A277749 -id:A277749 - cp's OEIS frontend

A277750 Denominators of rationals R_n associated with an analog of Stern's diatomic sequence for Z[sqrt(2)].

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

Offset: 1

N. J. A. Sloane, Nov 08 2016

			2, 1, 4, 3/2, 2/3, 3, 4/3, 1/2, 6, 5/3, 4/5, 7/2, 10/7, 3/5, 8/3, 5/4, 2/5, 5, 8/5, 3/4, ...

Gheorghe Coserea, Table of n, a(n) for n = 1..54321
Florin P. Boca, Christopher Linden, On Minkowski type question mark functions associated with even or odd continued fractions, arXiv:1705.01238 [math.DS], 2017. See p. 10.
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Lionel Ponton, Two trees enumerating the positive rationals, arXiv:1707.02366 [math.NT], 2017. See tree p. 4.
Lionel Ponton, Two trees enumerating the positive rationals, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.
Lukas Spiegelhofer, A Digit Reversal Property for an Analogue of Stern's Sequence, Journal of Integer Sequences, Vol. 20 (2017), #17.10.8.

Cf. A002487 (Stern's diatomic sequence), A277749 (numerators).

R[1] = 2; R[n_] := R[n] = 4 IntegerExponent[n, 3] + 2 - 2/R[n-1];
Table[R[n] // Denominator, {n, 1, 100}] (* Jean-François Alcover, Sep 03 2018, after Gheorghe Coserea *)

seq(N) = {
  my(v = vector(N)); v[1] = 2;
  for (n = 2, N, v[n] = 4*valuation(n,3) + 2 - 2 / v[n-1]);
  return(v);
};
apply(denominator, seq(88))

a(n) = denominator(R(n)), where R(n) = 4 * A007949(n) + 2 - 2/R(n-1), with R(1) = 2. - Gheorghe Coserea, Nov 11 2016

More terms from Gheorghe Coserea, Nov 11 2016

A289772 a(n) is the numerator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 5, 3, 4, 1, 5, 4, 7, 3, 2, 3, 7, 4, 5, 1, 4, 3, 5, 2, 1, 3, 8, 5, 7, 2, 11, 9, 16, 7, 5, 8, 19, 11, 14, 3, 13, 10, 17, 7, 4, 5, 11, 6, 7, 1, 8, 7, 13, 6, 5, 9, 22, 13, 17, 4, 19, 15, 26, 11, 7, 10, 23, 13, 16, 3, 11, 8, 13, 5, 2, 5, 13, 8, 11, 3

Offset: 0

Michel Marcus, Jul 12 2017

For n>0, a(n)/A289773(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.

			Tree of rationals begin:
0;
1/3;
1/2, 2/3, 1, 1/6, 2/5;
5/9, 3/4, 4/3, 1/5, 5/12, 4/7, 7/9, 3/2, 2/9, 3/7, 7/12, 4/5, 5/3, 1/4, 4/9, 3/5, 5/6, 2, 1/9, 3/8, 8/15, 5/7, 7/6, 2/11, 11/27;
...

Robert Israel, Table of n, a(n) for n = 0..10000
Lionel Ponton, Two trees enumerating the positive rationals, arXiv:1707.02366 [math.NT], 2017. See p. 7.
Lionel Ponton, Two trees enumerating the positive rationals, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

Cf. A002487, A277749, A277750, A289773.

b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:
b(0):= 0:
map(numer@b, [$0..100]); # Robert Israel, Jul 12 2017

a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Numerator@ a@ n, {n, 0, 80}] (* Michael De Vlieger, Jul 12 2017 *)

b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));
lista(nn) = for (n=0, nn, print1(numerator(b(n)), ", "));

A289773 a(n) is the denominator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.

Original entry on oeis.org

1, 3, 2, 3, 1, 6, 5, 9, 4, 3, 5, 12, 7, 9, 2, 9, 7, 12, 5, 3, 4, 9, 5, 6, 1, 9, 8, 15, 7, 6, 11, 27, 16, 21, 5, 24, 19, 33, 14, 9, 13, 30, 17, 21, 4, 15, 11, 18, 7, 3, 8, 21, 13, 18, 5, 27, 22, 39, 17, 12, 19, 45, 26, 33, 7, 30, 23, 39, 16, 9, 11, 24, 13, 15, 2, 15, 13

Offset: 0

Michel Marcus, Jul 12 2017

For n>0, A289772(n)/a(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.

			Tree of rationals begin:
0;
1/3;
1/2, 2/3, 1, 1/6, 2/5;
5/9, 3/4, 4/3, 1/5, 5/12, 4/7, 7/9, 3/2, 2/9, 3/7, 7/12, 4/5, 5/3, 1/4, 4/9, 3/5, 5/6, 2, 1/9, 3/8, 8/15, 5/7, 7/6, 2/11, 11/27;
...

Robert Israel, Table of n, a(n) for n = 0..10000
Lionel Ponton, Two trees enumerating the positive rationals, arXiv:1707.02366 [math.NT], 2017. See p. 7.
Lionel Ponton, Two trees enumerating the positive rationals, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

Cf. A002487, A277749, A277750, A289772.

b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:
b(0):= 0:
map(denom@b, [$0..100]); # Robert Israel, Jul 12 2017

a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Denominator@ a@ n, {n, 0, 76}] (* Michael De Vlieger, Jul 12 2017 *)

b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));
lista(nn) = for (n=0, nn, print1(denominator(b(n)), ", "));

cp's OEIS Frontend

A277750 Denominators of rationals R_n associated with an analog of Stern's diatomic sequence for Z[sqrt(2)].

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A289772 a(n) is the numerator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

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A289773 a(n) is the denominator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.

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cp's OEIS Frontend

A277750 Denominators of rationals R_n associated with an analog of Stern's diatomic sequence for Z[sqrt(2)].

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A289772 a(n) is the numerator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

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A289773 a(n) is the denominator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.

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Maple

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PARI

A289772 a(n) is the numerator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

A289773 a(n) is the denominator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.