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Search: id:A096975
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| A096975 |
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Trace sequence of a path graph plus loop. |
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+0
3
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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; listen)
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OFFSET
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0,1
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COMMENT
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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).
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REFERENCES
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R. Chapman, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441
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FORMULA
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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)=(2sqrt(7)sin(atan(sqrt(3)/9)/3)/3+1/3)^n+ (1/3- 2sqrt(7)sin(atan(sqrt(3)/9)/3+pi/3)/3)^n+ (2sqrt(7)cos(acot(-sqrt(3)/9)/3)/3+1/3)^n.
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A006053, A052547, A096976.
A033304(n)=a(-1-n). - Michael Somos Aug 03 2006.
Adjacent sequences: A096972 A096973 A096974 this_sequence A096976 A096977 A096978
Sequence in context: A096374 A007085 A094648 this_sequence A135184 A131304 A131302
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Jul 16 2004
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