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A020989
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a(n) = (5*4^n - 2)/3.
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25
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1, 6, 26, 106, 426, 1706, 6826, 27306, 109226, 436906, 1747626, 6990506, 27962026, 111848106, 447392426, 1789569706, 7158278826, 28633115306, 114532461226, 458129844906, 1832519379626, 7330077518506
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OFFSET
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0,2
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COMMENTS
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Let Zb[n](x) = polynomial in x whose coefficients are the corresponding digits of index n in base b. Then Z2[(5*4^k-2)/3](1/tau) = 1. - Marc LeBrun, Mar 01 2001
a(n)=number of derangements of [2n+2] with runs consisting of consecutive integers. E.g., a(1)=6 because the derangements of {1,2,3,4} with runs consisting of consecutive integers are 4|123, 34|12, 4|3|12, 4|3|2|1, 234|1 and 34|2|1 (the bars delimit the runs). - Emeric Deutsch, May 26 2003
Sum of n-th row of triangle of powers of 4: 1; 1 4 1; 1 4 16 4 1; 1 4 16 64 16 4 1; ... - Philippe Deléham, Feb 22 2014
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REFERENCES
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Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, at pp. 104 and 311 (for "Mr. Zanti's ants").
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
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FORMULA
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a(0) = 1, a(n) = 4*a(n-1) + 2; a(n) = a(n-1)+ 5*{4^(n-1)}; - Amarnath Murthy, May 27 2001
G.f.: (1+x)/((1-4*x)*(1-x)). - Zerinvary Lajos, Jan 11 2009; Philippe Deléham, Feb 22 2014
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 1, a(1) = 6. - Philippe Deléham, Feb 22 2014
a(n) = Sum_{k=0..n} A112468(n,k)*5^k. - Philippe Deléham, Feb 22 2014
a(n) = (A020988(n) + A020988(n+1))/2. - Yosu Yurramendi, Jan 23 2017
a(n) = A002450(n) + A002450(n+1). - Yosu Yurramendi, Jan 24 2017
a(n) = 10*A020988(n-1) + 6. - Yosu Yurramendi, Feb 19 2017
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EXAMPLE
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a(0) = 1;
a(1) = 1 + 4 + 1 = 6;
a(2) = 1 + 4 + 16 + 4 + 1 = 26;
a(3) = 1 + 4 + 16 + 64 + 16 + 4 + 1 = 106; etc. - Philippe Deléham, Feb 22 2014
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MATHEMATICA
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NestList[4#+2&, 1, 25] (* Harvey P. Dale, Jul 23 2011 *)
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PROG
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(MAGMA) [(5*4^n-2)/3: n in [0..25]]; // Vincenzo Librandi, Jul 24 2011
(PARI) a(n)=(5*4^n-2)/3 \\ Charles R Greathouse IV, Jul 02 2013
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CROSSREFS
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A column of A119726.
Sequence in context: A254317 A037545 A027996 * A079675 A113991 A267578
Adjacent sequences: A020986 A020987 A020988 * A020990 A020991 A020992
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Divided g.f. by x to match the offset. - Philippe Deléham, Feb 22 2014
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STATUS
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approved
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