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A096975
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Trace sequence of a path graph plus loop.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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).
(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). - L. Edson Jeffery, March 22, 2011. (End)
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REFERENCES
| R. Chapman, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441
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LINKS
| L. E. Jeffery, Unit-primitive matrix
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FORMULA
| G.f. : (3-2x-2x^2)/(1-x-2x^2+x^3);
a(n)=a(n-1)+2a(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). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 25 2010]
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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 */
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CROSSREFS
| Cf. A006053, A052547, A096976.
A033304(n)=a(-1-n). - Michael Somos Aug 03 2006.
Sequence in context: A096374 A007085 A094648 * A145174 A135184 A131304
Adjacent sequences: A096972 A096973 A096974 * A096976 A096977 A096978
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jul 16 2004
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