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A166578
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a(n) = a(n-3) + 2^(n-4) with a(1) = 1, a(2) = 2, a(3) = 1.
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1
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1, 2, 1, 2, 4, 5, 10, 20, 37, 74, 148, 293, 586, 1172, 2341, 4682, 9364, 18725, 37450, 74900, 149797, 299594, 599188, 1198373, 2396746, 4793492, 9586981, 19173962, 38347924, 76695845, 153391690, 306783380, 613566757, 1227133514, 2454267028
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OFFSET
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0,2
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COMMENTS
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a(n) and successive differences: 1,1,2,1,2,4,5,10,20,37; 0,1,-1,1,2,1,5,10,17,37; 1,-2,2,1,-1,4,5,7,20,37; -3,4,-1,-2,5,1,2,13,17,34; 7,-5,-1,7,-4,1,11,4,17,43; -12,4,8,-11,5,10,-7,13,26,25; Rows must be taken by pairs (companions because a(n)-a(n-3) alternatively gives A131577 and A011782 also companions). Note a(3n+2)=2*a(3n+1)=4*a(3n), n positive; see A113405. Sum of consecutive three terms of even rows gives 0,4,32,256.
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LINKS
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FORMULA
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For n > 4, a(n) = 2a(n-1) + a(n-3) - 2a(n-4).
a(3n) = (8^n - 8)/14 + 1, a(3n-1) = (8^n - 8)/28 + 2, a(3n-2) = (8^n - 8)/56 + 1.
G.f.: (1-3*x^2-x^3)/(1-2*x-x^3+2*x^4). - Colin Barker, Jan 25 2012
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MATHEMATICA
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RecurrenceTable[{a[1]==1, a[2]==2, a[3]==1, a[n]==a[n-3]+2^(n-4)}, a[n], {n, 40}] (* or *) LinearRecurrence[{2, 0, 1, -2}, {1, 2, 1, 2}, 40] (* Harvey P. Dale, Sep 14 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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