A124750 Expansion of (1 + x + x^2)/(1 - x^3 + x^4).

1, 1, 1, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, -1, 2, -1, -1, 3, -3, 0, 4, -6, 3, 4, -10, 9, 1, -14, 19, -8, -15, 33, -27, -7, 48, -60, 20, 55, -108, 80, 35, -163, 188, -45, -198, 351, -233, -153, 549, -584, 80, 702, -1133, 664, 622, -1835, 1797, -42, -2457, 3632, -1839

Offset

0,15

Comments

Row sums of number triangle A124749.

Let A(n) denote the n X n matrix with 1's along and everywhere above the main diagonal, 1's along the sub-sub-subdiagonal, and 0's everywhere else; for n>3, a(n) equals (-1)^(n+1) times the sum of the coefficients of the characteristic polynomial of A(n-3) (see Mathematica code below). - John M. Campbell, Mar 10 2012

Links

Table of n, a(n) for n=0..60.

Index entries for linear recurrences with constant coefficients, signature (0,0,1,-1).

Formula

G.f.: (1 + x + x^2)/(1 - x^3 + x^4).

a(n) = Sum_{k=0..n} binomial(floor(k/3), n-k) * (-1)^(n-k).

a(n) = a(n-3) - a(n-4). - Wesley Ivan Hurt, May 02 2021

Mathematica

A[n_] := Array[Sum[KroneckerDelta[#1, #2 - j], {j, 0, n}] + KroneckerDelta[#1, #2 + 3] &, {n, n}]; Table[(-1)^(r + 1)*Total[CoefficientList[CharacteristicPolynomial[A[r - 3], x],  x]], {r, 4, 60}] (* John M. Campbell, Mar 10 2012 *)
CoefficientList[Series[(1+x+x^2)/(1-x^3+x^4), {x, 0, 70}], x] (* or *) LinearRecurrence[{0, 0, 1, -1}, {1, 1, 1, 1}, 70] (* Harvey P. Dale, Jun 06 2018 *)

Crossrefs

Cf. A124749.

Sequence in context: A217765 A237928 A108482 * A275865 A136458 A048805

Adjacent sequences: A124747 A124748 A124749 * A124751 A124752 A124753

Keyword

easy,sign

Author

Paul Barry, Nov 06 2006

Status

approved