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A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2]. 2
3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].

From 7, the last digit is of period 4: repeat [7, 2, 5, 8].

(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)

Difference table of a(n):

3,   7, 12, 25, 48,  97, 192, ...

4,   5, 13, 23, 49,  95, 193, ...  = -(-1)^n* A140683(n)

1,   8, 10, 26, 46,  98, 190, ...  = A259713(n)

7,   2, 16, 20, 52,  92, 196, ...

-5, 14,  4, 32, 40, 104, 184, ...

... .

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

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

FORMULA

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.

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

a(n+2) = a(n) + 9*2^n.

a(n) = 2^(n+2) - A051049(n).

From Colin Barker, Jan 01 2017: (Start)

a(n) = 3*2^n for n even.

a(n) = 3*2^n + 1 for n odd.

G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).

(End)

Binomial transform of 3, followed by (-1)^n* A140657(n).

EXAMPLE

a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.

MATHEMATICA

a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)

CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)

PROG

(PARI) Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

CROSSREFS

Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

Sequence in context: A226229 A167491 A210185 * A062325 A301982 A196858

Adjacent sequences:  A280342 A280343 A280344 * A280346 A280347 A280348

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jan 01 2017

EXTENSIONS

More terms from Colin Barker, Jan 01 2017

STATUS

approved

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Last modified August 16 23:58 EDT 2018. Contains 313809 sequences. (Running on oeis4.)