A140725 Inverse binomial transform of (0 followed by A037481).

0, 1, 4, 10, 34, 94, 298, 862, 2650, 7822, 23722, 70654, 212986, 636910, 1914826, 5736286, 17225242, 51642958, 154994410, 464852158, 1394818618, 4183931566, 12552843274, 37656432670, 112973492314, 338912088334, 1016753042218

Offset: 0

Author

Paul Curtz, Jul 12 2008

Keywords

• nonn

Comments

From Sean A. Irvine, Jun 07 2025: (Start)

For n>=1, the number of walks of length n-1 starting at vertex 1 (or, by symmetry, vertex 4) in the graph K_{1,1,3}:

1---2

/|\ /

0 | X

\|/ \

4---3. (End)

Links

• Sean A. Irvine, Walks on Graphs.

Crossrefs

Cf. A083421 (bin. transform of (0 followed by A037481)).

Programs

Mathematica

Join[{0},LinearRecurrence[{1,6},{1,4},26]] (* or *) a[0]=0;a[n_]:= ((-2)^n+4*3^n)/10;Array[a,27,0] (* James C. McMahon, Jul 13 2025 *)

Formula

a(n)= (-1)^n*A091003(n), n>0.

a(n+1)-3*a(n) = (-1)^(n+1)*A000079(n-1), n>0.

|a(n+1)-3*a(n)| = A011782(n).

From R. J. Mathar, Jul 14 2008: (Start)

O.g.f.: (1+3*x)*x / ((1+2*x)*(1-3*x)).

a(n) = ((-2)^n+4*3^n)/10, n>0. (End)

a(n) = a(n-1)+6*a(n-2) for n>2, a(0)=0, a(1)=1, a(2)=4. - Philippe Deléham, Nov 17 2013

a(n) + a(n+1) = A140796(n). - Philippe Deléham, Nov 17 2013

a(n+1) = sum_{k=0..n} A108561(n,k)*(-3)^k. - Philippe Deléham, Nov 17 2013

Extensions

Edited and extended by R. J. Mathar, Jul 14 2008