A092184 -id:A092184 - cp's OEIS frontend

A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.

1, 2, 9, 32, 121, 450, 1681, 6272, 23409, 87362, 326041, 1216800, 4541161, 16947842, 63250209, 236052992, 880961761, 3287794050, 12270214441, 45793063712, 170902040409, 637815097922, 2380358351281, 8883618307200, 33154114877521, 123732841202882

Offset: 0

N. J. A. Sloane

Number of tilings of a box with sides 2 X 2 X n in R^3 by boxes of sides 2 X 1 X 1 (3-dimensional dominoes). - Frans J. Faase
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Also stacking bricks.
a(n)*(-1)^n = (1-T(n+1,-2))/3, n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-2 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. - Wolfdieter Lang, Oct 18 2004
Partial sums of A217233. - Bruno Berselli, Oct 01 2012
The sequence is the case P1 = 2, P2 = -8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

			G.f. = 1 + 2*x + 9*x^2 + 32*x^3 + 121*x^4 + 450*x^5 + ... - _Michael Somos_, Mar 17 2022

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 360.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Butler and S. Osborne, Counting tilings by taking walks, preprint, 2012; J. Combin. Math. Combin. Comput. 88 2014 17-25. - From _N. J. A. Sloane_, Dec 27 2012
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
D. Deford, Seating rearrangements on arbitrary graphs, preprint 2013; involve, Vol. 7 (2014), No. 6, 787-805.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
Charles H. Jepsen, Packing a box with bricks, Math. Mag. 64 (2) (1991) 92-97, Table 1
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (36).
Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.
László Németh, Tilings of (2 X 2 X n)-board with colored cubes and bricks, arXiv:1909.11729 [math.CO], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Radovan Potůček, The number of fillings a 2 X 2 X n prism with 1 X 1 X 2 prisms, Equations (2024) Vol. 3, 104-114.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to dominoes
Index entries for sequences related to bricks
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A002530, A004003, A006125, A217233 (first differences), A109437 (partial sums).
Column k=2 of A181206, A189650, A233308.
Cf. A100047.

a:=[1,2,9];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Nov 16 2018

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)/(1-3*x-3*x^2+x^3))); // G. C. Greubel, Nov 16 2018

CoefficientList[Series[(1 - x)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2012 *)
RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == BitXor[1, a[n - 1]]^2/a[n - 2]}, a, {n, 30}] (* Jon Maiga, Nov 16 2018 *)
LinearRecurrence[{3,3,-1}, {1,2,9}, 30] (* G. C. Greubel, Nov 16 2018 *)
a[ n_] := (-1)^n * ChebyshevU[n, Sqrt[-1/2]]^2; (* Michael Somos, Mar 17 2022 *)

a(n)=(sqrt(3)+2)^(n+1) \/ 6 \\ Charles R Greathouse IV, Aug 18 2016

a(n)=([0,1,0; 0,0,1; -1,3,3]^n*[1;2;9])[1,1] \\ Charles R Greathouse IV, Aug 18 2016

Vec((1 - x) / ((1 + x)*(1 - 4*x + x^2)) + O(x^40)) \\ Colin Barker, Dec 16 2017

{a(n) = simplify((-1)^n * polchebyshev(n, 2, quadgen(-8)/2)^2)}; /* Michael Somos, Mar 17 2022 */

s=((1-x)/(1-3*x-3*x^2+x^3)).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018

G.f.: (1-x)/((1+x)*(1-4*x+x^2)) = (1-x)/(1-3*x-3*x^2+x^3). - Simon Plouffe in his 1992 dissertation; typo corrected by Vincenzo Librandi, Oct 15 2012
Nearest integer to (1/6)*(2+sqrt(3))^(n+1). - Don Knuth, Jul 15 1995
For n >= 4, a(n) = 3a(n-1) + 3a(n-2) - a(n-3). - Avi Peretz (njk(AT)netvision.net.il), Mar 30 2001
For n >= 3, a(n) = 4a(n-1) - a(n-2) + 2*(-1)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 14 2001
From Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 11 2001: The values are a(1) = 2 * 1^2, a(2) = 3^2, a(3) = 2 * 4^2, a(4) = 11^2, a(5) = 2 * 15^2, ... and in general for odd n a(n) is twice a square, for even n a(n) is a square. If we define b(n) by b(n) = sqrt(a(n)) for even n, b(n) = sqrt(a(n)/2) for odd n then apart from the first 2 elements b(n) is A002530(n+1).
a(n) + a(n+1) = A001835(n+2). - R. J. Mathar, Dec 06 2013
From Peter Bala, Apr 03 2014: (Start)
a(n) = |U(n,i/sqrt(2))|^2 where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n-1) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 2; 1, 1] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = (2*(-1)^n + (2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n)) / 6. - Colin Barker, Dec 16 2017
a(n) = (1 XOR a(n-1))^2/a(n-2). - Jon Maiga, Nov 16 2018
a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 17 2022
INVERT transform of sequence p(n), n > 0, where p is the number of nonreducible tilings by height of 2 X 2 X n using dicubes; p is (2, 5, 4, 4, 4, 4...). - Nicolas Bělohoubek, Jun 04 2024

A079291 Squares of Pell numbers.

Original entry on oeis.org

0, 1, 4, 25, 144, 841, 4900, 28561, 166464, 970225, 5654884, 32959081, 192099600, 1119638521, 6525731524, 38034750625, 221682772224, 1292061882721, 7530688524100, 43892069261881, 255821727047184, 1491038293021225

Offset: 0

Ralf Stephan, Feb 08 2003

(-1)^(n+1)*a(n) is the r=-4 member of the r-" of sequences S_r(n), n>=1, defined in A092184 where more information can be found.
Binomial transform of A086346. - Johannes W. Meijer, Aug 01 2010
In general, squaring the terms of a Horadam sequence with signature (c,d) will result in a third-order recurrence with signature (c^2+d, c^2*d+d^2, -d^3). - Gary Detlefs, Nov 11 2021
(Conjectured) For any primitive Pythagorean triple of the form (X, Y, Z=Y+1), it appears that Y or Z will always be (and only be) a square Pell number if X = A001333(n), for n > 1. If n is even, Y is always a square Pell number, and if n is odd, then Z is always a square Pell number. For example: (3, 4, 5), (7, 24, 25), (17, 144, 145), (41, 840, 841), (99, 4900, 4901). - Jules Beauchamp, Feb 02 2022
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/2,1/2)-fences, black half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal), and white half-squares. A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using black (1/4,1/4)-fences, white (1/4,1/4)-fences, and (1/4,3/4)-fences. - Michael A. Allen, Dec 29 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Joerg Arndt, Matters Computational (The Fxtbook), sect. 32.1.5, pp. 626-627
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.
Toufik Mansour, A note on sum of k-th power of Horadam's sequence, arXiv:math/0302015 [math.CO], 2003.
Toufik Mansour, Squaring the terms of an ell-th order linear recurrence, arXiv:math/0303138 [math.CO], 2003.
Pantelimon Stanica, Generating functions, weighted and non-weighted sums for powers of second-order recurrence sequences, arXiv:math/0010149 [math.CO], 2000.
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000129, A001254, A001109, A001541, A001653, A174968.

Cf. A002203, A007598, A084158 (partial sums), A086346, A092184.

I:=[0,1,4]; [n le 3 select I[n] else 5*Self(n-1)+ 5*Self(n-2) - Self(n-3): n in [1..31]]; // Vincenzo Librandi, May 17 2013

with(combinat):seq(fibonacci(i,2)^2, i=0..31); # Zerinvary Lajos, Mar 20 2008

CoefficientList[Series[x(1-x)/((1+x)*(1-6x+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 17 2013 *)
LinearRecurrence[{5,5,-1},{0,1,4},40] (* Harvey P. Dale, Dec 20 2015 *)
Fibonacci[Range[0, 30], 2]^2 (* G. C. Greubel, Sep 17 2021 *)

[lucas_number1(n, 2, -1)^2 for n in (0..30)] # G. C. Greubel, Sep 17 2021

G.f.: x*(1-x)/((1+x)*(1-6*x+x^2)).
a(n) = (r^n + (1/r)^n - 2*(-1)^n)/8, with r = 3 + sqrt(8).
a(n+3) = 5*a(n+2) + 5*a(n+1) - a(n).
L.g.f.: (1/8)*log((1+2*x+x^2)/(1-6*x+x^2)) = Sum_{n>=0} (a(n)/n)*x^n, see p. 627 of the Fxtbook link; special case of the following: let v(0)=0, v(1)=1, and v(n) = u*v(n-1) + v(n-2), then (1/A)*log((1+2*x+x^2)/(1-(2-A)*x+x^2)) = Sum_{n>=0} v(n)^2/n*x^n where A = u^2 + 4. - Joerg Arndt, Apr 08 2011
a(n+1) = Sum_{k=0..n} ( (-1)^(n-k)*A001653(k) ); e.g., 144 = -1 + 5 - 29 + 169; 25 = 1 - 5 + 29. - Charlie Marion, Jul 16 2003
a(n) = A000129(n)^2.
a(n) = (T(n, 3) - (-1)^n)/4 with Chebyshev's polynomials of the first kind evaluated at x=3: T(n, 3) = A001541(n) = ((3 + 2*sqrt(2))^n + (3 - 2*sqrt(2))^n )/2. - Wolfdieter Lang, Oct 18 2004
a(n) is the rightmost term of M^n * [1 0 0] where M is the 3 X 3 matrix [4 4 1 / 2 1 0 / 1 0 0]. a(n+1) = leftmost term. E.g., a(6) = 4900, a(5) = 841 since M^5 * [1 0 0] = [4900 2030 841]. - Gary W. Adamson, Oct 31 2004
a(n) = ( (-1)^(n+1) + A001109(n+1) - 3*A001109(n) )/4. - R. J. Mathar, Nov 16 2007
a(n) = ( (((1 - sqrt(2))^n + (1 + sqrt(2))^n) /2 )^2 + (-1)^(n+1) )/2. - Antonio Pane (apane1(AT)spc.edu), Dec 15 2007
Lim_{k -> infinity} ( a(n+k)/a(k) ) = A001541(n) + 2*A001109(n)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
For n>0, a(2*n) = 6*a(2*n-1) - a(2*n-2) - 2, a(2*n+1) = 6*a(2*n) - a(2*n-1) + 2. - Charlie Marion, Sep 24 2011
a(n) = (1/8)*(A002203(2*n) - 2*(-1)^n). - G. C. Greubel, Sep 17 2021
Conjectured formula for (X, Y, Z) for primitive Pythagorean triple of the form (X, Y, Z=Y+1) is (A001333(n)^2, A079291(n)^2, A079291(n)^2-1) or (A001333(n)^2, A079291(n)^2-1, A079291(n)^2). As a closed formula (X, Y, Z) = (((1-sqrt(2))^n + (1+sqrt(2))^n)/2, (((1-sqrt(2))^n + (1+sqrt(2))^n)^2 - 4)/8, (((1-sqrt(2))^n + (1+sqrt(2))^n)^2 + 4)/8). - Jules Beauchamp, Feb 02 2022
From Michael A. Allen, Dec 29 2022: (Start)
a(n+1) = 6*a(n) - a(n-1) + 2*(-1)^n.
a(n+1) = (1 + (-1)^n)/2 + 4*Sum_{k=1..n} ( k*a(n+1-k) ). (End)
Product_{n>=2} (1 + (-1)^n/a(n)) = (1 + sqrt(2))/2 (A174968) (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A007877 Period 4 zigzag sequence: repeat [0,1,2,1].

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0

Offset: 0

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

Euler transform of finite sequence [2,-2,0,1]. - Michael Somos, Sep 17 2004
This is the r = 2 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found.
a(n+1) is the transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x) -> (1/(1 + x^2))A(x/(1 + x^2)). See also A084099. - Paul Barry, Oct 12 2004
Multiplicative with a(2) = 2, a(2^e) = 0 if e >= 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
The e.g.f. of 1, 2, 1, 0, 1, 2, 1, 0, ... (shifted left, offset zero) is exp(x) + sin(x).
Binomial transform is A000749(n+2). - Wesley Ivan Hurt, Dec 30 2015
Decimal expansion of 11/909. - David A. Corneth, Dec 12 2016
Ternary expansion of 1/5. - J. Conrad, Aug 14 2017

P. Liu, Efficient recognition of integer sequences, Master's Essay, University of Waterloo, Dec. 1994. (Annotated scanned copy)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000749, A021913, A049310, A056594, A063886, A084099, A092184, A181878.

Period k zigzag sequences: A000035 (k=2), this sequence (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

&cat [[0,1,2,1]^^25]; // Vincenzo Librandi, Dec 27 2015

A007877:=n->sqrt(n^2 mod 8); seq(A007877(n), n=0..100); # Wesley Ivan Hurt, Jan 01 2014

f[n_] := Mod[n, 4] - Mod[n^3, 4] + Mod[n^2, 4] (* Or *)
f[n_] := Mod[n, 2] + 2 Floor[Mod[n + 1, 4]/3] (* Or *)
f[n_] := Switch[Mod[n, 4], 0, 0, 1, 1, 2, 2, 3, 1]; Array[f, 105, 0] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Sqrt[Mod[n^2,8]], {n,0,100}] (* Wesley Ivan Hurt, Jan 01 2014 *)
LinearRecurrence[{1, -1, 1}, {0, 1, 2}, 80] (* Vincenzo Librandi, Dec 27 2015 *)
PadRight[{},100,{0,1,2,1}] (* Harvey P. Dale, Oct 24 2023 *)

a(n)=[0,1,2,1][1+n%4] \\ Jaume Oliver Lafont, Mar 27 2009

concat(0, Vec(x*(1+x)/(1-x+x^2-x^3) + O(x^100))) \\ Altug Alkan, Dec 29 2015

def A007877(n): return (0,1,2,1)[n&3] # Chai Wah Wu, Jan 26 2023

Multiplicative with a(p^e) = 2 if p = 2 and e = 0; 0 if p = 2 and e > 0; 1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = -Sum_{k=0..n} (-1)^C(k+2, 2) (Offset -1). - Paul Barry, Jul 07 2003
a(n) = 1 - cos(n*Pi/2); a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
a(n) = -a(n-2) + 2, n >= 2, a(0) = 0, a(1) = 1.
G.f.: x*(1+x)/((1-x)*(1+x^2)) = x*(1+x)/(1-x+x^2-x^3).
a(n) = 1 - T(n, 0) = 1 - A056594(n) with Chebyshev's polynomials T(n, x) of the first kind. Note that T(n, 0) = S(n, 0).
a(n) = b(n) + b(n-1), n >= 1, with b(n) := A021913(n+1) the partial sums of S(n,0) = U(n,0) = A056594(n) (Chebyshev's polynomials evaluated at x=0).
a(n) = 1 + (1/2){(-1)^[(n-1)/2] - (-1)^[n/2]}. - Ralf Stephan, Jun 09 2005
Non-reduced g.f.: x*(1+x)^2/(1-x^4). - Jaume Oliver Lafont, Mar 27 2009
a(n+1) = (S(n, sqrt(2)))^2, n >= 0, with the Chebyshev S-polynomials A049310. See the W. Lang link under A181878. - Wolfdieter Lang, Dec 15 2010
Dirichlet g.f. (1 + 1/2^s - 2/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (n mod 4) - (n^3 mod 4) + (n^2 mod 4). - Gary Detlefs, Apr 17 2011
a(n) = (n mod 2) + 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Jul 19 2011
a(n) = sqrt(n^2 mod 8). - Wesley Ivan Hurt, Jan 01 2014
a(n) = (n AND 4*k+2)-(n AND 4*k+1) + 2*floor(((n+2) mod 4)/3), for any k. - Gary Detlefs, Jun 08 2014
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/2). - Wesley Ivan Hurt, Dec 26 2015
a(n) = a(n-4) for n >= 4. - Wesley Ivan Hurt, Sep 07 2022
a(n) = n - 2*floor(n/4) - 2*floor((n+1)/4). - Ridouane Oudra, Jan 22 2024
E.g.f.: exp(x) - cos(x). - Stefano Spezia, Aug 04 2025

Chebyshev comments from Wolfdieter Lang, Sep 10 2004

A049684 a(n) = Fibonacci(2n)^2.

Original entry on oeis.org

0, 1, 9, 64, 441, 3025, 20736, 142129, 974169, 6677056, 45765225, 313679521, 2149991424, 14736260449, 101003831721, 692290561600, 4745030099481, 32522920134769, 222915410843904, 1527884955772561, 10472279279564025, 71778070001175616, 491974210728665289

Offset: 0

Clark Kimberling

This is the r=9 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

Apparently, this sequence consists of those nonnegative integers k for which x*(x^2-1)*y*(y^2-1) = k*(k^2-1) has a solution in nonnegative integers x, y. If k = a(n), x = A000045(2*n-1) and y = A000045(2*n+1) are a solution. See A374375 for numbers k*(k^2-1) that can be written as a product of two or more factors of the form x*(x^2-1). - Pontus von Brömssen, Jul 14 2024

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 27.
H. J. H. Tuenter, Fibonacci summation identities arising from Catalan's identity, Fib. Q., 60:4 (2022), 312-319.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Pridon Davlianidze, Problem B-1264, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 1 (2020), p. 82; It's All About Catalan, Solution to Problem B-1264, ibid., Vol. 59, No. 1 (2021), pp. 87-88.
E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 1, k=2.
R. Stephan, Boring proof of a nonlinearity
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

First differences give A033890.
First differences of A103434.
Bisection of A007598 and A064841.
a(n) = A064170(n+2) - 1 = (1/5) A081070.
Cf. A000032, A000045, A001622, A001906, A056854, A065563, A092184, A374375.

Join[{a=0, b=1}, Table[c=7*b-1*a+2; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
Fibonacci[Range[0, 40, 2]]^2 (* Harvey P. Dale, Mar 22 2012 *)
Table[Fibonacci[n - 1] Fibonacci[n + 1] - 1, {n, 0, 40, 2}] (* Bruno Berselli, Feb 12 2015 *)
LinearRecurrence[{8, -8, 1},{0, 1, 9},21] (* Ray Chandler, Sep 23 2015 *)

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

PARI
```
a(n)=fibonacci(2*n)^2
```

[fibonacci(2*n)^2 for n in range(0, 21)] # Zerinvary Lajos, May 15 2009

G.f.: (x+x^2) / ((1-x)*(1-7*x+x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) with n>2, a(0)=0, a(1)=1, a(2)=9.
a(n) = 7*a(n-1) - a(n-2) + 2 = A001906(n)^2.
a(n) = (A000032(4*n)-2)/5. [This is in Koshy's book (reference under A065563) on p. 88, attributed to Lucas 1876.] - Wolfdieter Lang, Aug 27 2012
a(n) = 1/5*(-2 + ( (7+sqrt(45))/2 )^n + ( (7-sqrt(45))/2 )^n). - Ralf Stephan, Apr 14 2004
a(n) = 2*(T(n, 7/2)-1)/5 with twice the Chebyshev polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2)= A056854(n). - Wolfdieter Lang, Oct 18 2004
a(n) = F(2*n-1)*F(2*n+1)-1, see A064170 - Bruno Berselli, Feb 12 2015
a(n) = Sum_{i=1..n} F(4*i-2) for n>0. - Bruno Berselli, Aug 25 2015
From Peter Bala, Nov 20 2019: (Start)
Sum_{n >= 1} 1/(a(n) + 1) = (sqrt(5) - 1)/2.
Sum_{n >= 1} 1/(a(n) + 4) = (3*sqrt(5) - 2)/16. More generally, it appears that
Sum_{n >= 1} 1/(a(n) + F(2*k+1)^2) = ((2*k+1)*F(2*k+1)*sqrt(5) - Lucas(2*k+1))/ (2*F(2*k+1)*F(4*k+2)) for k = 0,1,2,....
Sum_{n >= 2} 1/(a(n) - 1) = (8 - 3*sqrt(5))/9. (End)
E.g.f.: (1/5)*(-2*exp(x) + exp((16*x)/(1 + sqrt(5))^4) + exp((1/2)*(7 + 3*sqrt(5))*x)). - Stefano Spezia, Nov 23 2019
Product_{n>=2} (1 - 1/a(n)) = phi^2/3, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 01 2021
a(n) = A092521(n-1)+A092521(n). - R. J. Mathar, Nov 22 2024

Better description and more terms from Michael Somos

A092936 Area of n-th triple of hexagons around a triangle.

Original entry on oeis.org

1, 9, 100, 1089, 11881, 129600, 1413721, 15421329, 168220900, 1835008569, 20016873361, 218350598400, 2381839709041, 25981886201049, 283418908502500, 3091626107326449, 33724468272088441, 367877524885646400

Offset: 1

Peter J. C. Moses, Apr 18 2004

This is the unsigned member r=-9 of the family of Chebyshev sequences S_r(n) defined in A092184: ((-1)^(n+1))*a(n) = S_{-9}(n), n>=0.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/2,1/2)-fences, red half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal), green half-squares, and blue half-squares. A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,3/4)-fences, red (1/4,1/4)-fences, green (1/4,1/4)-fences, and blue (1/4,1/4)-fences. - Michael A. Allen, Dec 30 2022

			a(5) = 10*(1089+100)-9 = 11881. From A006190, a(5) = (3*33+10)^2 = 11881.

Muniru A Asiru, Table of n, a(n) for n = 1..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 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 (10,10,-1).

Cf. A005386, A006190, A057076, A092184, A176019.

a:=[1,9,100];; for n in [4..18] do a[n]:=10*(a[n-1]+a[n-2])-a[n-3]; od; a; # Muniru A Asiru, Feb 20 2018

seq(fibonacci(n,3)^2,n=1..18); # Zerinvary Lajos, Apr 05 2008

CoefficientList[Series[(1-x)*x/(1-10*x-10*x^2+x^3), {x, 0, 20}], x]
(CoefficientList[Series[x/(1-3*x-x^2), {x, 0, 20}], x])^2
Table[Round[((3+Sqrt[13])^n)^2/(13*4^n)], {n, 0, 20}]
LinearRecurrence[{10, 10, -1}, {1, 9, 100}, 18] (* Georg Fischer, Feb 22 2019 *)

a(n) = A006190(n)^2.
a(n) = 10*(a(n-1)+a(n-2)) - a(n-3).
G.f.: (1-x)*x/(1-10*x-10*x^2+x^3).
a(n) = ((3-sqrt(13))^n-(3+sqrt(13))^n)^2/(13*4^n).
a(n) = 2*(T(n, 11/2)-(-1)^n)/13 with twice the Chebyshev polynomials of the first kind evaluated at x = 11/2: 2*T(n, 11/2) = A057076(n) = ((11+3*sqrt(13))^n + (11-3*sqrt(13))^n)/2^n. - Wolfdieter Lang, Oct 18 2004
From Michael A. Allen, Dec 30 2022: (Start)
a(n+1) = 11*a(n) - a(n-1) + 2*(-1)^n.
a(n+1) = (1 + (-1)^n)/2 + 9*Sum_{k=1..n} ( k*a(n+1-k) ). (End)
Product_{n>=2} (1 + (-1)^n/a(n)) = (3 + sqrt(13))/6 (A176019) (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A054493 A Pellian-related recursive sequence.

Original entry on oeis.org

1, 7, 36, 175, 841, 4032, 19321, 92575, 443556, 2125207, 10182481, 48787200, 233753521, 1119980407, 5366148516, 25710762175, 123187662361, 590227549632, 2827950085801, 13549522879375, 64919664311076, 311048798676007, 1490324329068961, 7140572846668800

Offset: 0

Barry E. Williams, May 06 2000

This is the r=7 member in the r-family of sequences S_r(n+1) defined in A092184 where more information can be found.

Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 7, P2 = 10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

			G.f. = 1 + 7*x + 36*x^2 + 175*x^3 + 841*x^4 + 4032*x^5 + 19321*x^6 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
R. Stephan, Boring proof of a nonlinearity
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-6,1)

Cf. A004254, A100047, A030221 (first differences).

A054493 := proc(n)
    option remember;
    if n <= 1 then
        6*n+1 ;
    else
        5*procname(n-1)-procname(n-2)+2 ;
    end if ;
end proc:
seq(A054493(n),n=0..10) ; # R. J. Mathar, Apr 16 2018

LinearRecurrence[{6,-6,1},{1,7,36},30] (* Harvey P. Dale, Apr 15 2015 *)
a[ n_] := ChebyshevU[n, Sqrt[7]/2]^2; (* Michael Somos, Jan 22 2017 *)

{a(n) = simplify(polchebyshev(n, 2, quadgen(28)/2)^2)}; /* Michael Somos, Jan 22 2017 */

a(n) = 5*a(n-1) - a(n-2) + 2, a(0)=1, a(1)=7.
A004254 = sqrt{21*(A054493)^2+28*(A054493)}/7. - James Sellers, May 10 2000
a(n) = (1/3)*(-2 + ((5+sqrt(21))/2)^n + ((5-sqrt(21))/2)^n). - Ralf Stephan, Apr 14 2004
G.f.: (1+x)/((1-x)*(1 - 5*x + x^2)) = (1+x)/(1 - 6*x + 6*x^2 - x^3). From the R. Stephan link.
a(n) = 6*a(n-1) - 6*a(n-2) + a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=7.
a(n) = (2*T(n, 5/2)-2)/3, with twice the Chebyshev polynomials of the first kind, 2*T(n, x=5/2)=A003501(n).
a(n) = b(n) + b(n-1), n>=1, with b(n)=A089817(n) the partial sums of S(n, 5)= U(n, 5/2)=A004254(n+1), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind.
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
Let {u(n)} be the Lucas sequence in the quadratic integer ring Z[sqrt(7)] defined by the recurrence u(0) = 0, u(1) = 1 and u(n) = sqrt(7)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u(n)^2.
Equivalently, a(n) = U(n-1,sqrt(7)/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = 1/3*( ((sqrt(7) + sqrt(3))/2)^n - ((sqrt(7) - sqrt(3))/2)^n )^2.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -5/2; 1, 7/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(2*n - 1) = 7 * A004254(n)^2, a(2*n) = A030221(n)^2 for all n in Z. - Michael Somos, Jan 22 2017
a(n) = a(-2-n) for all n in Z. - Michael Somos, Jan 22 2017
0 = 1 + a(n)*(-2 + a(n) - 5*a(n+1)) + a(n+1)*(-2 + a(n+1)) for all n in Z. - Michael Somos, Jan 22 2017

Chebyshev comments from Wolfdieter Lang, Sep 10 2004

A005386 Area of n-th triple of squares around a triangle.

Original entry on oeis.org

1, 3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003, 638710426743841, 3060245505715200

Offset: 1

Jean Meeus

a(n)*(-1)^(n+1) is the r=-3 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.

The sequence is the case P1 = 3, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000
J. Meeus, Letter to N. J. A. Sloane with attachment, Mar 1975
J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 14 (1975-1976) 77-81.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

Essentially the same as A003769.
First differences of A099025.
Cf. A100047.

I:=[1, 3, 16]; [n le 3 select I[n] else 4*Self(n-1) +4*Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Nov 16 2022

A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
a:= n-> (Matrix([[0,1,3]]). Matrix(3, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,4,-1][i] else 0 fi)^(n))[1,1]: seq(a(n), n=1..25); # Alois P. Heinz, Aug 05 2008

a[n_]:= Module[{n1=1, n2=0}, Do[{n1, n2}={Sqrt[3]*n1+n2, n1}, {n-1}];n1^2];
Table[a[n], {n,30}]
a[n_]:= Round[((5+Sqrt[21])/2)^n/7]; Table[a[n], {n, 30}]
Rest@(CoefficientList[Series[x/(1-x*(Sqrt[3]+x)), {x, 0, 30}], x])^2
Abs[ChebyshevU[Range[1,40]-1, I*Sqrt[3]/2]]^2 (* G. C. Greubel, Nov 16 2022 *)

def A005386(n): return abs(chebyshev_U(n-1, i*sqrt(3)/2))^2
[A005386(n) for n in range(1,40)] # G. C. Greubel, Nov 16 2022

G.f.: x*(1-x)/((1+x)*(1-5*x+x^2)).
a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3), a(1)=1, a(2)=3, a(3)=16.
a(n) = (2/7)*(T(n, 5/2) - (-1)^n) with twice Chebyshev's polynomials of the first kind evaluated at x=5/2: 2*T(n, 5/2) = A003501(n) = ((5+sqrt(21))^n + (5-sqrt(21))^n)/2^n. - Wolfdieter Lang, Oct 18 2004
a(2*n) = A003690(n). a(2*n+1) = A004253(n)^2. - Alexander Evnin, Mar 11 2012
From Peter Bala, Apr 03 2014: (Start)
a(n) = |U(n-1, sqrt(3)*i/2)|^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

Edited by Peter J. C. Moses, Apr 23 2004

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

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

A049683 a(n) = (Lucas(6*n) - 2)/16.

Original entry on oeis.org

0, 1, 20, 361, 6480, 116281, 2086580, 37442161, 671872320, 12056259601, 216340800500, 3882078149401, 69661065888720, 1250017107847561, 22430646875367380, 402501626648765281, 7222598632802407680, 129604273763794572961, 2325654329115499905620

Offset: 0

Clark Kimberling

This is the r = 20 member of the r-family of sequences S_r(n), n >= 1, defined in A092184 where more information can be found.

Colin Barker, Table of n, a(n) for n = 0..750
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A000032, A004146, A023039, A049660, A049682, A049684, A092184.

List([0..30], n-> (Lucas(1,-1,6*n)[2] - 2)/16 ); # G. C. Greubel, Dec 14 2019

[(Lucas(6*n) -2)/16: n in [0..30]]; // G. C. Greubel, Dec 02 2017

with(combinat); seq( (5*fibonacci(3*n)^2 -2*(1-(-1)^n))/16, n=0..30); # G. C. Greubel, Dec 14 2019

LinearRecurrence[{19,-19,1}, {0,1,30}, 20] (* or *) Table[(LucasL[6*n] -2)/16, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-18*x+x^2)) + O(x^30))) \\ Colin Barker, Jun 03 2016

vector(31, n, (5*fibonacci(3*n-3)^2 -2*(1+(-1)^n))/16 ) \\ G. C. Greubel, Dec 14 2019

[(lucas_number2(6*n,1,-1) -2)/16 for n in (0..30)] # G. C. Greubel, Dec 14 2019

a(n) = (-2 + (9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/16. - Ralf Stephan, Apr 14 2004
a(n) = (T(n, 9) - 1)/8 with Chebyshev's polynomials of the first kind evaluated at x = 9: T(n, 9) = A023039(n). Wolfdieter Lang, Oct 18 2004
G.f.: x*(1 + x)/((1 - x)*(1 - 18*x + x^2)) = x*(1 + x)/(1 - 19*x + 19*x^2 - x^3). (from the Stephan link, see A092184).
exp( Sum_{n >= 1} 16*a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Fibonacci(6*n)*x^n. - Peter Bala, Jun 03 2016
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3) for n>2. - Colin Barker, Jun 03 2016

A078070 Expansion of (1-x)/(1 + 2x + 2x^2 + x^3).

Original entry on oeis.org

1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3

Offset: 0

N. J. A. Sloane, Nov 17 2002

Period 6: repeat [1, -3, 4, -3, 1, 0].
The unsigned sequence is the r=3 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found. |a(n)| = 2-2*T(n,1/2), with twice the Chebyshev polynomials of the first kind 2*T(n,x=1/2) = A057079(n+1) = S(n+1,1) + S(n,1) with S(n,1)= A010892(n).
Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = -3, P2 = 2, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

			G.f. = 1 - 3*x + 4*x^2 - 3*x^3 + x^4 + x^6 - 3*x^7 + 4*x^8 - 3*x^9 + x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A100047, A254745.

a[ n_] := {-3, 4, -3, 1, 0, 1}[[Mod[ n, 6, 1]]]; (* Michael Somos, Aug 05 2015 *)
CoefficientList[Series[(1-x)/(1+2x+2x^2+x^3),{x,0,120}],x] (* or *) PadRight[ {},120,{1,-3,4,-3,1,0}] (* or *) LinearRecurrence[{-2,-2,-1},{1,-3,4},120] (* Harvey P. Dale, Jan 06 2016 *)

Vec((1-x)/(1+2*x+2*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = [1, -3, 4, -3, 1, 0][n%6 + 1]}; /* Michael Somos, Aug 05 2015 */

abs(a(n)) = 2 + 2*cos(Pi*n/3 - 2*Pi/3). - Paul Barry, Mar 14 2004
Euler transform of finite sequence [-3, 1, 1]. - Michael Somos, Sep 17 2004
a(n) = (n+1)*(Sum_{k=0..floor((n+1)/2)} (-1)^k*binomial(n-k+1, k)*(-1)^(n-2k+1)/(n-k+1)) + 2*(-1)^n; a(n) = 2*T(n+1, -1/2) + 2(-1)^n. - Paul Barry, Dec 12 2004
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
Let {u_j(n)}, j = 0 or j = 1, be two Lucas sequences in the quadratic integer ring Z[w], where w = exp(2*Pi*i/3), defined by the recurrences u_j(0) = 0, u_j(1) = 1 and u_j(n) = (-1)^j*sqrt(3)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u_0(n)*u_1(n).
Equivalently, a(n) = U(n-1,sqrt(3)/2)*U(n-1,-sqrt(3)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = - ( ((sqrt(3) + i)/2)^n - ((sqrt(3) - i)/2)^n )*( ((-sqrt(3) + i)/2)^n - ((-sqrt(3) - i)/2)^n ) = w^n + w^(2*n) - 2*(-1)^n = 2*cos(2*n*Pi/3) - 2*(-1)^n.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -1/2; 1, -3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = a(n+6) = a(-2-n) for all n in Z. - Michael Somos, Aug 05 2015
a(n) = (-1)^n * A254745(n). - Michael Somos, Jul 16 2017

Chebyshev comment and related formulas from Wolfdieter Lang, Sep 10 2004

cp's OEIS Frontend

A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

PARI

PARI

Sage

Formula

A079291 Squares of Pell numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A007877 Period 4 zigzag sequence: repeat [0,1,2,1].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A049684 a(n) = Fibonacci(2n)^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

MuPAD

PARI

Sage

Formula

Extensions

A092936 Area of n-th triple of hexagons around a triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Formula

A054493 A Pellian-related recursive sequence.

Views

Author

A078070 Expansion of (1-x)/(1 + 2x + 2x^2 + x^3).