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A282153
Expansion of x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
1
0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, 4095, 8191, 16382, 32765, 65533, 131070, 262143, 524287, 1048574, 2097149, 4194301, 8388606, 16777215, 33554431, 67108862, 134217725, 268435453, 536870910, 1073741823, 2147483647, 4294967294, 8589934589
OFFSET
0,3
COMMENTS
After 0, partial sums of A281166.
Table of the first differences:
0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, ...
1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, ... A281166
0, 2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, ...
2, 3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, ...
repeat A281166.
FORMULA
From Colin Barker, Feb 10 2017: (Start)
G.f.: x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n>3. (End)
From Bruno Berselli, Feb 10 2017: (Start)
a(n) = 2^n + ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2 - 2. Therefore:
a(3*k) = 8^k + (-1)^k - 2,
a(3*k+1) = 2*8^k + (-1)^k - 2,
a(3*k+2) = 4*8^k - 2. (End)
a(n+6*h) = a(n) + 2^n*(64^h - 1) with h>=0. For h=1, a(n+6) = a(n) + 63*2^n.
a(n) - (a(n) mod 9) = A153237(n) = 9*A153234(n).
MATHEMATICA
LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* Robert P. P. McKone, Feb 07 2021 *)
PROG
(PARI) concat(0, Vec(x*(1 - 2*x + 3*x^2) / ((1 - x)*(1 - 2*x)*(1 - x + x^2)) + O(x^50))) \\ Colin Barker, Feb 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Feb 07 2017
EXTENSIONS
More terms from Colin Barker, Feb 10 2017
STATUS
approved