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
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).
Crossrefs
A033304(n) = a(-1-n). - Michael Somos, Aug 03 2006
Programs
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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 *) -
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 */
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
a(n) = (-1)^n*A094648(n). - R. J. Mathar, Nov 05 2024
Comments