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

0,3

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

Table of n, a(n) for n=0..26.

Sean A. Irvine, Walks on Graphs.

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

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 *)

Crossrefs

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

Sequence in context: A301595 A022445 A091003 * A005630 A100507 A384677

Adjacent sequences: A140722 A140723 A140724 * A140726 A140727 A140728

Keyword

nonn

Author

Paul Curtz, Jul 12 2008

Extensions

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

Status

approved