A083218 -id:A083218 - cp's OEIS frontend

A090307 a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.

2, 18, 326, 5886, 106274, 1918818, 34644998, 625528782, 11294163074, 203920464114, 3681862517126, 66477445772382, 1200275886420002, 21671443401332418, 391286257110403526, 7064824071388595886

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Lim_{n-> infinity} a(n)/a(n+1) = 0.0553851... = 1/(9+sqrt(82)) = (sqrt(82)-9).

Lim_{n-> infinity} a(n+1)/a(n) = 18.0553851... = (9+sqrt(82)) = 1/(sqrt(82)-9).

			a(4) = 18*a(3) + a(2) = 18*5886 + 326 = (9+sqrt(82))^4 + (9-sqrt(82))^4 = 106273.9999905903 + 0.000009406 = 106274.

G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (18,1).

Cf. A083218, A087215.

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), this sequence (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=18;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 30 2019

m:=18; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 30 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 9*I)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{18,1},{2,18},25] (* or *) CoefficientList[ Series[ (2-18x)/(1-18x-x^2),{x,0,25}],x] (* Harvey P. Dale, Apr 22 2011 *)
LucasL[Range[20]-1, 18] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 9*I) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 9*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
a(n) = (9+sqrt(82))^n + (9-sqrt(82))^n.
(a(n))^2 = a(2n) - 2 if n=1, 3, 5, ...
(a(n))^2 = a(2n) + 2 if n=2, 4, 6, ...
G.f.: (2-18*x)/(1-18*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 18) = 2*(-i)^n * ChebyshevT(n, 9*i). - G. C. Greubel, Dec 30 2019
E.g.f.: 2*exp(9*x)*cosh(sqrt(82)*x). - Stefano Spezia, Dec 31 2019

More terms from Ray Chandler, Feb 14 2004

A057237 Maximum k <= n such that 1, 2, ..., k are all relatively prime to n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 4, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 4, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 6, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1

Offset: 1

Leroy Quet, Sep 20 2000

nonn

In reduced residue system for n [=RRS(n)] the [initial] segment of consecutive integers, i.e. of which no number is missing is {1,2,....,a[n]}. The first missing term from RRS(n) is 1+a(n), the least prime divisor.. E.g. n=121 : RRS[121] = {1,2,3,4,5,6,7,8,9,10,lag,12,..}, i.e. no 11 is in RRS; a[n] is the length of longest lag-free number segment consisting of consecutive integers, since A020639[n] divides n. - Labos Elemer, May 14 2003

a(n) is also the difference between the smallest two divisors of n, (the column 1 of A193829), if n >= 2. - Omar E. Pol, Aug 31 2011

			a(25) = 4 because 1, 2, 3 and 4 are relatively prime to 25.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A020639, A083218.

Cf. A066169.

Join[{1},Table[Length[Split[Boole[CoprimeQ[n,Range[n-1]]]][[1]]],{n,2,100}]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = if (n==1, 1, factor(n)[1,1] - 1); \\ Michel Marcus, May 29 2015

For n >= 2, a(n) = (smallest prime dividing n) - 1 = A020639(n) - 1.

For n >= 2, a(n) = (n-1) mod (smallest prime dividing n); cf. A083218. - Reinhard Zumkeller, Apr 22 2003

cp's OEIS Frontend

A090307 a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.

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