A087798 -id:A087798 - cp's OEIS frontend

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

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).

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

A041150 Numerators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

9, 37, 46, 83, 378, 6887, 27926, 34813, 62739, 285769, 5206581, 21112093, 26318674, 47430767, 216041742, 3936182123, 15960770234, 19896952357, 35857722591, 163327842721, 2975758891569, 12066363408997

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 17 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A087798.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Cf. A010536, A041018, A041046, A041090, A041150, A041151, A041226, A041318, A041426,

A041550.

Numerator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

From Johannes W. Meijer, Jun 17 2010: (Start)
a(5*n) = A087798(3*n+1), a(5*n+1) = (A087798(3*n+2) - A087798(3*n+1))/2, a(5*n+2) = (A087798(3*n+2) + A087798(3*n+1))/2, a(5*n+3) = A087798(3*n+2) and a(5*n+4) = A087798(3*n+3)/2. (End)
G.f.: -(x^9-9*x^8+37*x^7-46*x^6+83*x^5+378*x^4+83*x^3+46*x^2+37*x+9) / (x^10+756*x^5-1). - Colin Barker, Nov 04 2013

A352362 Array read by ascending antidiagonals. T(n, k) = L(k, n) where L are the Lucas polynomials.

Original entry on oeis.org

2, 2, 0, 2, 1, 2, 2, 2, 3, 0, 2, 3, 6, 4, 2, 2, 4, 11, 14, 7, 0, 2, 5, 18, 36, 34, 11, 2, 2, 6, 27, 76, 119, 82, 18, 0, 2, 7, 38, 140, 322, 393, 198, 29, 2, 2, 8, 51, 234, 727, 1364, 1298, 478, 47, 0, 2, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 2

Offset: 0

Peter Luschny, Mar 18 2022

			Array starts:
n\k 0, 1,  2,   3,    4,     5,      6,       7,        8, ...
--------------------------------------------------------------
[0] 2, 0,  2,   0,    2,     0,      2,       0,        2, ... A010673
[1] 2, 1,  3,   4,    7,    11,     18,      29,       47, ... A000032
[2] 2, 2,  6,  14,   34,    82,    198,     478,     1154, ... A002203
[3] 2, 3, 11,  36,  119,   393,   1298,    4287,    14159, ... A006497
[4] 2, 4, 18,  76,  322,  1364,   5778,   24476,   103682, ... A014448
[5] 2, 5, 27, 140,  727,  3775,  19602,  101785,   528527, ... A087130
[6] 2, 6, 38, 234, 1442,  8886,  54758,  337434,  2079362, ... A085447
[7] 2, 7, 51, 364, 2599, 18557, 132498,  946043,  6754799, ... A086902
[8] 2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, ... A086594
[9] 2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, ... A087798
A007395|A059100|
    A001477 A079908

Eric Weisstein's World of Mathematics, Lucas Polynomial

Rows: A010673, A000032, A002203, A006497, A014448, A087130, A085447, A086902, A086594, A087798.
Columns: A007395, A001477, A059100, A079908.
Cf. A320570 (main diagonal), A114525, A309220 (variant), A117938 (variant), A352361 (Fibonacci polynomials), A350470 (Jacobsthal polynomials).

T := (n, k) -> (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k:
seq(seq(simplify(T(n - k, k)), k = 0..n), n = 0..10);

Table[LucasL[k, n], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[ 0, k_] := 2 Mod[k+1, 2]; T[n_, 0] := 2;
T[n_, k_] := n^k Hypergeometric2F1[1/2 - k/2, -k/2, 1 - k, -4/n^2];
Table[T[n, k], {n, 0, 9}, {k, 0, 8}] // TableForm

T(n, k) = ([0, 1; 1, k]^n*[2; k])[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*(k/(k-j))*n^(k-2*j) for k >= 1.
T(n, k) = (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k.
T(n, k) = [x^k] ((2 - n*x)/(1 - n*x - x^2)).
T(n, k) = n^k*hypergeom([1/2 - k/2, -k/2], [1 - k], -4/n^2) for n,k >= 1.

cp's OEIS Frontend

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

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

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

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

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

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