A088316 -id:A088316 - cp's OEIS frontend

A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

2, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, 17555720002, 125348805407, 894997357851, 6390330310364, 45627309530399, 325781497023157, 2326097788692498, 16608466017870643, 118585359913786999, 846705985414379636

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003

a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.
Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.
From Johannes W. Meijer, Jun 12 2010: (Start)
In general sequences with recurrence a(n) = k*a(n-1)+a(n-2) with a(0)=2 and a(1)=k [and a(-1)=0] have generating function (2-k*x)/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n+1) = b(5n), a(3n+2)=b(5*n+3), a(3n+3)=2*b(5n+4) where b(n) is the sequence of numerators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n)/2, for n>=1, is the sequence of numerators of continued fraction convergents to sqrt(k^2/4+1).]
For the sequence given above k=7 which implies that it is associated with A041090.
For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0)=1 [and a(-1)=0] see A054413; a sequence that is associated with A041091.
For more information follow the Khovanova link and see A087130, A140455 and A178765.
(End)

			a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (7,1).

Cf. A058316, A176439.

Cf. A000032 (k=1), A006497 (k=3), A087130 (k=5), A086902 (k=7), A087798 (k=9), A001946 (k=11), A088316 (k=13), A090301 (k=15), A090306 (k=17). - Johannes W. Meijer, Jun 12 2010

I:=[2,7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{7,1},{2,7},30] (* Harvey P. Dale, May 25 2023 *)

a(n)=([0,1; 1,7]^n*[2;7])[1,1] \\ Charles R Greathouse IV, Apr 06 2016

a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.
E.g.f. : 2exp(7x/2)cosh(sqrt(53)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)53^k7^(n-2k)}. a(n)=2T(n, 7i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
G.f.: (2-7x)/(1-7x-x^2). - Philippe Deléham, Nov 16 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 7*A097837(n), a(2n) = A099368(n).
a(3n+1) = A041090(5n), a(3n+2) = A041090(5*n+3), a(3n+3) = 2*A041090(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53). (End)

A114525 Triangle of coefficients of the Lucas (w-)polynomials.

Original entry on oeis.org

2, 0, 1, 2, 0, 1, 0, 3, 0, 1, 2, 0, 4, 0, 1, 0, 5, 0, 5, 0, 1, 2, 0, 9, 0, 6, 0, 1, 0, 7, 0, 14, 0, 7, 0, 1, 2, 0, 16, 0, 20, 0, 8, 0, 1, 0, 9, 0, 30, 0, 27, 0, 9, 0, 1, 2, 0, 25, 0, 50, 0, 35, 0, 10, 0, 1, 0, 11, 0, 55, 0, 77, 0, 44, 0, 11, 0, 1, 2, 0, 36, 0, 105, 0, 112, 0, 54, 0, 12, 0, 1

Offset: 0

Eric W. Weisstein, Dec 06 2005

Unsigned version of A108045.

The row reversed triangle is A162514. - Paolo Bonzini, Jun 23 2016

			2, x, 2 + x^2, 3*x + x^3, 2 + 4*x^2 + x^4, 5*x + 5*x^3 + x^5, ... give triangle
  n\k   0  1  2  3  4  5  6  7  8  9 10 ...
  0:    2
  1:    0  1
  2:    2  0  1
  3:    0  3  0  1
  4:    2  0  4  0  1
  5:    0  5  0  5  0  1
  6:    2  0  9  0  6  0  1
  7:    0  7  0 14  0  7  0  1
  8:    2  0 16  0 20  0  8  0  1
  9:    0  9  0 30  0 27  0  9  0  1
  10:   2  0 25  0 50  0 35  0 10  0  1
  n\k   0  1  2  3  4  5  6  7  8  9 10 ...
  .... reformatted by _Wolfdieter Lang_, Feb 10 2023

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
B. Johnson, Fibonacci numbers and matrices, 2009.
Jong Hyun Kim, Hadamard products and tilings, JIS 12 (2009) 09.7.4.
M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial
Eric Weisstein's World of Mathematics, Lucas Polynomial

Cf. A108045 (signed version).
Cf. A034807, A162514.
Cf. Sequences L(n,x): A000032(x = 1), A002203 (x = 2), A006497 (x = 3), A014448 (x = 4), A087130 (x = 5), A085447 (x = 6), A086902 (x = 7), A086594 (x = 8), A087798 (x = 9), A086927 (x = 10), A001946 (x = 11), A086928 (x = 12), A088316 (x = 13), A090300 (x = 14), A090301 (x = 15), A090305 (x = 16), A090306 (x = 17), A090307 (x = 18), A090308 (x = 19), A090309 (x = 20), A090310 (x = 21), A090313 (x = 22), A090314 (x = 23), A090316 (x = 24), A087281 (x = 29), A087287 (x = 76), A089772 (x = 199).

Lucas := proc(n,x)
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
end proc:
A114525 := proc(n,k)
    coeftayl(Lucas(n,x),x=0,k) ;
end proc:
seq(seq(A114525(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 16 2019

row[n_] := CoefficientList[LucasL[n, x], x];
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 11 2018 *)

From Peter Bala, Mar 18 2015: (Start)
The Lucas polynomials L(n,x) satisfy the recurrence L(n+1,x) = x*L(n,x) + L(n-1,x) with L(0,x) = 2 and L(1,x) = x.
O.g.f.: Sum_{n >= 0} L(n,x)*t^n = (2 - x*t)/(1 - t^2 - x*t) = 2 + x*t + (x^2 + 2)*t^2 + (3*x + x^3)*t^3 + ....
L(n,x) = trace( [ x, 1; 1, 0 ]^n ).
exp( Sum_{n >= 1} L(n,x)*t^n/n ) = Sum_{n >= 0} F(n+1,x)*t^n, where F(n,x) denotes the n-th Fibonacci polynomial. (see Appendix A3 in Johnson).
exp( Sum_{n >= 1} L(n,x)*L(2*n,x)*t^n/n ) = 1/( F(1,x)*F(2*x)*F(3,x) ) * Sum_{n >= 0} F(n+1,x)*F(n+2,x)*F(n+3,x)*t^n.
exp( Sum_{n >= 1} L(3*n,x)/L(n,x)*t^n/n ) = Sum_{n >= 0} L(2*n + 1,x)*t^n.
L(n,1) = Lucas(n) = A000032(n); L(n,4) = Lucas(3*n) = A014448(n); L(n,11) = Lucas(5*n) = A001946(n); L(n,29) = Lucas(7*n) = A087281(n); L(n,76) = Lucas(9*n) = A087287(n); L(n,199) = Lucas(11*n) = A089772(n). The general result is L(n,Lucas(2*k + 1)) = Lucas((2*k + 1)*n). (End)
From Jeremy Dover, Jun 10 2016: (Start)
Read as a triangle T(n,k), n >= 0, n >= k >= 0, T(n,k) = (Binomial((n+k)/2,k) + Binomial((n+k-2)/2,k))*(1+(-1)^(n-k))/2.
T(n,k) = A046854(n-1,k-1) + A046854(n-1,k) + A046854(n-2,k) for even n+k with n+k > 0, assuming A046854(n,k) = 0 for n < 0, k < 0, k > n.
T(n,k) is the number of binary strings of length n with exactly k pairs of consecutive 0's and no pair of consecutive 1's, where the first and last bits are considered consecutive. (End)
From Peter Bala, Sep 03 2022: (Start)
L(n,x) = 2*(i)^n*T(n,-i*x/2), where i = sqrt(-1) and T(n,x) is the n-th Chebyshev polynomial of the first kind.
d/dx(L(n,x)) = n*F(n,x), where F(n,x) denotes the n-th Fibonacci polynomial.
Let P_n(x,y) = (L(n,x) - L(n,y))/(x - y). Then {P_n(x,y): n >= 1} is a fourth-order linear divisibility sequence of polynomials in the ring Z[x,y]: if m divides n in Z then P_m(x,y) divides P_n(x,y) in Z[x,y].
P_n(1,1) = A045925(n); P_n(1,4) = A273622; P_n(2,2) = A093967(n).
L(2*n,x)^2 - L(2*n-1,x)*L(2*n+1,x) = x^2 + 4 for n >= 1.
Sum_{n >= 1} L(2*n,x)/( L(2*n-1,x) * L(2*n+1,x) ) = 1/x^2 and
Sum_{n >= 1} (-1)^(n+1)/( L(2*n,x) + x^2/L(2*n,x) ) = 1/(x^2 + 4), both valid for all nonzero real x. (End)
From Peter Bala, Nov 18 2022: (Start)
L(n,x) = Sum_{k = 0..floor(n/2)} (n/(n-k))*binomial(n-k,k)*x^(n-2*k) for n >= 1.
For odd m, L(n, L(m,x)) = L(n*m, x).
For integral x, the sequence {u(n)} := {L(n,x)} satisfies the Gauss congruences: u(m*p^r) == u(m*p^(r-1)) (mod p^r) for all positive integers m and r and all primes p.
Let p be an odd prime and let 0 <= k <= p - 1. Let alpha_k = the p-adic limit_{n -> oo} L(p^n,k). Then alpha_k is a well-defined p-adic-integer and the polynomial L(p,x) - x of degree p factorizes as L(p,x) - x = Product_{k = 0..p-1} (x - alpha_k). For example, L(5,x) - x = x^5 + 5*x^3 + 4*x = x*(x - A269591)*(x - A210850)*(x - A210851)*(x - A269592) in the ring of 5-adic integers. (End)
The formula for L(n,x) given in the first line of the preceding section, with L(0, x) = 2, is rewritten L(n, x) = Sum_{k = 0..floor(n/2)} A034807(n, k)*x^(n - 2*k). See the formula by Alexander Elkins in A034807. - Wolfdieter Lang, Feb 10 2023

A090301 a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.

Original entry on oeis.org

2, 15, 227, 3420, 51527, 776325, 11696402, 176222355, 2655031727, 40001698260, 602680505627, 9080209282665, 136805819745602, 2061167505466695, 31054318401746027, 467875943531657100, 7049193471376602527

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.066372... = 2/(15+sqrt(229)) = (sqrt(229)-15)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 15.066372... = (15+sqrt(229))/2 = 2/(sqrt(229)-15).
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

			a(4) = 15*a(3) + a(2) = 15*3420 + 227 = ((15+sqrt(229))/2)^4 + ((15-sqrt(229))/2)^4 = 51526.9999805 + 0.0000194 = 51527.

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 (15,1).

Cf. A058087, A071416.
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), this sequence (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), A089772 (m=199).

m:=15;; 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:=15; 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, 15*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019

LucasL[Range[20]-1, 15] (* G. C. Greubel, Dec 31 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 15*I/2) ) \\ G. C. Greubel, Dec 31 2019

[2*(-I)^n*chebyshev_T(n, 15*I/2) for n in (0..20)] # G. C. Greubel, Dec 31 2019

a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
a(n) = ((15+sqrt(229))/2)^n + ((15-sqrt(229))/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.: (2-15*x)/(1-15*x-x^2). - Philippe Deléham, Nov 02 2008
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
Lim_{k-> infinity} a(n+k)/a(k) = (A090301(n) + A154597(n)*sqrt(229))/2.
Lim_{n-> infinity} A090301(n)/ A154597(n) = sqrt(229).
a(2n+1) = 15*A098246(n).
a(3n+1) = A041426(5n), a(3n+2) = A041426(5n+3), a(3n+3) = 2*A041426(5n+4).
(End)
a(n) = Lucas(n, 15) = 2*(-i)^n * ChebyshevT(n, 15*i/2). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(15*x/2)*cosh(sqrt(229)*x/2). - Stefano Spezia, Jan 01 2020

More terms from Ray Chandler, Feb 14 2004

A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

Original entry on oeis.org

2, 17, 291, 4964, 84679, 1444507, 24641298, 420346573, 7170533039, 122319408236, 2086600473051, 35594527450103, 607193567124802, 10357885168571737, 176691241432844331, 3014108989526925364, 51416544063390575519

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.058621... = 2/(17+sqrt(293)) = (sqrt(293)-17)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 17.058621... = (17+sqrt(293))/2 = 2/(sqrt(293)-17).
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

			a(4) = 17*a(3) + a(2) = 17*4964 + 291=((17+sqrt(293))/2)^4 + ((17-sqrt(293))/2)^4 = 84678.999988190 + 0.000011809 = 84679.

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 (17,1).

Cf. A005074.

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), this sequence (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).

m:=17;; 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:=17; 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, 17*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{17,1},{2,17},30] (* Harvey P. Dale, Jan 24 2018 *)
LucasL[Range[20]-1, 17] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 17*I/2) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 17*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.
a(n) = ((17+sqrt(293))/2)^n + ((17-sqrt(293))/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.: (2-17*x)/(1-17*x-x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 17*A098249(n).
a(3n+1) = A041550(5n), a(3n+2) = A041550(5n+3), a(3n+3) = 2*A041550(5n+4).
Lim_{k-> infinity} a(n+k)/a(k) = (A090306(n) + A178765(n)*sqrt(293))/2.
Lim_{n-> infinity} A090306(n)/A178765(n) = sqrt(293). (End)
a(n) = Lucas(n, 17) = 2*(-i)^n * ChebyshevT(n, 17*i/2). - G. C. Greubel, Dec 30 2019
E.g.f.: 2*exp(17*x/2)*cosh(sqrt(293)*x/2). - Stefano Spezia, Dec 31 2019

More terms from Ray Chandler, Feb 14 2004

A090309 a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.

Original entry on oeis.org

2, 20, 402, 8060, 161602, 3240100, 64963602, 1302512140, 26115206402, 523606640180, 10498248010002, 210488566840220, 4220269584814402, 84615880263128260, 1696537874847379602, 34015373377210720300

Offset: 0

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

Lim_{n-> infinity} a(n)/a(n+1) = 0.0498756... = 1/(10+sqrt(101)) = (sqrt(101)-10).

Lim_{n-> infinity} a(n+1)/a(n) = 20.0498756... = (10+sqrt(101)) = 1/(sqrt(101)-10).

			a(4) = 20*a(3) + a(2) = 20*8060 + 402 = (10+sqrt(101))^4 + (10-sqrt(101))^4 = 161601.999993811 + 0.000006188 = 161602.

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 (20,1).

Cf. A002116.

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

m:=20;; 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:=20; 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, 10*I)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{20,1},{2,20},20] (* Harvey P. Dale, Nov 19 2015 *)
LucasL[Range[20]-1,20] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 10*I) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 10*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.
a(n) = (10 + sqrt(101))^n + (10 - sqrt(101))^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.: (2-20*x)/(1-20*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 20) = 2*(-i)^n * ChebyshevT(n, 10*i). - G. C. Greubel, Dec 30 2019

More terms from Ray Chandler, Feb 14 2004

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

Original entry on oeis.org

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

A140455 13-Fibonacci sequence.

Original entry on oeis.org

0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309

Offset: 0

R. J. Mathar, Jul 22 2008

The k-Fibonacci sequences for k=2..12 are A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061. This here is k=13. k=14 is A041085, k=16 A041113, k=18 A041145, k=20 A041181, k=22 A041221.
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 13's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,13} avoiding runs of zeros of odd length. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 21 2023: (Start)
Also called the 13-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 13 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Sergio Falcon and Angel Plaza, The k-Fibonacci sequence and Pascal 2-triangle, Chaos, Solit. Fract. 33 (2007) 38-49.
Tanya Khovanova, Recursive sequences.
Index entries for linear recurrences with constant coefficients, signature (13,1).

Row n=13 of A073133, A172236 and A352361 and column k=13 of A157103.

F := proc(n,k) coeftayl( x/(1-k*x-x^2),x=0,n) ; end: for n from 0 to 20 do printf("%d,",F(n,13)) ; od:

LinearRecurrence[{13, 1}, {0, 1}, 30] (* Vincenzo Librandi, Nov 17 2012 *)

[lucas_number1(n,13,-1) for n in range(0, 18)] # Zerinvary Lajos, Apr 29 2009

O.g.f.: x/(1-13*x-x^2).
a(n) = 13*a(n-1) + a(n-2).
a(n-r)*a(n+r) - a(n)^2 = (-1)^(n+1-r)*a(r)^2.
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n,2i+1)*13^(n-1-2*i)*(13^2+4)^i/2^(n-1).
a(n) = ((13+sqrt(173))^n - (13-sqrt(173))^n)/(2^n*sqrt(173)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2*n) = 13*A097844(n), a(2*n+1) = A098244(n).
a(3*n+1) = A041319(5*n), a(3*n+2) = A041319(5*n+3), a(3*n+3) = 2*A041319(5*n+4).
Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.
Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)

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

cp's OEIS Frontend

A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

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A114525 Triangle of coefficients of the Lucas (w-)polynomials.

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A090301 a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.

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A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

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A090309 a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.

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

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