with i >= 0, by 2^i or -2^i,
- the various values obtained make up the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  ----------------------------------
   0  0
   1  -1, 1
   2  -2, 2
   3  -4, 4
   4  -5, -3, 3, 5
   5  -8, 8
   6  -9, -7, 7, 9
   7  -10, -6, 6, 10
   8  -16, 16
   9  -17, -15, 15, 17
  10  -18, -14, 14, 18
  11  -20, -12, 12, 20
  12  -21, -19, -13, -11, 11, 13, 19, 21
  13  -32, 32
  14  -33, -31, 31, 33
  15  -34, -30, 30, 34

Rémy Sigrist, Table of n, a(n) for n = 0..10922 (rows for n = 0..609 flattened)
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A003714, A007895, A184617, A379175.

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
row(n) = { my (z = tozeck(n), r = [0], b); while (z, z -= b = 2^valuation(z, 2); r = concat([v - b | v <- r], [v + b | v <- r]);); return (r); }

T(n, 1) = -A003714(n).
T(n, 2^A007895(n)) = A003714(n).
T(n, k) = -T(n, 2^A007895(n)+1-k) for k = 1..2^A007895(n).

A379213 a(n) is the number of nonnegative integers m such that A184615(m) = A003714(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 5, 2, 1, 2, 1, 8, 3, 2, 2, 1, 3, 1, 1, 13, 5, 3, 4, 2, 3, 1, 1, 5, 2, 1, 2, 1, 21, 8, 5, 6, 3, 6, 2, 2, 5, 2, 1, 2, 1, 8, 3, 2, 2, 1, 3, 1, 1, 34, 13, 8, 10, 5, 9, 3, 3, 10, 4, 2, 4, 2, 8, 3, 2, 2, 1, 3, 1, 1, 13, 5, 3, 4, 2, 3, 1, 1, 5

Offset: 0

Rémy Sigrist, Dec 18 2024

To compute a(n) for n > 0:
- consider the runs in the binary expansion A003714(n), say (r_1, ..., r_w) (with w = A005811(A003714(n))),
- then a(n) = A000045(r_1) * ... * A000045(r_{w-1}) * A000045(r_w + 1).

Cf. A000045, A003714, A005811, A101211, A184615.

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
a(n) = { my (z = tozeck(n), r = toruns(z), v = 1); forstep (i = 2, #r, 2, v *= fibonacci(r[i] + if (i==#r, 1, 0));); return (v); }

a(n) = Product_{k = 1..A005811(A003714(n))} A000045(A101211(A003714(n), k) + [k = A005811(A003714(n))]) (where [] is the Iverson bracket).

A379261 Irregular triangle T(n, k), n >= 0, k = 1..A379213(n), read by rows; the n-th row lists the nonnegative integers m such that A184615(m) = A003714(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 13, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 25, 29, 33, 26, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 54, 55, 56, 59, 60, 62, 63, 64, 45, 49, 57, 61, 65, 50, 58, 66, 51, 52, 67, 68

Offset: 0

Rémy Sigrist, Dec 19 2024

As a flat sequence, this is a permutation of the nonnegative integers with inverse A379262.

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------------
   0  0
   1  1
   2  2
   3  3, 4
   4  5
   5  6, 7, 8
   6  9
   7  10
   8  11, 12, 14, 15, 16
   9  13, 17
  10  18
  11  19, 20
  12  21
  13  22, 23, 24, 27, 28, 30, 31, 32
  14  25, 29, 33
  15  26, 34

Rémy Sigrist, Table of n, a(n) for n = 0..10922
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A184615, A379213, A379262 (inverse).

PARI
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\\ See Links section.
```

T(n, A379213(n)) = A003714(n).

A089939 T(i,j) = 1 if F(i) AND F(j) = 0, otherwise 0, where F is A003714 and AND is the bitwise logical-and operation. Table read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1

Offset: 0

Marc LeBrun, Nov 15 2003

Encodes which row/column patterns may be adjacent in 01-matrices where no two 0 elements may be adjacent. Contains many interesting recursive patterns such as Fibonacci-sized blocks of 0's along main diagonal.

			T(3,4) = 0 because F(3) AND F(4) = 4 AND 5 = 1, which is nonzero.

Cf. A003714 (Fibbinary), A005614 (row or column 1).
Cf. A000045, A047999.
Cf. A001333, A051736, A051737, A089934, A089935, A089936, A089937, A089938.

Name clarified by Jon E. Schoenfield, Aug 19 2022

A144759 Primes the squares of which are Fibbinary numbers (A003714).

Original entry on oeis.org

2, 3, 13, 17, 23, 47, 137, 257, 521, 577, 727, 739, 1033, 1153, 1181, 1471, 2081, 2113, 3251, 4129, 4253, 8209, 8329, 8353, 11597, 11677, 11779, 11971, 12503, 16417, 17053, 18433, 18583, 24799, 26317, 32801, 32833, 35267, 35393, 46703, 52177, 65537, 66569, 74257, 92801, 98327

Offset: 1

Vladimir Shevelev, Sep 20 2008

			a(1)=2 since 2^2 is in A003714.

Robert Israel, Table of n, a(n) for n = 1..505

Cf. A003714, A001248.

fibbin:= proc(n) Bits:-Xor(n,2*n) = 3*n end proc:
select(t -> isprime(t) and fibbin(t^2), [2,seq(i,i=3..10^5,2)]); # Robert Israel, Feb 15 2016

msb(n)=my(k=1); while(k<=n, k<<=1); k>>1;
lista(nn) = {forprime(p=1, nn, n=p^2; k=bitand(n, n<<1); if(k, n=bitor(n, msb(k)-1), print1(p, ", ")););} \\ Michel Marcus, Feb 15 2016

More terms from Michel Marcus, Feb 15 2016

A300896 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n) + a(n+1) + a(n+2) is a Fibbinary number (A003714).

Original entry on oeis.org

1, 2, 5, 3, 8, 6, 4, 7, 9, 16, 11, 10, 12, 14, 15, 13, 36, 17, 19, 28, 18, 20, 26, 22, 21, 23, 24, 25, 31, 29, 68, 32, 30, 66, 33, 34, 61, 35, 37, 56, 39, 38, 51, 40, 41, 47, 42, 43, 44, 45, 48, 52, 46, 50, 49, 62, 27, 55, 54, 53, 57, 58, 141, 59, 60, 137, 63

Offset: 1

Rémy Sigrist, Mar 14 2018

This sequence has similarities with A266191: here we consider the sum of triples of consecutive terms, there their product.
This sequence is conjectured to be a permutation of the natural numbers.
See A300890 for a similar sequence.

			The first terms, alongside the binary representation of triples of consecutive terms, are:
  n   a(n)    binary(a(n) + a(n+1) + a(n+2))
  --  ----    ------------------------------
   1     1       1000
   2     2       1010
   3     5      10000
   4     3      10001
   5     8      10010
   6     6      10001
   7     4      10100
   8     7     100000
   9     9     100100
  10    16     100101
  11    11     100001
  12    10     100100
  13    12     101001
  14    14     101010
  15    15    1000000
  16    13    1000010
  17    36    1001000
  18    17    1000000
  19    19    1000001
  20    28    1000010

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A300896

Cf. A003714, A266191, A300890.

PARI
```
See Links section.
```

A301572 a(n) = distance from n to nearest Fibbinary number (A003714).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 1, 0, 0

Offset: 0

Altug Alkan and Rémy Sigrist, Mar 23 2018

This sequence is unbounded.

			a(12) = a(14) = 2 because 10 is the nearest Fibbinary number to 12 and 16 is the nearest Fibbinary number to 14.

Cf. A003714.

a(n) = for (k=0, oo, if (bitand(n-k, 2*(n-k))==0 || bitand(n+k, 2*(n+k))==0, return (k)))

a(n) = 0 iff n belongs to A003714.

0 <= 2 * a(n) - a(2 * n) <= 1.

A307591 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, A000196(2*a(n) + a(n+1)) is a Fibbinary number (A003714).

Original entry on oeis.org

1, 2, 3, 10, 4, 8, 5, 6, 7, 9, 11, 12, 40, 13, 38, 14, 36, 15, 34, 16, 32, 17, 30, 18, 28, 19, 26, 20, 24, 21, 22, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 174, 52, 152, 42, 172, 56, 144, 43, 170, 60, 136, 44, 168, 64, 128, 45, 166, 68, 120, 46, 164, 72, 112

Offset: 1

Rémy Sigrist, Apr 19 2019

This sequence is a variant of A300890 and has interesting graphical features.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (amended by _Georg Fischer_, May 05 2019)
Rémy Sigrist, Scatterplot of the first 150000 terms

Cf. A000196, A003714, A300890.

s=0; v=1; for (n=1, 64, print1 (v ", "); s+=2^v; for (o=1, oo, if (!bittest(s, o), f=sqrtint(2*v+o); if (bitand(f, 2*f)==0, v=o; break))))

A344180 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 0, where f(n) = 0 if n is a Fibbinary number (A003714), otherwise f(n) = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 5, 6, 7, 8, 9, 1, 1, 1, 10, 1, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 1, 1, 21, 1, 1, 22, 23, 1, 1, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 1, 1, 1, 45, 1, 1, 46, 47, 1, 1, 1, 48, 49, 50, 51, 52, 1, 1, 1, 53, 1, 1, 54, 55, 56, 57

Offset: 0

Antti Karttunen, May 16 2021

nonn

For all i, j:
  a(i) = a(j) => A085357(i) = A085357(j),
  a(i) = a(j) => A213370(i) = A213370(j),
  a(i) = a(j) => A344182(i) = A344182(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A003714 (positions of 1's), A085357, A213370, A344182.

Cf. also A324400.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux344180(n) = if(!bitand(n,n+n),0,n);
v344180 = rgs_transform(vector(1+up_to,n,Aux344180(n-1)));
A344180(n) = v344180[1+n];

A000695 Moser-de Bruijn sequence: sums of distinct powers of 4.

Original entry on oeis.org

0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, 85, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 1024, 1025, 1028, 1029, 1040, 1041, 1044, 1045, 1088, 1089, 1092, 1093, 1104, 1105, 1108, 1109, 1280, 1281, 1284, 1285

Offset: 0

N. J. A. Sloane

Although this is a list, it has offset 0 for both historical and mathematical reasons.
Numbers whose set of base-4 digits is a subset of {0,1}. - Ray Chandler, Aug 03 2004, corrected by M. F. Hasler, Oct 16 2018
Numbers k such that the sum of the base-2 digits of k = sum of the base-4 digits of k. - Clark Kimberling
Numbers having the same representation in both binary and negabinary (A039724). - Eric W. Weisstein
This sequence has many other interesting and useful properties. Every term k corresponds to a unique pair i,j with k = a(i) + 2*a(j) (i=A059905(n), j=A059906(n)) -- see A126684. Every list of numbers L = [L1,L2,L3,...] can be encoded uniquely by "recursive binary interleaving", where f(L) = a(L1) + 2*a(f([L2,L3,...])) with f([])=0. - Marc LeBrun, Feb 07 2001
This may be described concisely using the "rebase" notation b[n]q, which means "replace b with q in the expansion of n", thus "rebasing" n from base b into base q. The present sequence is 2[n]4. Many interesting operations (e.g., 10[n](1/10) = digit reverse, shifted) are nicely expressible this way. Note that q[n]b is (roughly) inverse to b[n]q. It's also natural to generalize the idea of "basis" so as to cover the likes of F[n]2, the so-called "fibbinary" numbers (A003714) and provide standard ready-made images of entities obeying other arithmetics, say like GF2[n]2 (e.g., primes = A014580, squares = the present sequence, etc.). - Marc LeBrun, Mar 24 2005
a(n) is also equal to the product n X n formed using carryless binary multiplication (A059729, A063010). - Henry Bottomley, Jul 03 2001
Numbers k such that A004117(k) is odd. - Pontus von Brömssen, Nov 25 2008
Fixed point of the morphism: 0 -> 01; 1 -> 45; 2 -> 89; ...; n -> (4n)(4n+1), starting from a(0)=0. - Philippe Deléham, Oct 22 2011
If n is even and present, so is n+1. - Robert G. Wilson v, Oct 24 2014
Also: interleave binary digits of n with 0's. (Equivalent to the "rebase" interpretation above.) - M. F. Hasler, Oct 16 2018
Named after the Austrian-Canadian mathematician Leo Moser (1921-1970) and the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - Amiram Eldar, Jun 19 2021
Conjecture: The sums of distinct powers of k > 2 can be constructed as the following (k-1)-ary rooted tree. For each n the tree grows and a(n) is then the total number of nodes. For n = 1, the root of the tree is added. For n > 1, if n is odd one leaf of depth n-2 grows one child. If n is even all leaves of depth >= (n - 1 - A000225(A001511(n/2))) grow the maximum number of children. An illustration is provided in the links. - John Tyler Rascoe, Oct 09 2022

			G.f.: x + 4*x^2 + 5*x^3 + 16*x^4 + 17*x^5 + 20*x^6 + 21*x^7 + 64*x^8 + ...
If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = 4^4 + 4^3 + 4 + 1 = 325; k = b_0 + b_2*2 + b_4*2^2 = 5, l = b_1 + b_3*2 = 3, such that a(5)=17, a(3)=5 and 27 = 17 + 2*5. - _Vladimir Shevelev_, Nov 10 2008

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1023
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197.
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197.
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), pp. 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
David Applegate, Marc LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq., Vol. 14 (2011), Article 11.9.8.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 59-60, pp. 750-751.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
N. G. de Bruijn, Some direct decompositions of the set of integers, Math. Comp., Vol. 18, No. 88 (1964), pp. 537-546.
Karl Dilcher and Larry Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, Vol. 22, No. 2 (2015), #P2.24.
Roger B. Eggleton, Maximal Midpoint-Free Subsets of Integers, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
S. J. Eigen, Y. Ito, and V. S. Prasad, Universally bad integers and the 2-adics, J. Number Theory, Vol. 107, No. 2 (2004), pp. 322-334.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 44.
Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, Electronic Journal of Combinatorial Number Theory, Volume 15 #A23.
Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See m(n) p. 11.
Leo Moser, An application of generating series, Math. Mag., Vol. 35, No. 1 (1962), pp. 37-38.
Leo Moser, An application of generating series, Math. Mag., Vol. 35, No. 1 (1962), pp. 37-38. [Annotated scanned copy]
John Tyler Rascoe, Illustration of terms.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016-2017.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Stephen Nicholas Swatman, Ana-Lucia Varbanescu, Andy D. Pimentel, Andreas Salzburger, and Attila Krasznahorkay, Finding Morton-Like Layouts for Multi-Dimensional Arrays Using Evolutionary Algorithms, arXiv:2309.07002 [cs.NE], 2023.
Eric Weisstein's World of Mathematics, Moser-de Bruijn Sequence.
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Morton code. (also known as Z-order curve. Cf. Marc LeBrun's comments about binary interleaving.)
Index entries for 2-automatic sequences.

For generating functions Product_{k>=0} (1 + a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Main diagonal of A048720, second column of A048723.
Cf. A000225, A000302, A001511, A007583, A059884, A059901, A059904, A059905, A059906, A007088, A033042-A033052, A126684, A145812.
A062880(n) = 2*a(n); A001196(n) = 3*a(n).
Row 4 of array A104257.

uint32_t a_next(uint32_t a_n) { return (a_n + 0xaaaaaaab) & 0x55555555; } /* Falk Hüffner, Jan 24 2022 */

a000695 n = if n == 0 then 0 else 4 * a000695 n' + b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014, Dec 03 2011

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 4
    end
r end; [a(n) for n in 0:51] |> println # Peter Luschny, Jan 03 2021

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( (&+[4^k*x^(2^k)/(1+x^(2^k)): k in [0..20]])/(1-x) )); // G. C. Greubel, Dec 06 2018

a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*4 od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 16 2013

Table[FromDigits[Riffle[IntegerDigits[n, 2], 0], 2], {n, 0, 51}] (* Jacob A. Siehler, Jun 30 2010 *)
Table[FromDigits[IntegerDigits[n, 2], 4], {n, 0, 51}] (* IWABUCHI Yu(u)ki, Apr 06 2013 *)
Union@ Flatten@ NestList[ Join[ 4#, 4# + 1] &, {0}, 6] (* Robert G. Wilson v, Aug 30 2014 *)
Select[ Range[0, 1320], Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#, 4] &] (* Robert G. Wilson v, Oct 24 2014 *)
Union[FromDigits[#,4]&/@Flatten[Table[Tuples[{0,1},n],{n,6}],1]] (* Harvey P. Dale, Oct 03 2015 *)
a[ n_] := Which[n < 1, 0, EvenQ[n], a[n/2] 4, True, a[n - 1] + 1]; (* Michael Somos, Nov 30 2016 *)

a(n)=n=binary(n);sum(i=1,#n,n[i]*4^(#n-i)) \\ Charles R Greathouse IV, Mar 04 2013

{a(n) = if( n<1, 0, n%2, a(n-1) + 1, a(n/2) * 4)}; /* Michael Somos, Nov 30 2016 */

A000695(n)=fromdigits(binary(n),4) \\ M. F. Hasler, Oct 16 2018

def a(n):
    n = bin(n)[2:]
    x = len(n)
    return sum(int(n[i]) * 4**(x - 1 - i) for i in range(x))
[a(n) for n in range(101)] # Indranil Ghosh, Jun 25 2017

def a():
    x = 0
    while True:
        yield x
        y = ~(x << 1)
        x = (x - y) & y # Falk Hüffner, Dec 21 2021

from itertools import count, islice
def A000695_gen(): # generator of terms
    yield (a:=0)
    for n in count(1):
        yield (a := a+((1<<((~n & n-1).bit_length()<<1)+1)+1)//3)
A000695_list = list(islice(A000695_gen(),30)) # Chai Wah Wu, Feb 22 2023

def A000695(n): return int(bin(n)[2:],4) # Chai Wah Wu, Aug 21 2023

s=(sum(4^k*x^(2^k)/(1+x^(2^k)) for k in range(10))/(1-x)).series(x, 60); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018

G.f.: 1/(1-x) * Sum_{k>=0} 4^k*x^2^k/(1+x^2^k). - Ralf Stephan, Apr 27 2003
Numbers k such that the coefficient of x^k is > 0 in Product_{n>=0} 1+x^(4^n). - Benoit Cloitre, Jul 29 2003
For n >= 1, a(n) = a(n-1) + (4^t+2)/6, where t is such that 2^t||2n,or t=A007814(2n). a(n) = (A145812(n+1) - 1)/2. - Vladimir Shevelev, Nov 07 2008
To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_j*2^(2j). The Diophantine equation a(k)+2a(l)=n has the unique solution: k=Sum b_(2j)*2^j, l=Sum b_(2j+1)*2^j. - Vladimir Shevelev, Nov 10 2008
If a(k)*a(l)=a(m), then k*l=m (the inverse, generally speaking, is not true). - Vladimir Shevelev, Nov 21 2008
Let F(x) be the generating function, then F(x)*F(x^2) = 1/(1-x). - Joerg Arndt, May 12 2010
a(n+1) = (a(n) + 1/3) & -1/3, where & is bitwise AND, -1/3 is represented as the infinite dyadic ...010101 (just as -1 is ...111111 in two's complement) and +1/3 is ...101011. - Marc LeBrun, Sep 30 2010
a(n) = Sum_{k>=0} {A030308(n,k)*b(k)} with b(k) = 4^k = A000302(k). - Philippe Deléham, Oct 18 2011
A182560(6*a(n)) = 0. - Reinhard Zumkeller, May 05 2012
G.f.: x/(1-x^2) + 4*x^2/((1-x)*(W(0) - 4*x - 4*x^2)), where W(k) = 1 + 4*x^(2^k) + 5*x^(2^(k+1)) - 4*x^(2^(k+1))*(1 + x^(2^(k+1)))^2/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 04 2014
liminf a(n)/n^2 = 1/3 and limsup a(n)/n^2 = 1. - Gheorghe Coserea, Sep 15 2015
Let f(x) = (Sum_{k=-oo..oo} floor(x*2^k)/4^k)/2. Then f(x) is a real-valued extension of a(n), which a(n) approximates in the sense that f(x) = lim_{k->oo} a(floor(x*2^k))/a(2^k). - Velin Yanev, Nov 28 2016
G.f. A(x) satisfies x/(1 - x^2) = A(x) - 4 * (1+x) * A(x^2). - Michael Somos, Nov 30 2016
a(2^k) = 4^k = A000302(k). a(n + 2^k) = a(n) + a(2^k) for 2^k > n >= 1. - David A. Corneth, Oct 16 2018
Sum_{n>=1} 1/a(n) = 1.886176434476107244547259512076353532930680508099044818673061351780360211128... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

cp's OEIS Frontend

A379147 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the integers m such that A184617(abs(m)) = A003714(n).

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A379213 a(n) is the number of nonnegative integers m such that A184615(m) = A003714(n).

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A379261 Irregular triangle T(n, k), n >= 0, k = 1..A379213(n), read by rows; the n-th row lists the nonnegative integers m such that A184615(m) = A003714(n).

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A089939 T(i,j) = 1 if F(i) AND F(j) = 0, otherwise 0, where F is A003714 and AND is the bitwise logical-and operation. Table read by antidiagonals.

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A144759 Primes the squares of which are Fibbinary numbers (A003714).

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A300896 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n) + a(n+1) + a(n+2) is a Fibbinary number (A003714).

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A301572 a(n) = distance from n to nearest Fibbinary number (A003714).

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A307591 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, A000196(2*a(n) + a(n+1)) is a Fibbinary number (A003714).

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