A096975 Trace sequence of a path graph plus loop.

3, 1, 5, 4, 13, 16, 38, 57, 117, 193, 370, 639, 1186, 2094, 3827, 6829, 12389, 22220, 40169, 72220, 130338, 234609, 423065, 761945, 1373466, 2474291, 4459278, 8034394, 14478659, 26088169, 47011093, 84708772, 152642789, 275049240

Offset

0,1

Comments

Let A be the adjacency matrix of the graph P_3 with a loop added at the end. Then a(n) = trace(A^n). A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) = abs(A094648(n)).

From L. Edson Jeffery, Mar 22 2011: (Start)

Let A be the unit-primitive matrix (see [Jeffery])

A = A_(7,1) =

(0 1 0)

(1 0 1)

(0 1 1).

Then a(n) = Trace(A^n). (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 0..3910

A. Akbary, Q. Wang, A generalized Lucas sequence and permutations binomials, Proc. Am. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=7.

Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 441.

Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.

L. E. Jeffery, Unit-primitive matrix

Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 1 (k=3).

Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Formula

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

a(n) = a(n-1) + 2*a(n-2) - a(n-3);

a(n) = (2*sqrt(7)*sin(atan(sqrt(3)/9)/3)/3+1/3)^n + (1/3-2*sqrt(7)*sin(atan(sqrt(3)/9)/3+Pi/3)/3)^n + (2*sqrt(7)*cos(acot(-sqrt(3)/9)/3)/3+1/3)^n.

a(n) = 2^n*((cos(Pi/7))^n+(cos(3*Pi/7))^n+(cos(5*Pi/7))^n). - Vladimir Shevelev, Aug 25 2010

Mathematica

CoefficientList[Series[(3 - 2 x - 2 x^2)/(1 - x - 2 x^2 + x^3), {x, 0, 33}], x] (* Michael De Vlieger, Aug 21 2019 *)

Prog

(PARI) {a(n)=if(n>=0, n+=1; polsym(x^3-x^2-2*x+1, n-1)[n], n=1-n; polsym(1-x-2*x^2+x^3, n-1)[n])} /* Michael Somos, Aug 03 2006 */
(PARI) a(n)=trace([0, 1, 0; 1, 0, 1; 0, 1, 1]^n); /* Joerg Arndt, Apr 30 2011 */

Crossrefs

Cf. A006053, A052547, A096976.

A033304(n) = a(-1-n). - Michael Somos, Aug 03 2006

Sequence in context: A096374 A007085 A094648 * A145174 A372283 A351965

Adjacent sequences: A096972 A096973 A096974 * A096976 A096977 A096978

Keyword

easy,nonn

Author

Paul Barry, Jul 16 2004

Status

approved