A108765 Expansion of g.f. (1 - x + x^2)/((1-3*x)*(x-1)^2).

1, 4, 14, 45, 139, 422, 1272, 3823, 11477, 34440, 103330, 310001, 930015, 2790058, 8370188, 25110579, 75331753, 225995276, 677985846, 2033957557, 6101872691, 18305618094, 54916854304, 164750562935, 494251688829, 1482755066512

Offset: 0

Author

Creighton Dement, Jun 24 2005

Keywords

• easy
• nonn

Comments

Superseeker suggests a(n+2) - 2*a(n+1) + a(n) = 7*3^n = A005032(n).

Inverse binomial transform gives match with first differences of A026622.

Floretion Algebra Multiplication Program, FAMP Code: kbasefor[(- 'j + 'k - 'ii' - 'ij' - 'ik')], vesfor = A000004, Fortype: 1A, Roktype (leftfactor) is set to:Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code)

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Crossrefs

Cf. A005032, A026622.

Programs

Mathematica

s=1;lst={s};Do[s+=(s+(n+=s));AppendTo[lst, s], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 11 2008 *)
CoefficientList[Series[(1-x+x^2)/((1-3x)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[{5,-7,3},{1,4,14},40] (* Harvey P. Dale, Dec 11 2012 *)

Formula

From Rolf Pleisch, Feb 10 2008: (Start)

a(0) = 1; a(n) = 3*a(n-1) + n.

a(n) = (7*3^n - 2*n - 3)/4. (End)

a(0)=1, a(1)=4, a(2)=14, a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Harvey P. Dale, Dec 11 2012