A178765 - OEIS

%I #59 May 01 2023 10:01:18

%S 0,1,17,290,4947,84389,1439560,24556909,418907013,7145976130,

%T 121900501223,2079454496921,35472626948880,605114112627881,

%U 10322412541622857,176086127320216450,3003786576985302507,51240457936070359069,874091571490181406680,14910797173269154272629

%N a(n) = 17*a(n-1) + a(n-2), with a(-1) = 0 and a(0) = 1.

%C The numerators and the denominators of continued fraction convergents to (17+sqrt(293))/2 lead to the sequence given above.

%C 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

%C 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

%C From _Michael A. Allen_, May 01 2023: (Start)

%C Also called the 17-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

%C 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)

%H G. C. Greubel, <a href="/A178765/b178765.txt">Table of n, a(n) for n = -1..800</a>

%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.

%H Dale Gerdemann, <a href="https://www.youtube.com/watch?v=dfsR7wk-wVM">Fractal images from (17,1) recursion</a>, YouTube Video, Nov 08 2014

%H Dale Gerdemann, <a href="https://www.youtube.com/watch?v=DXJViUA7pu4">Fractal images from (17,1) recursion: Selected image in detail</a>, YouTube Video, Nov 08 2014

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,1).

%F a(n) = 17*a(n-1) + a(n-2) with a(-1) = 0, a(0) = 1.

%F G.f.: 1/(1 - 17*x - x^2).

%F E.g.f.: exp(17*x/2)*sinh(sqrt(293)*x/2)/(sqrt(293)/2).

%F a(n) = ( (17+sqrt(17^2+4))^(n+1) - (17-sqrt(17^2+4))^(n+1) )/(2^(n+1)*sqrt(17^2+4)).

%F a(n) = (Sum_{i=0..floor(n/2)} binomial(n+1,2*i+1)*17^(n-2*i)*293^i)/2^n.

%F a(n) = Fibonacci(n+1,17), the (n+1)-th Fibonacci polynomial evaluated at x=17.

%F a(n) = U(n, 17*i/2)*(-i)^n with i^2=(-1) and U(n, x/2)=S(n, x), see A049310.

%F 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.

%F a(n-1) + a(n+1) = A090306(n+1); A090306(n)^2 - 293*a(n-1)^2 - 4*(-1)^n = 0.

%F a(p-1) == 293^((p-1)/2)) (mod p) for odd primes p.

%F a(2n+1) = 17*A098248(n) (S(n,291)), a(2n) = A098250(n) (first differences of S(n,291)).

%F a(3n) = A041551(5n), a(3n+1) = A041551(5n+3), a(3n+2) = 2*A041551(5n+4).

%F Limit_{k -> oo}(a(n+k)/a(k)) = (A090306(n) + a(n)*sqrt(293))/2.

%F Limit_{n -> oo)(A090306(n)/a(n)) = sqrt(293).

%e a(2) = 17*a(1) + a(0) = 289 + 1 = 290.

%p 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);

%t Join[{a=0,b=1},Table[c=17*b+1*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)

%t Join[{0}, LinearRecurrence[{17,1},{1,17},30]] (* _Harvey P. Dale_, Jan 29 2014 *)

%t CoefficientList[Series[x/(1-17x-x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 08 2014 *)

%o (Magma) [n le 2 select (n-1) else 17*Self(n-1)+Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Nov 08 2014

%o (PARI) my(x='x+O('x^30)); concat([0], Vec(1/(1-17*x-x^2))) \\ _G. C. Greubel_, Jan 24 2019

%o (Sage) (x/(1-17*x-x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 24 2019

%o (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

%Y Cf. A000045 (k=1), A006190 (k=3), A052918 (k=5), A054413 (k=7), A099371 (k=9), A049666 (k=11), A140455 (k=13), A154597 (k=15), this sequence (k=17).

%Y Cf. A086902, A087130.

%Y Cf. A243399.

%Y Row n=17 of A073133, A172236 and A352361 and column k=17 of A157103.

%K nonn,easy

%O -1,3

%A _Johannes W. Meijer_, Jun 12 2010, Jul 09 2011

%E Changed name from defining a(1)=17. - _Jon Perry_, Nov 08 2014

%E More terms from _Vincenzo Librandi_, Nov 08 2014