A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward.

1, 0, 3, 1, 7, 3, 15, 7, 31, 15, 63, 31, 127, 63, 255, 127, 511, 255, 1023, 511, 2047, 1023, 4095, 2047, 8191, 4095, 16383, 8191, 32767, 16383, 65535, 32767, 131071, 65535, 262143, 131071, 524287, 262143, 1048575, 524287, 2097151, 1048575, 4194303, 2097151, 8388607, 4194303

Offset

0,3

Links

Table of n, a(n) for n=0..45.

Jim Singh and others, Fascinating periodic sequence pairs, Mersenne Forum thread, July 2018.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n>2, a(0)=1, a(1)=0, a(2)=3.

From Bruno Berselli, Jul 12 2018: (Start)

G.f.: (1 - x + x^2)/((1 - x)*(1 - 2*x^2)).

a(n) = 2*a(n-2) + 1 for n>1, a(0)=1, a(1)=0.

a(n) = (1 + (-1)^n)*(2^(n/2) - 2^((n-3)/2)) + 2^((n-1)/2) - 1.

Therefore: a(4*k) = 2*4^k - 1, a(4*k+1) = 4^k - 1, a(4*k+2) = 4^(k+1) - 1, a(4*k+3) = 2*4^k - 1. (End)

Example

Let 1. The first four terms are 1, (1-1)/2 = 0, 2*1+1 = 3, 1.
Let 4*1+3 = 7. The next four terms are 7, (7-1)/2 = 3, 2*7+1 = 15, 7.
Let 4*7+3 = 31. The next four terms are 31, (31-1)/2 = 15, 2*31+1 = 63, 31; etc.

Maple

seq(coeff(series((1-x+x^2)/((1-x)*(1-2*x^2)), x, n+1), x, n), n=0..45); # Muniru A Asiru, Jul 14 2018

Mathematica

CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - 2 x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Jul 13 2018 *)
LinearRecurrence[{1, 2, -2}, {1, 0, 3}, 46] (* Robert G. Wilson v, Jul 21 2018 *)

Prog

(GAP) a:=[1, 0, 3];; for n in [4..45] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # Muniru A Asiru, Jul 14 2018

Crossrefs

Cf. A000225, A024036, A083420.

Sequence in context: A083239 A333847 A342268 * A189050 A095868 A140962

Adjacent sequences: A316739 A316740 A316741 * A316743 A316744 A316745

Keyword

nonn,easy,less

Author

Jim Singh, Jul 12 2018

Status

approved