A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.
2, 1, 11, 7, 47, 31, 191, 127, 767, 511, 3071, 2047, 12287, 8191, 49151, 32767, 196607, 131071, 786431, 524287, 3145727, 2097151, 12582911, 8388607, 50331647, 33554431, 201326591, 134217727, 805306367, 536870911, 3221225471, 2147483647, 12884901887, 8589934591
Offset: 2
Links
- Olivier Rozier, Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding, arXiv:1805.00133 [math.DS], 2018-2025, see Definition 4.4 on p. 21.
- Index entries for linear recurrences with constant coefficients, signature (1,4,-4).
Programs
-
Magma
[IsOdd(n) select 2^(n-2)-1 else 3*2^(n-2)-1: n in [2..40]];
-
Magma
I:=[2,1,11]; [n le 3 select I[n] else Self(n-1)+4*Self(n-2)-4*Self(n-3): n in [1..35]];
-
Mathematica
Table[If[OddQ[n], 2^(n - 2) - 1, 3 2^(n - 2) - 1], {n, 2, 80}] LinearRecurrence[{1, 4, -4}, {2, 1, 11}, 30] -
PARI
a(n) = if (n%2, 2^(n-2) - 1, 3*2^(n-2) - 1); \\ Michel Marcus, May 30 2018
Formula
a(n) = 2^(n-2) - 1 for odd n, otherwise a(n) = 3*2^(n-2) - 1, with n>1.
From Bruno Berselli, May 07 2018: (Start)
O.g.f.: x^2*(2 - x + 2*x^2)/((1 - x)*(1 - 2*x)*(1 + 2*x)).
E.g.f.: (1 + 2*x - 4*exp(x) + exp(-2*x) + 2*exp(2*x))/4.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
a(n) = (2 + (-1)^n)*2^(n-2) - 1. (End)
Comments