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A178765
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a(n) = 17*a(n-1) + a(n-2), with a(-1) = 0 and a(0) = 1.
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15
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0, 1, 17, 290, 4947, 84389, 1439560, 24556909, 418907013, 7145976130, 121900501223, 2079454496921, 35472626948880, 605114112627881, 10322412541622857, 176086127320216450, 3003786576985302507, 51240457936070359069, 874091571490181406680, 14910797173269154272629
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OFFSET
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-1,3
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COMMENTS
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The numerators and the denominators of continued fraction convergents to (17+sqrt(293))/2 lead to the sequence given above.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 17's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n>=0, a(n) equals the number of words of length n on alphabet {0,1,...,17} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
Also called the 17-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 17 kinds of squares available. (End)
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LINKS
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FORMULA
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a(n) = 17*a(n-1) + a(n-2) with a(-1) = 0, a(0) = 1.
G.f.: 1/(1 - 17*x - x^2).
E.g.f.: exp(17*x/2)*sinh(sqrt(293)*x/2)/(sqrt(293)/2).
a(n) = ( (17+sqrt(17^2+4))^(n+1) - (17-sqrt(17^2+4))^(n+1) )/(2^(n+1)*sqrt(17^2+4)).
a(n) = (Sum_{i=0..floor(n/2)} binomial(n+1,2*i+1)*17^(n-2*i)*293^i)/2^n.
a(n) = Fibonacci(n+1,17), the (n+1)-th Fibonacci polynomial evaluated at x=17.
a(n) = U(n, 17*i/2)*(-i)^n with i^2=(-1) and U(n, x/2)=S(n, x), see A049310.
a(n-r-1)*a(n+r-1) - a(n-1)^2 + (-1)^(n-r)*a(r-1)^2 = 0; a(-1) = 0 and n >= r+1.
a(n-1) + a(n+1) = A090306(n+1); A090306(n)^2 - 293*a(n-1)^2 - 4*(-1)^n = 0.
a(p-1) == 293^((p-1)/2)) (mod p) for odd primes p.
a(2n+1) = 17*A098248(n) (S(n,291)), a(2n) = A098250(n) (first differences of S(n,291)).
Limit_{k -> oo}(a(n+k)/a(k)) = (A090306(n) + a(n)*sqrt(293))/2.
Limit_{n -> oo)(A090306(n)/a(n)) = sqrt(293).
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EXAMPLE
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a(2) = 17*a(1) + a(0) = 289 + 1 = 290.
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MAPLE
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A178765:=proc(n): if n=0 then 1 elif n=1 then 17 elif n>=2 then 17*procname(n-1)+procname(n-2) fi: end: seq(A178765(n), n=0..15);
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MATHEMATICA
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Join[{0}, LinearRecurrence[{17, 1}, {1, 17}, 30]] (* Harvey P. Dale, Jan 29 2014 *)
CoefficientList[Series[x/(1-17x-x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
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PROG
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(Magma) [n le 2 select (n-1) else 17*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Nov 08 2014
(PARI) my(x='x+O('x^30)); concat([0], Vec(1/(1-17*x-x^2))) \\ G. C. Greubel, Jan 24 2019
(Sage) (x/(1-17*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 24 2019
(GAP) a:=[1, 17];; for n in [3..30] do a[n]:=17*a[n-1]+a[n-2]; od; Concatenation([0], a); # G. C. Greubel, Jan 24 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Changed name from defining a(1)=17. - Jon Perry, Nov 08 2014
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STATUS
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approved
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