A005059 a(n) = (5^n - 3^n)/2.

0, 1, 8, 49, 272, 1441, 7448, 37969, 192032, 966721, 4853288, 24325489, 121804592, 609554401, 3049366328, 15251614609, 76272421952, 381405156481, 1907154922568, 9536162033329, 47681972428112, 238413348924961, 1192077204978008, 5960417405949649, 29802181172927072

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of lines passing through 3 points of an n-dimensional grid of points of side 3. - David W. Wilson, c. 1999

a(n) is also the total number of words of length n, over an alphabet of five letters, one of them appearing an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment under A006516 (4-letter words), and the Balakrishnan reference there. See A003462 for the analogous 3-letter words problem. - Wolfdieter Lang, Jul 16 2017

Examples

			For the fifth formula: a(4) = 1*125 + 3*25 + 9*5 + 27*1 = 272. [_Bruno Berselli_, Aug 07 2013]
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. A. Alekseyev and T. Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
• Index entries for linear recurrences with constant coefficients, signature (8,-15).

Crossrefs

Cf. A081199 (binomial transform), A006516 (inverse binomial transform, and special 4-letter words), A003462 (special 3-letter words).

Programs

Magma

[(5^n - 3^n)/2: n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011

Maple

A005059:=n->(5^n-3^n)/2: seq(A005059(n), n=0..30); # Wesley Ivan Hurt, Nov 18 2014

Mathematica

Join[{a=0,b=1},Table[c=8*b-15*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8, -15}, {0, 1}, 50] (* Sture Sjöstedt, Oct 17 2012 *)
Table[(5^n - 3^n)/2, {n, 0, 23}] (* Michael De Vlieger, Jul 16 2017 *)

PARI

a(n)=(5^n-3^n)/2 \\ Charles R Greathouse IV, Jun 11 2013

Sage

[lucas_number1(n,8,15) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

From Paul Barry, Mar 03 2003: (Start)

a(n) = 8*a(n-1) - 15*a(n-2).

G.f.: x/((1-3*x)*(1-5*x)). (End)

a(n) = Sum_{k=1..n} 2^(k-1)*3^(n-k)*binomial(n,k). - Zerinvary Lajos, Sep 24 2006

a(n) = (r^n-s^n)/(r-s) with r=5 and s=3. - Sture Sjöstedt, Oct 17 2012

a(n) = Sum_{k=0..n-1} 3^k*5^(n-k-1) for n>0, a(0)=0. - Bruno Berselli, Aug 07 2013

E.g.f.: exp(3*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024