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A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Jim Singh and others, Fascinating periodic sequence pairs, Mersenne Forum thread, July 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)
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
Sequence in context: A083239 A333847 A342268 * A189050 A095868 A140962
KEYWORD
nonn,easy,less
AUTHOR
Jim Singh, Jul 12 2018
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)