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A141775
Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).
1
1, 3, 5, 8, 15, 31, 64, 129, 257, 512, 1023, 2047, 4096, 8193, 16385, 32768, 65535, 131071, 262144, 524289, 1048577, 2097152, 4194303, 8388607, 16777216, 33554433, 67108865, 134217728, 268435455, 536870911, 1073741824, 2147483649, 4294967297, 8589934592, 17179869183
OFFSET
0,2
COMMENTS
From Paul Curtz, Jun 15 2011: (Start)
A square array of a(n) and its higher order differences is defined by T(0,k) = a(k) and T(n,k) = T(n-1,k+1)-T(n-1,k):
1, 3, 5, 8, 15, 31,
2, 2, 3, 7, 16, 33,
0, 1, 4, 9, 17, 32, see A130785(n).
1, 3, 5, 8, 15, 31,
2, 2, 3, 7, 16, 33,
a(n) is identical to its third differences: T(n+3,k) = T(n,k).
The main diagonal is T(n,n) = 2^n. Subdiagonals are T(n,n-1) = A014551(n) and T(n,n-2) = A062510(n).
(End)
FORMULA
From Paul Curtz, Jun 15 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^n - A128834(n).
a(n) - 2a(n-1)= A057079(n+1).
a(n) + a(n+3) = 9*2^n.
a(n+6) - a(n) = 63*2^n.
a(n) = A130785(n) - A130785(n-1). (End)
G.f.: (x-1)*(1+x) / ( (2*x-1)*(x^2-x+1) ). - R. J. Mathar, Jun 22 2011
a(n) = 2^n + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Feb 10 2017
EXAMPLE
a(4) = 8 = (1, 2, 0, 1) dot (1, 3, 3, 1) = (1 + 6 + 0 + 1).
MATHEMATICA
LinearRecurrence[{3, -3, 2}, {1, 3, 5}, 40] (* Harvey P. Dale, May 29 2012 *)
PROG
(PARI) x='x+O('x^30); Vec((x-1)*(1+x)/((2*x-1)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018
(Magma) I:=[1, 3, 5]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018
CROSSREFS
Sequence in context: A285010 A352917 A099846 * A056765 A080006 A374680
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jul 03 2008
STATUS
approved