A099374 -id:A099374 - cp's OEIS frontend

A041041 Denominators of continued fraction convergents to sqrt(26).

1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, 10928351501, 110365635060, 1114584702101, 11256212656070, 113676711262801, 1148023325284080, 11593909964103601, 117087122966320090, 1182465139627304501, 11941738519239365100

Offset: 0

N. J. A. Sloane

Generalized Fibonacci sequence.
Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - Gary W. Adamson, Jun 13 2008
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 10's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0, 1, ..., 10} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Bruno Berselli, May 03 2018: (Start)
Numbers k for which m*k^2 + (-1)^k is a perfect square:
m =  2: 0, 1, 2, 5, 12, 29, 70, 169, ...   (A000129);
m =  3: 0, 4, 56, 780, 10864, 151316, ...  (4*A007655);
m =  5: 0, 1, 4, 17, 72, 305, 1292, ...    (A001076);
m =  6: 0, 2, 20, 198, 1960, 19402, ...    (A001078);
m =  7: 0, 48, 12192, 3096720, ...         (2*A175672);
m =  8: 0, 6, 204, 6930, 235416, ...       (A082405);
m = 10: 0, 1, 6, 37, 228, 1405, 8658, ...  (A005668);
m = 11: 0, 60, 23880, 9504180, ...         [°];
m = 12: 0, 2, 28, 390, 5432, 75658, ...    (A011944);
m = 13: 0, 5, 180, 6485, 233640, ...       (5*A041613);
m = 14: 0, 4, 120, 3596, 107760, ...       (A068204);
m = 15: 0, 8, 496, 30744, 1905632, ...     [°];
m = 17: 0, 1, 8, 65, 528, 4289, 34840, ... (A041025);
m = 18: 0, 4, 136, 4620, 156944, ...       (A202299);
m = 19: 0, 13260, 1532829480, ...          [°];
m = 20: 0, 2, 36, 646, 11592, 208010, ...  (A207832);
m = 21: 0, 12, 1320, 145188, ...           (A174745);
m = 22: 0, 42, 16548, 6519870, ...         (A174766);
m = 23: 0, 240, 552480, 1271808720, ...    [°];
m = 24: 0, 10, 980, 96030, 9409960, ...    (A168520);
m = 26: 0, 1, 10, 101, 1020, 10301, ...    (this sequence);
m = 27: 0, 260, 702520, 1898208780, ...    [°];
m = 28: 0, 24, 6096, 1548360, ...          (A175672);
m = 29: 0, 13, 1820, 254813, 35675640, ... [°];
m = 30: 0, 2, 44, 966, 21208, 465610, ...  (2*A077421), etc.
[°] apparently without related sequences in the OEIS.
(End)
From Michael A. Allen, Mar 12 2023: (Start)
Also called the 10-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 10 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.
Sergio Falcón and Ángel Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007).
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,1).

Cf. A099374 (squares).
Cf. A041040.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A243399.
Row n=10 of A073133, A172236 and A352361.

I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

seq(combinat:-fibonacci(n+1, 10), n=0..19); # Peter Luschny, May 04 2018

Denominator[Convergents[Sqrt[26], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{10,1}, {1,10}, 30] (* G. C. Greubel, Jan 24 2018 *)

x='x+O('x^30); Vec(1/(1-10*x-x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,10,-1) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009

G.f.: 1/(1 - 10*x - x^2).
a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.
a(n) = S(n, 10*i)*(-i)^n where i^2:=-1 and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = 5+sqrt(26), am = -1/ap = 5-sqrt(26).
a(n) = F(n+1, 10), the (n+1)-th Fibonacci polynomial evaluated at x=10. - T. D. Noe, Jan 19 2006
a(n) = Sum_{i=0..floor(n/2)} binomial(n-i,i)*10^(n-2*i). - Sergio Falcon, Sep 24 2007

Extended by T. D. Noe, May 23 2011

A099372 a(n) = A099371(n)^2.

Original entry on oeis.org

0, 1, 81, 6724, 558009, 46308025, 3843008064, 318923361289, 26466795978921, 2196425142889156, 182276820063821025, 15126779640154255921, 1255340433312739420416, 104178129185317217638609, 8645529381948016324584129, 717474760572500037722844100, 59541759598135555114671476169

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=9.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 9 kinds of half-square available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 9 kinds of (1/4,1/4)-fence available. - Michael A. Allen, Mar 21 2024

Stefano Spezia, Table of n, a(n) for n = 0..500
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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (82,82,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A099371, A099373.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, A099367, A099369, this sequence, A099374.

LinearRecurrence[{82,82,-1},{0,1,81},17] (* Stefano Spezia, Apr 06 2023 *)

a(n) = A099371(n)^2.
a(n) = 82*a(n-1) + 82*a(n-2) - a(n-3), n>=3; a(0)=0, a(1)=1, a(2)=81.
a(n) = 83*a(n-1) - a(n-2) - 2*(-1)^n, n>=2; a(0)=0, a(1)=1.
a(n) = 2*(T(n, 83/2)-(-1)^n)/85 with twice the Chebyshev polynomials of the first kind: 2*T(n, 83/2) = A099373(n).
G.f.: x*(1-x)/((1-83*x+x^2)*(1+x)) = x*(1-x)/(1-82*x-82*x^2+x^3).
E.g.f.: 2*exp(-x)*(exp(85*x/2)*cosh(9*sqrt(85)*x/2) - 1)/85. - Stefano Spezia, Apr 06 2023
a(n) = (1 - (-1)^n)/2 + 81*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Mar 21 2024
Product_{n>=2} (1 + (-1)^n/a(n)) = (9 + sqrt(85))/18 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099279 Squares of A001076.

Original entry on oeis.org

0, 1, 16, 289, 5184, 93025, 1669264, 29953729, 537497856, 9645007681, 173072640400, 3105662519521, 55728852710976, 1000013686278049, 17944517500293904, 322001301319012225, 5778078906241926144, 103683419011035658369, 1860523463292399924496, 33385738920252162982561

Offset: 0

Wolfdieter Lang, Oct 18 2004

For the generalized Fibonacci sequences U(n-1;a) = (ap(a)^n - am(a)^n)/(ap(a) - am(a)) with ap(a) = (a + sqrt(a^2+4))/2, am(a) = (a - sqrt(a^2+4))/2, a from the integers, one has for the squared sequences U(n-1;a)^2 = (2*T(n,(a^2+2)/2) - 2*(-1)^n)/(a^2+4). Here T(n,x) are Chebyshev's polynomials of the first kind (see A053120). Therefore the o.g.f. for the squared sequence is x*(1-x)/((1+x)*(1-(a^2+2)*x+x^2)) = x*(1-x)/(1 - (a^2+1)*x - (a^2+1)*x^2 + x^3). For this example a=4.
Unsigned member r=-16 of the family of Chebyshev sequences S_r(n) defined in A092184.
(-1)^(n+1)*a(n) = S_{-16}(n), n >= 0, defined in A092184.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 4 kinds of half-squares available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 4 kinds of (1/4,1/4)-fences available. - Michael A. Allen, Mar 12 2023

G. C. Greubel, Table of n, a(n) for n = 0..750
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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (17,17,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A001076, A001654, A023039, A053120, A092184.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, this sequence, A099365, A099366, A099367, A099369, A099372, A099374.

[Fibonacci(3*n)^2/4: n in [0..30]]; // G. C. Greubel, Aug 18 2022

with (combinat):seq(fibonacci(n,4)^2,n=0..16); # Zerinvary Lajos, Apr 09 2008
nmax:=48: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax/3 do a(n):=a(n-1)+A001654(3*n-2) od: seq(a(n),n=0..nmax/3); # Johannes W. Meijer, Sep 22 2010

LinearRecurrence[{17,17,-1},{0,1,16},30] (* Harvey P. Dale, Mar 26 2012 *)
Fibonacci[3*Range[0, 30]]^2/4 (* G. C. Greubel, Aug 18 2022 *)

numlib::fibonacci(3*n)^2/4 $ n = 0..35; // Zerinvary Lajos, May 13 2008

my(x='x+O('x^99)); concat([0], Vec(x*(1-x)/((1-18*x+x^2)*(1+x)))) \\ Altug Alkan, Dec 17 2017

[(fibonacci(3*n))^2/4 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

a(n) = A001076(n)^2.
a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=16.
a(n) = 18*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2, a(0)=0, a(1)=1.
a(n) = (T(n, 9) - (-1)^n)/10 with Chebyshev's T(n, x) polynomials of the first kind. T(n, 9) = A023039(n).
G.f.: x*(1-x)/((1+x)*(1-18*x+x^2)) = x*(1-x)/(1-17*x-17*x^2+x^3).
a(n) = a(n-1) + A001654(3*n-2) with a(0)=0, where A001654 are the golden rectangle numbers. - Johannes W. Meijer, Sep 22 2010
a(n+1) = (1 + (-1)^n)/2 + 16*Sum_{r=1..n} ( r*a(n+1-r) ). - Michael A. Allen, Mar 12 2023
E.g.f.: exp(-x)*(exp(10*x)*cosh(4*sqrt(5)*x) - 1)/10. - Stefano Spezia, Apr 06 2023
Product_{n>=2} (1 + (-1)^n/a(n)) = (2 + sqrt(5))/4 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099365 Squares of A052918(n-1) (generalized Fibonacci).

Original entry on oeis.org

0, 1, 25, 676, 18225, 491401, 13249600, 357247801, 9632441025, 259718659876, 7002771375625, 188815108482001, 5091005157638400, 137268324147754801, 3701153746831741225, 99793882840309258276, 2690733682941518232225

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=5.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 5 kinds of half-squares available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 5 kinds of (1/4,1/4)-fences available. - Michael A. Allen, Mar 30 2023

G. C. Greubel, Table of n, a(n) for n = 0..650
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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (26,26,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, this sequence, A099366, A099367, A099369, A099372, A099374.

[(2/29)*(Evaluate(ChebyshevFirst(n), 27/2) -(-1)^n): n in [0..30]]; // G. C. Greubel, Aug 21 2022

with (combinat):seq(fibonacci(n,5)^2,n=0..16); # Zerinvary Lajos, Apr 09 2008

LinearRecurrence[{26,26,-1},{0,1,25},30] (* Harvey P. Dale, Sep 25 2019 *)

def A099365(n): return (2/29)*(chebyshev_T(n, 27/2) - (-1)^n)
[A099365(n) for n in (0..30)] # G. C. Greubel, Aug 21 2022

a(n) = A052918(n-1)^2, n >= 1, a(0) = 0.
a(n) = 26*a(n-1) + 26*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=25.
a(n) = 27*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
a(n) = 2*(T(n, 27/2) - (-1)^n)/29 with twice the Chebyshev's T(n, x) polynomials of the first kind. 2*T(n, 27/2) = A090248(n).
G.f.: x*(1-x)/((1-27*x+x^2)*(1+x)) = x*(1-x)/(1-26*x-26*x^2+x^3).
a(n) = (1 - (-1)^n)/2 + 25*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Mar 30 2023
Product_{n>=2} (1 + (-1)^n/a(n)) = (5 + sqrt(29))/10 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099369 Squares of A041025(n-1), n>=1, (generalized Fibonacci).

Original entry on oeis.org

0, 1, 64, 4225, 278784, 18395521, 1213825600, 80094094081, 5284996383744, 348729667233025, 23010873040995904, 1518368891038496641, 100189335935499782400, 6610977802851947141761, 436224345652293011573824

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=8.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 8 kinds of half-squares available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 8 kinds of (1/4,1/4)-fences available. - Michael A. Allen, Apr 30 2023

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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (65,65,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, A099367, this sequence, A099372, A099374.

LinearRecurrence[{65,65,-1},{0,1,64},20] (* Harvey P. Dale, Oct 05 2021 *)

a(n) = A041025(n-1)^2, n >= 1, a(0)=0.
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=64.
a(n) = 66*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
a(n) = (T(n, 33) - (-1)^n)/34 with the Chebyshev polynomials of the first kind: T(n, 33) = A099370(n).
G.f.: x*(1-x)/((1-66*x+x^2)*(1+x)) = x*(1-x)/(1-65*x-65*x^2+x^3).
a(n) = (1 - (-1)^n)/2 + 64*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Apr 30 2023
Product_{n>=2} (1 + (-1)^n/a(n)) = (4 + sqrt(17))/8 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099367 a(n) = A054413(n-1)^2, n >= 1.

Original entry on oeis.org

0, 1, 49, 2500, 127449, 6497401, 331240000, 16886742601, 860892632649, 43888637522500, 2237459621014849, 114066552034234801, 5815156694124960000, 296458924848338725201, 15113590010571150025249, 770496631614280312562500

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=7.

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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (50,50,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A054413.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, this sequence, A099369, A099372, A099374.

LinearRecurrence[{50,50,-1},{0,1,49},20] (* Harvey P. Dale, Jul 27 2023 *)

a(n) = A054413(n-1)^2, n >= 1. a(0)=0.
a(n) = 50*a(n-1) + 50*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=49.
a(n) = 51*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
a(n) = 2*(T(n, 51/2) - (-1)^n)/53 with twice the Chebyshev polynomials of the first kind: 2*T(n, 51/2) = A099368(n).
G.f.: x*(1-x)/((1-51*x+x^2)*(1+x)) = x*(1-x)/(1-50*x-50*x^2+x^3).
a(n+1) = (1 + (-1)^n)/2 + 49*Sum_{k=1..n} k*a(n+1-k). - Michael A. Allen, Feb 21 2023
Product_{n>=2} (1 + (-1)^n/a(n)) = (7 + sqrt(53))/14 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099366 Squares of A005668.

Original entry on oeis.org

0, 1, 36, 1369, 51984, 1974025, 74960964, 2846542609, 108093658176, 4104712468081, 155870980128900, 5918992532430121, 224765845252215696, 8535183127051766329, 324112192982714904804, 12307728150216114616225

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=6.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 6 kinds of half-squares available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 6 kinds of (1/4,1/4)-fences available. - Michael A. Allen, Apr 21 2023

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, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (37,37,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, this sequence, A099367, A099369, A099372, A099374.

with (combinat):seq(fibonacci(n,6)^2,n=0..15); # Zerinvary Lajos, Apr 09 2008

LinearRecurrence[{37,37,-1},{0,1,36},20] (* Harvey P. Dale, Sep 23 2018 *)

a(n) = A005668(n)^2.
a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=36.
a(n) = 38*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
a(n) = (T(n, 19) - (-1)^n)/20 with the Chebyshev polynomials of the first kind: T(n, 19) = A078986(n).
G.f.: x*(1-x)/((1 - 38*x + x^2)*(1+x)) = x*(1-x)/(1 - 37*x - 37*x^2 + x^3).
a(n) = (1 - (-1)^n)/2 + 36*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Apr 21 2023
Product_{n>=2} (1 + (-1)^n/a(n)) = (3 + sqrt(10))/6 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A099397 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=51.

Original entry on oeis.org

1, 51, 5201, 530451, 54100801, 5517751251, 562756526801, 57395647982451, 5853793337683201, 597029524795704051, 60891157735824130001, 6210301059529265556051, 633389816914249262587201

Offset: 0

Wolfdieter Lang, Oct 18 2004

Used in A099374.

Numbers n such that 26*(n^2-1) is square. - Vincenzo Librandi, Nov 17 2010

Indranil Ghosh, Table of n, a(n) for n = 0..497
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102, -1).

Row 5 of array A188645.

[n: n in [1..1000] |IsSquare(26*(n^2-1))] // Vincenzo Librandi, Nov 17 2010

LinearRecurrence[{102, -1},{1, 51},13] (* Ray Chandler, Aug 11 2015 *)

a(n) = polchebyshev(n, 1, 51); \\ Michel Marcus, Jan 20 2018

a(n) = 102*a(n-1) - a(n-2), n>=1; a(-1):= 51, a(0)=1.
a(n) = T(n, 51) = (S(n, 102)-S(n-2, 102))/2 = S(n, 102)-51*S(n-1, 102) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 102)=(n).
a(n) = (ap^n + am^n)/2 with ap := 51+10*sqrt(26) and am := 51-10*sqrt(26).
a(n) = Sum_{k=0..floor(n/2)} (((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*51)^(n-2*k)), n >= 1. a(0):=1.
G.f.: (1 - 51*x)/(1 - 102*x + x^2).

cp's OEIS Frontend

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