A020989 a(n) = (5*4^n - 2)/3.

1, 6, 26, 106, 426, 1706, 6826, 27306, 109226, 436906, 1747626, 6990506, 27962026, 111848106, 447392426, 1789569706, 7158278826, 28633115306, 114532461226, 458129844906, 1832519379626, 7330077518506, 29320310074026, 117281240296106, 469124961184426

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

			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
		

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
• John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, Illinois Journal of Mathematics, volume 22, issue 1, 1978, pages 126-148. See Satz 9(a) page 132 and Satz 21 page 144 m_k = a(k).
• John Brillhart and Patrick Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869, see page 858 m_k = a(k).
• Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index "m_k".
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Cf. A002450, A020988, A112468.

A column of A119726.

Programs

Magma

[(5*4^n-2)/3: n in [0..25]]; // Vincenzo Librandi, Jul 24 2011

Mathematica

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

PARI

a(n)=(5*4^n-2)/3 \\ Charles R Greathouse IV, Jul 02 2013

Formula

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; adapted to offset by 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

E.g.f.: exp(x)*(5*exp(3*x) - 2)/3. - Stefano Spezia, Apr 10 2022