

A020989


a(n) = (5*4^n  2)/3.


25



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

0,2


COMMENTS

Let Zb[n](x) = polynomial in x whose coefficients are the corresponding digits of index n in base b. Then Z2[(5*4^k2)/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 4123, 3412, 4312, 4321, 2341 and 3421 (the bars delimit the runs).  Emeric Deutsch, May 26 2003
Sum of nth 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


REFERENCES

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, at pp. 104 and 311 (for "Mr. Zanti's ants").


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Brillhart and P. Morton, A case study in mathematical research: the GolayRudinShapiro sequence, Amer. Math. Monthly, 103 (1996) 854869.


FORMULA

a(0) = 1, a(n) = 4*a(n1) + 2; a(n) = a(n1)+ 5*{4^(n1)};  Amarnath Murthy, May 27 2001
G.f.: (1+x)/((14*x)*(1x)).  Zerinvary Lajos, Jan 11 2009; Philippe Deléham, Feb 22 2014
a(n) = 5*a(n1)  4*a(n2), 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(n1) + 6.  Yosu Yurramendi, Feb 19 2017


EXAMPLE

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


MATHEMATICA

NestList[4#+2&, 1, 25] (* Harvey P. Dale, Jul 23 2011 *)


PROG

(MAGMA) [(5*4^n2)/3: n in [0..25]]; // Vincenzo Librandi, Jul 24 2011
(PARI) a(n)=(5*4^n2)/3 \\ Charles R Greathouse IV, Jul 02 2013


CROSSREFS

A column of A119726.
Sequence in context: A254317 A037545 A027996 * A079675 A113991 A267578
Adjacent sequences: A020986 A020987 A020988 * A020990 A020991 A020992


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Divided g.f. by x to match the offset.  Philippe Deléham, Feb 22 2014


STATUS

approved



