A277749 Numerators of rationals R_n associated with an analog of Stern's diatomic sequence for Z[sqrt(2)].

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

Offset: 1

Author

N. J. A. Sloane, Nov 08 2016

Keywords

• nonn
• frac

Comments

"At the end of Section 6, the sequences of numerators and denominators of R_n were considered. What do these sequences count? Further, is there some kind of combinatorial reciprocity [Beck] occurring here?" - [S. Northshield]

Examples

			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, ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..54321
• Matthias Beck, Combinatorial Reciprocity Theorems, arXiv:1201.2212 [math.CO], 2012.
• 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.

Crossrefs

Cf. A002487 (Stern's diatomic sequence), A277750 (denominators).

Programs

Mathematica

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

PARI

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(numerator, seq(88))  \\ Gheorghe Coserea, Nov 11 2016

Formula

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

Extensions

More terms from Gheorghe Coserea, Nov 11 2016