A007306 -id:A007306 - cp's OEIS frontend

A284266 Odd bisection of A277700, binary weight of terms of A283975.

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

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A007306, A277700, A283484, A283975, A284265.

A264977[n_]:= If[n<2, n, If[EvenQ[n], 2 A264977[n/2], BitXor[A264977[(n - 1)/2], A264977[(n + 1)/2]]]]; Table[DigitCount[A264977[2n + 1], 2, 1], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

a(n) = if(n<2, n, if(n%2, bitxor(a((n - 1)/2), a((n + 1)/2)), 2*a(n/2)));
b(n) = if(n<1, 0, b(n\2) + n%2);
for(n=0, 150, print1(b(a(2*n + 1)),", ")) \\ Indranil Ghosh, Mar 28 2017

(define (A284266 n) (A277700 (+ n n 1)))
(define (A284266 n) (A000120 (A283975 n)))

a(n) = A277700((2*n)+1).
a(n) = A000120(A283975(n)).
Other identities. For all n >= 0:
A007306(1+n) = a(n) + 2*A284265(n).

A057431 Obtained by reading first the numerator then the denominator of fractions in full Stern-Brocot tree (A007305/A047679).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 3, 2, 3, 1, 1, 4, 2, 5, 3, 5, 3, 4, 4, 3, 5, 3, 5, 2, 4, 1, 1, 5, 2, 7, 3, 8, 3, 7, 4, 7, 5, 8, 5, 7, 4, 5, 5, 4, 7, 5, 8, 5, 7, 4, 7, 3, 8, 3, 7, 2, 5, 1, 1, 6, 2, 9, 3, 11, 3, 10, 4, 11, 5, 13, 5, 12, 4, 9, 5, 9, 7, 12, 8, 13, 7, 11, 7, 10, 8, 11, 7, 9, 5, 6, 6, 5

Offset: 0

N. J. A. Sloane, Sep 08 2000

When presented in this way, every row (e.g. row 3, 1 3 2 3 3 2 3 1) is a palindrome. - Joshua Zucker, May 11 2006

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences

Cf. A007305, A047679, A007306, A002487, A057432.

F:= proc(n) option remember; local t;
t:= L -> [[L[1], [L[1][1]+L[2][1], L[1][2]+L[2][2]], L[2]],
           [L[2], [L[2][1]+L[3][1], L[2][2]+L[3][2]], L[3]]][];
      if n=0 then [[[ ], [0, 1], [ ]], [[ ], [1, 0], [ ]]]
    elif n=1 then [[[0, 1], [1, 1], [1, 0]]]
             else map(t, F(n-1))
      fi
    end:
aa:= n-> map(x-> x[], [seq(map(x-> x[2], F(j))[], j=0..n)])[]:
aa(7);   # aa(n) gives the first 2^(n+1)+2 terms
# Alois P. Heinz, Jan 13 2011

sbt[n_] := Module[{R, L, Y, w, u},
   R = {{1, 0}, {1, 1}};
   L = {{1, 1}, {0, 1}};
   Y = {{1, 0}, {0, 1}};
   w[b_] := Fold[#1.If[#2 == 0, L, R]&, Y, b];
   u[a_] := {a[[2, 1]] + a[[2, 2]], a[[1, 1]] + a[[1, 2]]};
   Map[u, Map[w, Tuples[{0, 1}, n]]]];
Join[{0, 1, 1, 0}, Table[sbt[n], {n, 0, 5}]] // Flatten (* Jean-François Alcover, Sep 06 2022, after Peter Luschny in A007305 *)

More terms from Joshua Zucker, May 11 2006

A133404 Table of sum of numerator and denominator of Farey sequences, read by rows.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Nov 24 2007

Start with the Farey sequence F(n) of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Each row begins with the sum 1 from {0/1}. Each row ends with the sum 2 from {1/1}. The number of elements of the n-th row is A005728(n).

			F(1) = (0/1, 1/1) -> (0+1=1, 1+1=2).
F(2) = (0/1, 1/2, 1/1) -> (0+1=1, 1+2=3, 1+1=2).
F(3) = (0/1, 1/3, 1/2, 2/3, 1/1) -> (0+1=1, 1+3=4, 1+2=3, 2+3=5, 1+1=2).
F(4) = (0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1) -> (0+1=1, 1+4=5, 1+3=4, 1+2=3, 2+3=5, 3+4=7, 1+1=2).
The 5th row is formed from the 5th row of the table of Farey fractions:
F(5) = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1) whose sum of numerators and denominators is (1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2).
F(6) = (0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1) whose sums are (1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2).
F(7) = (0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1) whose sums are (1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2).
F(8) = (0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1) whose sums are (1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A005728, A007305, A007306, A049448.

Farey := proc(n) option remember: local j,s: if(n=1)then return {0,1}: else s:=procname(n-1): for j from 1 to n-1 do s := s union {j/n}: od: fi: end:
for n from 1 to 8 do F:=sort(convert(Farey(n),list)): nF:=nops(F): for m from 1 to nF do printf("%d, ",numer(F[m])+denom(F[m])): od: printf("\n"): od: # Nathaniel Johnston, Apr 27 2011

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Table[ Numerator[Farey[n]] + Denominator[Farey[n]], {n, 8}] // Flatten (* Robert G. Wilson v, Jun 10 2011 *)

A007305/A007306 maps to A007305+A007306 as shown in examples.

a(17) inserted by Nathaniel Johnston, Apr 27 2011

A174981 Numerators of the L-tree, left-to-right enumeration.

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 3, 5, 2, 5, 3, 4, 1, 3, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 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, 1, 5, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, 7, 16

Offset: 0

Peter Luschny, Apr 03 2010

a(n) is a subsequence of A174980. a(n)/A002487(n+2) enumerates all the reduced nonnegative rational numbers exactly once (L-tree).

			The sequence splits into rows of length 2^k:
0,
1, 1,
2, 3, 1, 2,
3, 5, 2, 5, 3, 4, 1, 3,
4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4,
...
The fractions are
0/1,
1/2, 1/1,
2/3, 3/2, 1/3, 2/1,
3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 3/1,
4/5, 7/4, 3/7, 8/3, 5/8, 7/5, 2/7, 7/2, 5/7, 8/5, 3/8, 7/3, 4/7, 5/4, 1/5, 4/1,
...

Edsger Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. EWD 578: More about the function fusc.
Peter Luschny, Rational Trees and Binary Partitions.
Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

Cf. A002487, A070879, A047679, A007306, A174980.

SternDijkstra := proc(L, p, n) local k, i, len, M; len := nops(L); M := L; k := n; while k > 0 do M[1+(k mod len)] := add(M[i], i = 1..len); k := iquo(k, len); od; op(p, M) end:
Ltree := proc(n) 5*2^ilog2(n+1); SternDijkstra([0,1], 1, n + 2 + %) / SternDijkstra([1,0], 2, n + 2) end:
a := proc(n) 5*2^ilog2(n+1); SternDijkstra([0,1], 1, n + 2 + %) end:
seq(a(n), n=0..90);

SternDijkstra[L_, p_, n_] := Module[{k, i, len, M}, len := Length[L]; M = L; k = n; While[k > 0, M[[1 + Mod[k, len]]] = Sum[M[[i]], {i, 1, len}]; k = Quotient[k, len]]; M[[p]]]; Ltree[n_] := With[{k = 5*2^Simplify[ Floor[ Log[2, n + 1]]]}, SternDijkstra[{0, 1}, 1, n + 2 + k]/ SternDijkstra[{1, 0}, 2, n + 2]]; a[0] = 0; a[n_] := With[{k = 5*2^Simplify[ Floor[ Log[2, n + 1]]]}, SternDijkstra[{1, 0}, 1, n + 2 + k]]; row[0] = {a[0]}; row[n_] := Table[a[k], {k, 2^n - 3, 2^(n+1) - 4}] // Reverse; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

A282715 Number of nonzero entries in row n of the base-3 generalized Pascal triangle P_3.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 6, 7, 7, 6, 4, 6, 5, 7, 6, 7, 5, 6, 4, 5, 7, 8, 8, 7, 10, 10, 11, 9, 7, 8, 10, 7, 5, 8, 11, 10, 9, 10, 13, 12, 13, 10, 12, 11, 11, 8, 5, 8, 7, 10, 9, 11, 8, 10, 7, 10, 12, 13, 11, 8, 11, 13, 12, 10, 7, 10, 8, 11, 9, 10

Offset: 0

N. J. A. Sloane, Mar 02 2017

nonn

It would be nice to have an entry for the triangle P_3 itself (compare A282714 which gives the base-2 triangle P_2).

			The number of nonzero entries in the n-th row of the following triangle:
  1
  1 1
  1 0 1
  1 1 0 1
  1 2 0 0 1
  1 1 1 0 0 1
  1 0 1 0 0 0 1
  1 1 1 0 0 0 0 1
  1 0 2 0 0 0 0 0 1
  1 1 0 2 0 0 0 0 0 1
  1 2 0 1 1 0 0 0 0 0 1
  1 1 1 1 0 1 0 0 0 0 0 1
  1 2 0 2 1 0 0 0 0 0 0 0 1
  1 3 0 0 3 0 0 0 0 0 0 0 0 1

Lars Blomberg, Table of n, a(n) for n = 0..10000
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881, Section 7.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

Cf. A007306, A282714.

# reuses code snippets of A282714
A282715 := proc(n)
    add(min(P(n,k,3),1),k=0..n) ;
end proc:
seq(A282715(n),n=0..100) ; # R. J. Mathar, Mar 03 2017

row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 3], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];
a[n_] := Count[row[n], _?Positive];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 28 2017 *)

Leroy et al. (2017) state some conjectured recurrences.

More terms from Lars Blomberg, Mar 03 2017

A282720 Number of nonzero terms in first n rows of the base-2 generalized Pascal triangle P_2 (see A282714).

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 18, 23, 27, 32, 39, 47, 54, 61, 69, 76, 81, 87, 96, 107, 117, 128, 141, 153, 162, 171, 183, 196, 207, 217, 228, 237, 243, 250, 261, 275, 288, 303, 321, 338, 351, 365, 384, 405, 423, 440, 459, 475, 486, 497, 513, 532, 549, 567

Offset: 0

N. J. A. Sloane, Mar 03 2017

nonn

Same as partial sums of (A007306 with initial 1 omitted).

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Julien Leroy, Michel Rigo, and Manon Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

Cf. A282714, A007306.

Accumulate@ Prepend[Array[Sum[Mod[Binomial[# + k - 1, 2 k], 2], {k, 0, #}] &, 53], 0] (* Michael De Vlieger, Sep 04 2018, after Jean-François Alcover at A007306 *)

f(n) = n--; sum(k=0, n, binomial(n+k, n-k)%2);
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, Oct 29 2023

A283974 Numbers n for which A002487(n-1) AND A002487(n) > 0 where AND is bitwise-and (A004198).

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 14, 17, 18, 19, 20, 23, 24, 25, 26, 29, 30, 31, 32, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 86, 89, 92, 95, 96, 97, 98, 101, 104, 107, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120

Offset: 1

Antti Karttunen, Mar 21 2017

Numbers n such that the binary representations of A002487(n-1) and A002487(n) have at least one 1-bit in a common shared position.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A283973 (complement).
Cf. A002487, A004198, A007306, A283986, A283987.
Positions of nonzeros in A283988.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Flatten@ Position[Table[BitAnd[a[n - 1], a@ n], {n, 120}], k_ /; k > 0] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
D(n) = if(n<1, 1, sum(k=0, n, binomial(n + k - 1, 2*k)%2))
for(n=1, 120, if(bitor(A(n - 1), A(n)) != D(n), print1(n, ", "))) \\ Indranil Ghosh, Mar 23 2017

A284009 Number of primes (counted with multiplicity) dividing lcm(A260443(n), A260443(n+1)): a(n) = A001222(A284008(n)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 4, 5, 5, 6, 6, 6, 6, 4, 5, 6, 6, 8, 9, 10, 9, 8, 8, 8, 8, 9, 9, 8, 8, 5, 6, 7, 7, 10, 10, 13, 13, 11, 12, 13, 12, 15, 16, 14, 13, 10, 10, 10, 10, 13, 14, 15, 14, 12, 12, 11, 11, 12, 12, 10, 10, 6, 7, 8, 8, 12, 12, 15, 15, 14, 15, 17, 17, 20, 21, 20, 19, 14, 15, 16, 15, 21, 22, 24, 24, 20, 21, 20, 18, 21, 21, 17, 17, 12, 12, 12, 12, 17, 18

Offset: 0

Antti Karttunen, Mar 22 2017

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

Cf. A001222, A007306, A277328, A284008.

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ ( a[First[#]] ^ Last[#] & ) /@ FactorInteger[n]; (* after Jean-François Alcover, in A003961 *) A[n_]:= If[n<2, n + 1, If[EvenQ[n], a[A[n/2]], A[(n - 1)/2] A[(n + 1)/2]]] ;Table[A[n], {n, 0, 51}]  (* sequence A260443 *) Table[PrimeOmega[LCM[A[n], A[n + 1]]], {n, 0, 101}] (* Indranil Ghosh, Mar 22 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A284008(n) = lcm(A260443(n),A260443(1+n));
A284009(n) = bigomega(A284008(n));

(define (A284009 n) (A001222 (A284008 n)))

a(n) = A001222(A284008(n)).
Other identities. For all n >= 0:
a(n) + A277328(n) = A007306(1+n).

A287731 Bisection of A287729.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 3, 4, 5, 5, 4, 3, 3, 2, 1, 1, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6, 5, 7, 8, 7, 7, 8, 7, 5, 4, 5, 5, 4, 3, 3, 2, 1

Offset: 1

I. V. Serov, Jun 01 2017

A287732(n)/a(n) enumerates all reduced fractions along the Stern-Brocot Tree. See the Serov link below.

I. V. Serov, Table of n, a(n) for n = 1..8192
I. V. Serov, The Stern-Brocot Tree as Sequence A287732/A287731
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences
Index entries for fraction trees

Cf. A002487, A007306, A287729, A287730, A287732.

def c(n): return 1 if n==1 else s(n/2) if n%2==0 else s((n - 1)/2) + s((n + 1)/2)
def s(n): return 0 if n==1 else c(n/2) if n%2==0 else c((n - 1)/2) + c((n + 1)/2)
def a(n): return c(2*n - 1) # Indranil Ghosh, Jun 08 2017

a(n) = A287729(2*n-1), n > 0.
a(n) = A287730(n-1) + A287730(n), n > 0.
a(n) = A007306(n) - A287732(n) .
Consider for n > 1 the binary expansion b(1:t) of n-1 without the leading 1.
Recurse: c=s=1; for j=1:t {if b(t-j+1) == mod(t,2) s = s+c; else c = c+s;}
Then: c = a(n) and s = A287732(n);

A287732 Bisection of A287730.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 5, 4, 5, 5, 4, 3, 3, 2, 1, 1, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6

Offset: 1

I. V. Serov, Jun 01 2017

a(n)/A287731(n) enumerates all reduced fractions along the Stern-Brocot Tree. See the Serov link below.

I. V. Serov, Table of n, a(n) for n = 1..8192
I. V. Serov, OEIS: The Stern-Brocot Tree as Sequence A287732/A287731
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for fraction trees
Index entries for sequences related to Stern's sequences

Cf. A002487, A007306, A287729, A287730, A287731.

def c(n): return 1 if n==1 else s(n/2) if n%2==0 else s((n - 1)/2) + s((n + 1)/2)
def s(n): return 0 if n==1 else c(n/2) if n%2==0 else c((n - 1)/2) + c((n + 1)/2)
def a(n): return s(2*n - 1) # Indranil Ghosh, Jun 08 2017

a(n) = A287730(2*n-1), n > 0.
a(n) = A287730(n-1) + A287730(n), n > 0.
a(n) = A007306(n) - A287732(n).
Consider for n > 1 the binary expansion b(1:t) of n-1 without the leading 1.
Recurse: c=s=1; for j=1:t {if b(t-j+1) == mod(t,2) s = s+c; else c = c+s;}
Then: c = A287731(n) and s = a(n);

cp's OEIS Frontend

A284266 Odd bisection of A277700, binary weight of terms of A283975.

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A057431 Obtained by reading first the numerator then the denominator of fractions in full Stern-Brocot tree (A007305/A047679).

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A133404 Table of sum of numerator and denominator of Farey sequences, read by rows.

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A174981 Numerators of the L-tree, left-to-right enumeration.

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A282715 Number of nonzero entries in row n of the base-3 generalized Pascal triangle P_3.

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A282720 Number of nonzero terms in first n rows of the base-2 generalized Pascal triangle P_2 (see A282714).

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A283974 Numbers n for which A002487(n-1) AND A002487(n) > 0 where AND is bitwise-and (A004198).

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