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A125819
a(n) = ((1 + 7*sqrt(2))^n - (1 - 7*sqrt(2))^n)/(14*sqrt(2)).
2
0, 1, 2, 101, 396, 10589, 59590, 1146313, 8072856, 127338073, 1037743178, 14427279437, 129515647140, 1658477399669, 15879972571918, 192632252911729, 1925621845299504, 22536572223036721, 231858463440125330, 2649764432514812597, 27789799818721782204
OFFSET
0,3
LINKS
FORMULA
From Philippe Deléham, Dec 12 2006: (Start)
a(0)=0, a(1)=1, a(n) = 2*a(n-1) + 97*a(n-2) for n>=2.
G.f.: x/(1 - 2*x - 97*x^2). (End)
a(n) = -(-97)^n * a(-n) for all n in Z. - Michael Somos, Aug 17 2018
EXAMPLE
G.f. = x + 2*x^2 + 101*x^3 + 396*x^4 + 10589*x^5 + 59590*x^6 + ... - Michael Somos, Aug 17 2018
MATHEMATICA
Expand[Table[((1+7Sqrt[2])^n -(1-7Sqrt[2])^n)/(14Sqrt[2]), {n, 0, 30}]] (* Artur Jasinski *)
LinearRecurrence[{2, 97}, {0, 1}, 30] (* T. D. Noe, Mar 28 2012 *)
a[ n_] := Sqrt[-97]^(n - 1) ChebyshevU[n - 1, 1 / Sqrt[-97]]; (* Michael Somos, Aug 17 2018 *)
PROG
(PARI) x='x+O('x^30); concat([0], Vec(x/(1 - 2*x - 97*x^2))) \\ G. C. Greubel, Aug 17 2018
(PARI) {a(n) = my(t = quadgen(-97*4)); simplify( polchebyshev(n-1, 2, 1/t) * t^(n-1))}; /* Michael Somos, Aug 17 2018 */
(Magma) I:=[0, 1]; [n le 2 select I[n] else 2*Self(n-1) + 97*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 17 2018
(Sage) (x/(1-2*x-97*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=2*a[n-1]+97*a[n-2]; od; a; # G. C. Greubel, Aug 0232019
CROSSREFS
Cf. A125820.
Sequence in context: A064152 A162353 A088272 * A179998 A072383 A037053
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Dec 10 2006, corrected Jan 12 2007
EXTENSIONS
Offset corrected to 0. - Michael Somos, Aug 17 2018
STATUS
approved