A001946 -id:A001946 - cp's OEIS frontend

A090310 a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

2, 21, 443, 9324, 196247, 4130511, 86936978, 1829807049, 38512885007, 810600392196, 17061121121123, 359094143935779, 7558038143772482, 159077895163157901, 3348193836570088403, 70471148463135014364

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.0475115... = 2/(21+sqrt(445)) = (sqrt(445)-21)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 21.0475115... = (21+sqrt(445))/2 = 2/(sqrt(445)-21).
a(2) = 443 divides a(14) = 3348193836570088403. Does this relate to the sequence being the (21,1)-weighted Fibonacci sequence with seed (2,21) and both 14 and 21 being multiples of 7? Primes in this sequence include: a(0) = 2, a(2) = 443, a(4) = 196247 Semiprimes in this sequence include: a(8) = 38512885007 = 97967 * 393121, a(14) = 3348193836570088403 = 443 * 7557999631083721. - Jonathan Vos Post, Feb 10 2005

			a(4) = 21*a(3) + a(2) = 21*9324 + 443 = ((21+sqrt(445))/2)^4 + ((21-sqrt(445))/2)^4 = 196246.9999949043 + 0.0000050956 = 196247.

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 (21,1).

Cf. A014842, A057031.

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), this sequence (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=21;; 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:=21; 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, 21*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{21,1},{2,21},40] (* or *) CoefficientList[ Series[ (2-21x)/(1-21x-x^2),{x,0,40}],x]  (* Harvey P. Dale, Apr 24 2011 *)
LucasL[Range[20]-1,21] (* G. C. Greubel, Dec 30 2019 *)

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

[2*(-I)^n*chebyshev_T(n, 21*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.
a(n) = ((21+sqrt(445))/2)^n + ((21-sqrt(445))/2)^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.: (2-21*x)/(1-21*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 21) = 2*(-i)^n * ChebyshevT(n, 21*i/2). - G. C. Greubel, Dec 30 2019

More terms from Ray Chandler, Feb 14 2004

A090305 a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

Original entry on oeis.org

2, 16, 258, 4144, 66562, 1069136, 17172738, 275832944, 4430499842, 71163830416, 1143051786498, 18359992414384, 294902930416642, 4736806879080656, 76083812995707138, 1222077814810394864, 19629328849962024962

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.0622577... = 1/(8+sqrt(65)) = (sqrt(65)-8).

Lim_{n-> infinity} a(n+1)/a(n) = 16.0622577... = (8+sqrt(65)) = 1/(sqrt(65)-8).

			a(4) = 16*a(3) + a(2) = 16*4144 + 258 = (8+sqrt(65))^4 + (8-sqrt(65))^4 = 66561.99998497... + 0.00001502... = 66562.

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 (16,1).

Cf. A002070, A015910.
Lucas polynomials: A114525.
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), this sequence (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25), A087281 (m=29), A087287 (m=76), A089772 (m=199).

m:=16;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019

m:=16; 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 31 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 8*I)), n = 0..20); # G. C. Greubel, Dec 31 2019

LinearRecurrence[{16,1},{2,16},40] (* or *) With[{c=Sqrt[65]}, Simplify/@ Table[(c-8)((8+c)^n-(8-c)^n (129+16c)),{n,20}]] (* Harvey P. Dale, Oct 31 2011 *)
LucasL[Range[20]-1, 16] (* G. C. Greubel, Dec 31 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 8*I) ) \\ G. C. Greubel, Dec 31 2019

[2*(-I)^n*chebyshev_T(n, 8*I) for n in (0..20)] # G. C. Greubel, Dec 31 2019

a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.
a(n) = (8+sqrt(65))^n + (8-sqrt(65))^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.: (2-16*x)/(1-16*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 16) = 2*(-i)^n * ChebyshevT(n, 8*i). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(8*x)*cosh(sqrt(65)*x). - Stefano Spezia, Jan 01 2020

More terms from Ray Chandler, Feb 14 2004

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

Original entry on oeis.org

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

A090308 a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.

Original entry on oeis.org

2, 19, 363, 6916, 131767, 2510489, 47831058, 911300591, 17362542287, 330799604044, 6302555019123, 120079344967381, 2287810109399362, 43588471423555259, 830468767156949283, 15822495047405591636

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.052486... = 2/(19+sqrt(365)) = (sqrt(365)-19)/2.

Lim_{n-> infinity} a(n+1)/a(n) = 19.052486... = (19+sqrt(365))/2 = 2/(sqrt(365)-19).

			a(4) = 19*a(3) + a(2) = 19*6916 + 363 = ((19+sqrt(365))/2)^4 + ((19-sqrt(365))/2)^4 = 131766.9999924108 + 0.0000075891 = 131767.

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 (19,1).

Cf. A049270.

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), A090307 (m=18), this sequence (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=19;; 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:=19; 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, 19*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

LucasL[Range[20]-1,20] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 19*I/2) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 19*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.
a(n) = ((19+sqrt(365))/2)^n + ((19-sqrt(365))/2)^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.: (2-19*x)/(1-19*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 19) = 2*(-i)^n * ChebyshevT(n, 19*i/2). - G. C. Greubel, Dec 30 2019

More terms from Ray Chandler, Feb 14 2004

A090313 a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.

Original entry on oeis.org

2, 22, 486, 10714, 236194, 5206982, 114789798, 2530582538, 55787605634, 1229857906486, 27112661548326, 597708411969658, 13176697724880802, 290485058359347302, 6403847981630521446, 141175140654230819114

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.045361... = 1/(11+sqrt(122)) = (sqrt(122)-11).

Lim_{n-> infinity} a(n+1)/a(n) = 22.045361... = (11+sqrt(122)) = 1/(sqrt(122)-11).

			a(4) = 236194 = 22*a(3) + a(2) = 22*10714 + 486 = (11 + sqrt(122))^4 + (11 - sqrt(122))^4 = 236193.999995766 + 0.000004233 = 236194.

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 (22,1).

Cf. A079219.

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), this sequence (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=22;; 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:=22; 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, 11*I)), n = 0..20); # G. C. Greubel, Dec 30 2019

LucasL[Range[20]-1,22] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 11*I) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 11*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.
a(n) = (11+sqrt(122))^n + (11-sqrt(122))^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.: (2-22*x)/(1-22*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 22) = 2*(-i)^n * ChebyshevT(n, 11*i). - G. C. Greubel, Dec 30 2019

More terms from Ray Chandler, Feb 14 2004

A090314 a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.

Original entry on oeis.org

2, 23, 531, 12236, 281959, 6497293, 149719698, 3450050347, 79500877679, 1831970236964, 42214816327851, 972772745777537, 22415987969211202, 516540496037635183, 11902847396834820411, 274282030623238504636, 6320389551731320427039, 145643241720443608326533, 3356114949121934311937298

Offset: 0

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

Lim_{n -> infinity} a(n)/a(n+1) = 0.04339638... = 2/(23+sqrt(533)) = (sqrt(533)-23)/2.

Lim_{n -> infinity} a(n+1)/a(n) = 23.04339638... = (23+sqrt(533))/2 = 2/(sqrt(533) - 23).

			a(4) = 281959 = 23*a(3) + a(2) = 23*12236 + 531 = ((23 + sqrt(533))/2)^4 + ((23 - sqrt(533))/2)^4 = 281958.999996453 + 0.000003546 = 281959.

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 (23,1).

Cf. A089141, A051502.

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), this sequence (m=23), A090316 (m=24), A330767 (m=25).

a:=[2,23];; for n in [3..20] do a[n]:=23*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019

I:=[2,23]; [n le 2 select I[n] else 23*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 23*I/2)), n = 0..20); # G. C. Greubel, Dec 29 2019

LinearRecurrence[{23,1},{2,23},20] (* Harvey P. Dale, Jul 11 2014 *)
LucasL[Range[20]-1,23] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 23*I/2) ) \\ G. C. Greubel, Dec 29 2019

[2*(-I)^n*chebyshev_T(n, 23*I/2) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
a(n) = ((23 + sqrt(533))/2)^n + ((23 - sqrt(533))/2)^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.: (2-23*x)/(1-23*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 23) = 2*(-i)^n * ChebyshevT(n, 23*i/2). - G. C. Greubel, Dec 29 2019

More terms from Ray Chandler, Feb 14 2004

Terms a(16) onward added by G. C. Greubel, Dec 29 2019

A090316 a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.

Original entry on oeis.org

2, 24, 578, 13896, 334082, 8031864, 193098818, 4642403496, 111610782722, 2683301188824, 64510839314498, 1550943444736776, 37287153512997122, 896442627756667704, 21551910219673022018, 518142287899909196136, 12456966819817493729282, 299485345963519758698904

Offset: 0

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

Lim_{n->infinity} a(n)/a(n+1) = 0.0415945... = 1/(12+sqrt(145)) = sqrt(145) - 12.

Lim_{n->infinity} a(n+1)/a(n) = 24.0415945... = 12+sqrt(145) = 1/(sqrt(145)-12).

			a(4) =334082 = 24a(3) + a(2) = 24*13896+ 578 = (12+sqrt(145))^4 + (12-sqrt(145))^4 = 334081.99999700672 + 0.00000299327 = 334082.

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 (24,1).

Cf. A041264, A058168, A056949.

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), this sequence (m=24), A330767 (m=25).

a:=[2,24];; for n in [3..20] do a[n]:=24*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019

I:=[2,24]; [n le 2 select I[n] else 24*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 12*I)), n = 0..20); # G. C. Greubel, Dec 29 2019

LinearRecurrence[{24,1},{2,24},20] (* Harvey P. Dale, Aug 30 2015 *)
LucasL[Range[20]-1,24] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 12*I) ) \\ G. C. Greubel, Dec 29 2019

[2*(-I)^n*chebyshev_T(n, 12*I) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.
a(n) = (12+sqrt(145))^n + (12-sqrt(145))^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*(1-12*x)/(1-24*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = 2*(-i)^n * ChebyshevT(n, 12*i) = Lucas(n, 24). - G. C. Greubel, Dec 29 2019
a(n) = 2 * A041264(n-1) for n>0. - Alois P. Heinz, Dec 29 2019

More terms from Ray Chandler, Feb 14 2004

Corrected by T. D. Noe, Nov 07 2006

A330767 a(n) = 25*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 25.

Original entry on oeis.org

2, 25, 627, 15700, 393127, 9843875, 246490002, 6172093925, 154548838127, 3869893047100, 96901875015627, 2426416768437775, 60757321085960002, 1521359443917437825, 38094743419021905627, 953889944919465078500, 23885343366405648868127, 598087474105060686781675, 14976072195992922818410002

Offset: 0

G. C. Greubel, Dec 29 2019

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (25,1).

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), this sequence (m=25).

a:=[2,25];; for n in [3..25] do a[n]:=25*a[n-1]+a[n-2]; od; a;

I:=[2,25]; [n le 2 select I[n] else 25*Self(n-1) +Self(n-2): n in [1..25]];

seq(simplify(2*(-I)^n*ChebyshevT(n, 25*I/2)), n = 0..25);

Mathematica
```
LucasL[Range[25] -1, 25]
```

vector(26, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 25*I/2) )

[2*(-I)^n*chebyshev_T(n, 25*I/2) for n in (0..25)]

a(n) = ( (25 + sqrt(629))^n + (25 - sqrt(629))^n )/2^n.
G.f.: (2 - 25*x)/(1-25*x-x^2).
a(n) = Lucas(n, 25) = 2*(-i)^n * ChebyshevT(n, 25*i/2).

A087265 Lucas numbers L(8*n).

Original entry on oeis.org

2, 47, 2207, 103682, 4870847, 228826127, 10749957122, 505019158607, 23725150497407, 1114577054219522, 52361396397820127, 2459871053643326447, 115561578124838522882, 5428934300813767249007, 255044350560122222180447

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003

a(n+1)/a(n) converges to (47+sqrt(2205))/2 = 46.9787137... a(0)/a(1)=2/47; a(1)/a(2)=47/2207; a(2)/a(3)=2207/103682; a(3)/a(4)=103682/4870847; etc. Lim_{n->infinity} a(n)/a(n+1) = 0.02128623625... = 2/(47+sqrt(2205)) = (47-sqrt(2205))/2.
a(n) = a(-n). - Alois P. Heinz, Aug 07 2008
From Peter Bala, Oct 14 2019: (Start)
Let F(x) = Product_{n >= 0} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let Phi = 1/2*(sqrt(5) - 1). This sequence gives the partial denominators in the simple continued fraction expansion of the number F(Phi^8) = 1.0212763906... = 1 + 1/(47 + 1/(2207 + 1/(103682 + ...))).
Also F(-Phi^8) = 0.9787231991... has the continued fraction representation 1 - 1/(47 - 1/(2207 - 1/(103682 - ...))) and the simple continued fraction expansion 1/(1 + 1/((47 - 2) + 1/(1 + 1/((2207 - 2) + 1/(1 + 1/((103682 - 2) + 1/(1 + ...))))))).
F(Phi^8)*F(-Phi^8) = 0.9995468962... has the simple continued fraction expansion 1/(1 + 1/((47^2 - 4) + 1/(1 + 1/((2207^2 - 4) + 1/(1 + 1/((103682^2 - 4) + 1/(1 + ...))))))).
1/2 + 1/2*F(Phi^8)/F(-Phi^8) = 1.0217391349... has the simple continued fraction expansion 1 + 1/((47 - 2) + 1/(1 + 1/((103682 - 2) + 1/(1 + 1/(228826127 - 2) + 1/(1 + ...))))). (End)

			a(4) = 4870847 = 47*a(3) - a(2) = 47*103682 - 2207=((47+sqrt(2205))/2)^4 + ( (47-sqrt(2205))/2)^4 =4870846.999999794696 + 0.000000205303 = 4870847.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 91.
R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

Indranil Ghosh, Table of n, a(n) for n = 0..596
Tanya Khovanova, Recursive Sequences
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (47,-1).

Cf. A000032. Cf. Lucas(k*n): A005248 (k = 2), A014448 (k = 3), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087287 (k = 9), A065705 (k = 10), A089772 (k = 11), A089775 (k = 12).

a(n) = A000032(8n).

[ Lucas(8*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

a:= n-> (Matrix([[2,47]]). Matrix([[47,1],[ -1,0]])^(n))[1,1]:
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 07 2008

LucasL[8*Range[0,20]] (* or *) LinearRecurrence[{47,-1},{2,47},20] (* Harvey P. Dale, Oct 23 2017 *)

a(n) = 47*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 47.
a(n) = ((47+sqrt(2205))/2)^n + ((47-sqrt(2205))/2)^n
(a(n))^2 = a(2n)+2.
G.f.: (2-47*x)/(1-47*x+x^2). - Alois P. Heinz, Aug 07 2008
From Peter Bala, Oct 14 2019: (Start)
a(n) = F(8*n+8)/F(8) - F(8*n-8)/F(8) = A049668(n+1) - A049668(n-1).
a(n) = trace(M^n), where M is the 2 X 2 matrix [0, 1; 1, 1]^8 = [13, 21; 21, 34].
Consequently the Gauss congruences hold: a(n*p^k) = a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. See Zarelua and also Stanley (Ch. 5, Ex. 5.2(a) and its solution).
45*Sum_{n >= 1} 1/(a(n) - 49/a(n)) = 1: (49 = Lucas(8) + 2 and 45 = Lucas(8) - 2)
49*Sum_{n >= 1} (-1)^(n+1)/(a(n) + 45/a(n)) = 1.
x*exp(Sum_{n >= 1} a(n)*x^/n) = x + 47*x^2 + 2208*x^3 + ... is the o.g.f. for A049668. (End)
E.g.f.: 2*exp(47*x/2)*cosh(21*sqrt(5)*x/2). - Stefano Spezia, Oct 18 2019
From Peter Bala, Apr 16 2025: (Start)
a(n) = Lucas(2*n)^4 - 4*Lucas(2*n)^2 + 2 = 2*T(4, (1/2)*Lucas(2*n)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind; more generally, for k >= 0, Lucas(2*k*n) = 2*T(k, Lucas(2*n)/2).
Sum_{n >= 1} 1/a(n) = (1/4) * (theta_3( (47 - sqrt(2205))/2 )^2 - 1) and
Sum_{n >= 1} (-1)^(n+1)/a(n) = (1/4) * (1 - theta_3( (sqrt(2205) - 47)/2 )^2),
where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122). See Borwein and Borwein, Proposition 3.5 (i), p. 91. Cf. A153415 and A003499. (End)

Terms a(22)-a(27) from John W. Layman, Jun 14 2004

A089772 a(n) = Lucas(11*n).

Original entry on oeis.org

2, 199, 39603, 7881196, 1568397607, 312119004989, 62113250390418, 12360848946698171, 2459871053643326447, 489526700523968661124, 97418273275323406890123, 19386725908489881939795601, 3858055874062761829426214722

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004

Lim_{n-> infinity} a(n+1)/a(n) = 199.00502499874... = (199 + sqrt(39605))/2.

Lim_{n->infinity} a(n)/a(n+1) = 0.00502499874... = 2/(199 + sqrt(39605)) = (sqrt(39605) - 199)/2.

			a(4) = 1568397607 = 199*a(3) + a(2) = 199*7881196 + 39603 = ((199 + sqrt(39605) )/2)^4 + ((199 - sqrt(39605))/2)^4 = 1568397606.9999999993624065... + 0.0000000006375934...

Nathaniel Johnston, Table of n, a(n) for n = 0..100
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 (199,1).

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), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25), A087281 (m=29), A087287 (m=76), this sequence (m=199).

List([0..20], n-> Lucas(1,-1,11*n)[2] ); # G. C. Greubel, Dec 30 2019

[Lucas(11*n): n in [0..20]]; // Vincenzo Librandi, Apr 15 2011

seq(simplify(2*(-I)^n*ChebyshevT(n, 199*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019

LucasL[11*Range[0,20]] (* or *) LinearRecurrence[{199,1},{2,199},20] (* Harvey P. Dale, Dec 23 2015 *)
LucasL[Range[20]-1,199] (* G. C. Greubel, Dec 31 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 199*I/2) ) \\ G. C. Greubel, Dec 31 2019

[lucas_number2(11*n,1,-1) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 199*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 199.
a(n) = ((199 + sqrt(39605))/2)^n + ((199 - sqrt(39605))/2)^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 - 199*x)/(1 - 199*x - x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 199) = 2*(-i)^n * ChebyshevT(n, 199*i/2). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(199*x/2)*cosh(sqrt(39605)*x/2). - Stefano Spezia, Jan 01 2020

cp's OEIS Frontend

A090310 a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

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A090305 a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

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

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A090308 a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.

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A090313 a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.

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