A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

1, 6, 18, 42, 90, 186, 378, 762, 1530, 3066, 6138, 12282, 24570, 49146, 98298, 196602, 393210, 786426, 1572858, 3145722, 6291450, 12582906, 25165818, 50331642, 100663290, 201326586, 402653178, 805306362, 1610612730, 3221225466, 6442450938, 12884901882

Offset: 1

Author

R. H. Hardin, Feb 24 2002

Keywords

• nonn
• easy

Comments

1/4 the number of colorings of an n X n octagonal array with 4 colors.

Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - N. J. A. Sloane, Feb 20 2016

From Gary W. Adamson, Jan 13 2009: (Start)

Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).

Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)

For n > 1, number of ternary strings of length n with exactly 2 different digits. - Enrique Navarrete, Nov 20 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Ana Rechtman, Février 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
• N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A068239-A068305, A000332, A002417, A027441.

Cf. A091344. - Gary W. Adamson, Jan 13 2009

Programs

Magma

[1] cat [6*(2^(n-1)-1): n in [2..40]]; // Vincenzo Librandi, Feb 20 2016

Mathematica

a=0; lst={1}; k=6; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
Transpose[NestList[{First[#]+1,6(2^First[#]-1)}&,{1,1},30]][[2]] (* or *) Join[{1},LinearRecurrence[{3,-2},{6,18},30]] (* Harvey P. Dale, Nov 27 2011 *)

PARI

a(n)=polcoeff(prod(i=1,2,(1+i*x))/(prod(i=1,2,(1-i*x))+x*O(x^n)),n)
for(n=0,50,print1(a(n),","))

Formula

G.f.: (1+x)*(1+2*x)/((1-x)*(1-2*x)). - Benoit Cloitre, Apr 13 2002

a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Nov 27 2011

E.g.f.: 1 - 6*exp(x)*(exp(x) - 1). - Stefano Spezia, May 18 2024

Extensions

More terms from Benoit Cloitre, Apr 13 2002

Old definition (which is now a comment) replaced with explicit formula by N. J. A. Sloane, May 12 2010