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

Author

Jim Singh, Jul 12 2018

Keywords

• nonn
• easy
• less

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A000225, A024036, A083420.

Programs

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

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 *)

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)