A100047 -id:A100047 - cp's OEIS frontend

A100051 A Chebyshev transform of 1,1,1,...

1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1

Offset: 0

Paul Barry, Oct 31 2004

1, followed by period 6: repeat [1, -1, -2, -1, 1, 2]. - Joerg Arndt, Aug 28 2024
A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - Paul Barry, Dec 01 2004
Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - David W. Wilson, Jun 10 2005

			G.f. = 1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A011655, A057079, A087204, A099837, A099443, A100047, A100048, A100050.

Row sums of array A127677.

CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x,0,50}], x] (* G. C. Greubel, May 03 2017 *)
LinearRecurrence[{1,-1},{1,1,-1},80] (* Harvey P. Dale, Mar 25 2019 *)

{a(n) = - (n == 0) + [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* Michael Somos, Mar 21 2011 */

From Paul Barry, Dec 01 2004: (Start)
G.f.: (1-x^2)/(1-x+x^2).
a(n) = a(n-1) - a(n-2), n>2.
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)/(n-k).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(2n/(n+k))*(-1)^k, n>1. (End)
Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6].
Euler transform of length 6 sequence [1, -2, -1, 0, 0, 1].
a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n) = A087204(n), n>0. - R. J. Mathar, Sep 02 2008
a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - R. J. Mathar, Apr 11 2011

A152090 a(n) = 2^nProduct_{k=1..floor((n-1)/2)} (1 + 2cos(kPi/n)^2 + 4cos(k*Pi/n)^4).

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 39, 91, 217, 513, 1216, 2881, 6825, 16171, 38311, 90768, 215047, 509491, 1207089, 2859841, 6775552, 16052673, 38032081, 90105811, 213479175, 505776016, 1198287271, 2838988683, 6726147337, 15935624641, 37754768064

Offset: 0

Roger L. Bagula and Gary W. Adamson, Nov 23 2008

Limiting ratio after n=30 terms is 2.369205407038926.
With a(0)=0, this is a divisibility sequence with g.f. x(1-x^2)/(1 - x - 3x^2 - x^3 + x^4). The limiting ratio is the largest zero of 1 - x - 3x^2 - x^3 + x^4. - T. D. Noe, Dec 22 2008
The sequence is the case P1 = 1, P2 = -5, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

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 (1,3,1,-1).

Cf. A100047.

bb = Table[FullSimplify[ExpandAll[Product[1 + 4*Cos[k*Pi/n]^2 + 16*Cos[k*Pi/n]^4, {k, 1, (n - 1)/2}]]], {n, 0, 30}]
LinearRecurrence[{1, 3, 1, -1}, {1, 1, 1, 3, 7}, 50] (* G. C. Greubel, Aug 08 2017 *)

Vec((x^4-2*x^3-3*x^2+1)/(x^4-x^3-3*x^2-x+1) + O(x^100)) \\ Colin Barker, Jan 05 2014

From Colin Barker, Jan 05 2014: (Start)
a(n) = a(n-1) +3*a(n-2) +a(n-3) -a(n-4) for n>4.
G.f.: (x^4-2*x^3-3*x^2+1) / (x^4-x^3-3*x^2-x+1). (End)
From Peter Bala, Mar 25 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(1 + sqrt(21)), beta = 1/4*(1 - sqrt(21)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = U(n-1,1/4*(1 + sqrt(-3)))*U(n-1,1/4*(1 - sqrt(-3))) for n >= 1, 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, 5/4; 1, 1/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

A075796 Numbers k such that 5*k^2 + 5 is a square.

Original entry on oeis.org

2, 38, 682, 12238, 219602, 3940598, 70711162, 1268860318, 22768774562, 408569081798, 7331474697802, 131557975478638, 2360712083917682, 42361259535039638, 760141959546795802, 13640194012307284798, 244763350261984330562, 4392100110703410665318, 78813038642399407645162

Offset: 1

Gregory V. Richardson, Oct 13 2002

Bisection of A001077; a(n) = A001077(2*n-1). - Greg Dresden, Jun 08 2021
From Peter Bala, Aug 25 2022: (Start)
The aerated sequence (b(n))n>=1 = [2, 0, 38, 0, 682, 0, 1238, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). The sequence (1/2)*(b(n))n>=1 is the case P1 = 0, P2 = -16, Q = -1  of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. (End)

A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A000290, A306380, A001077.

I:=[2,38]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 30 2011

[Lucas(6*n-3)/2: n in [1..20]]; // G. C. Greubel, Feb 13 2019

with(combinat); A075796:=n->fibonacci(6*n+3)+fibonacci(6*n)/2; seq(A075796(n), n=1..50); # Wesley Ivan Hurt, Nov 29 2013

LinearRecurrence[{18, -1}, {2, 38}, 50] (* Sture Sjöstedt, Nov 29 2011; typo fixed by Vincenzo Librandi, Nov 30 2011 *)
LucasL[6*Range[20]-3]/2 (* G. C. Greubel, Feb 13 2019 *)
CoefficientList[Series[2*(1+x)/( 1-18*x+x^2 ), {x,0,20}],x] (* Stefano Spezia, Mar 02 2019 *)

vector(20, n, (fibonacci(6*n-2) + fibonacci(6*n-4))/2) \\ G. C. Greubel, Feb 13 2019

[(fibonacci(6*n-2) + fibonacci(6*n-4))/2 for n in (1..20)] # G. C. Greubel, Feb 13 2019

a(n) = (((9 + 4*sqrt(5))^n - (9 - 4*sqrt(5))^n) + ((9 + 4*sqrt(5))^(n-1) - (9 - 4*sqrt(5))^(n-1)))/(4*sqrt(5)).
a(n) = 18*a(n-1) - a(n-2).
a(n) = 2*A049629(n-1).
Limit_{n->oo} a(n)/a(n-1) = 8*phi + 1 = 9 + 4*sqrt(5).
a(n+1) = 9*a(n) + 4*sqrt(5)*sqrt((a(n)^2+1)). - Richard Choulet, Aug 30 2007
G.f.: 2*x*(1 + x)/(1 - 18*x + x^2). - Richard Choulet, Oct 09 2007
From Johannes W. Meijer, Jul 01 2010: (Start)
a(n) = A000045(6*n+3) + A000045(6*n)/2.
a(n) = 2*A167808(6*n+4) - A167808(6*n+6).
Limit_{k->oo} a(n+k)/a(k) = A023039(n)*A060645(n)*sqrt(5).
(End)
5*A007805(n)^2 - 1 = a(n+1)^2. - Sture Sjöstedt, Nov 29 2011
From Peter Bala, Nov 29 2013: (Start)
a(n) = Lucas(6*n - 3)/2.
Sum_{n >= 1} 1/(a(n) + 5/a(n)) = 1/4. Compare with A002878, A005248, A023039. (End)
Limit_{n->oo} a(n)/A007805(n-1) = sqrt(5). - A.H.M. Smeets, May 29 2017
E.g.f.: (exp((9 - 4*sqrt(5))*x)*(- 5 + 2*sqrt(5) + (5 + 2*sqrt(5))*exp(8*sqrt(5)*x)))/(2*sqrt(5)). - Stefano Spezia, Feb 13 2019
Sum_{n > 0} 1/a(n) = (1/log(9 - 4*sqrt(5)))*(- 17 - 38/sqrt(5))*sqrt(5*(9 - 4*sqrt(5)))*(- 9 + 4*sqrt(5))*(psi_{9 - 4*sqrt(5)}(1/2) - psi_{9 - 4*sqrt(5)}(1/2 - (I*Pi)/log(9 - 4*sqrt(5)))) approximately equal to 0.527868600269500798938265500122302016..., where psi_q(x) is the q-digamma function. - Stefano Spezia, Feb 25 2019
a(n) = sinh((6*n - 3)*arccsch(2)). - Peter Luschny, May 25 2022

A097783 Chebyshev polynomials S(n,11) + S(n-1,11) with Diophantine property.

Original entry on oeis.org

1, 12, 131, 1429, 15588, 170039, 1854841, 20233212, 220710491, 2407582189, 26262693588, 286482047279, 3125039826481, 34088956044012, 371853476657651, 4056299287190149, 44247438682433988, 482665526219583719, 5265073349732986921, 57433141320843272412

Offset: 0

Wolfdieter Lang, Aug 31 2004

All positive integer solutions of Pell equation (3*a(n))^2 - 13*b(n)^2 = -4 together with b(n)=A078922(n+1), n>=0.

			All positive solutions to the Pell equation x^2 - 13*y^2 = -4 are (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...

Colin Barker, Table of n, a(n) for n = 0..963
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial
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 (11,-1).

Cf. S(n, 11) = A004190(n).

Cf. A000045, A002315, A004146, A006190, A006497, A100047, A113224, A192425.

I:=[1,12]; [n le 2 select I[n] else 11*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

CoefficientList[Series[(1 + x) / (1 - 11 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 22 2015 *)

Vec((1+x)/(1-11*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 22 2015

[(lucas_number2(n,11,1)-lucas_number2(n-1,11,1))/9 for n in range(1, 19)] # Zerinvary Lajos, Nov 10 2009

a(n) = S(n, 11) + S(n-1, 11) = S(2*n, sqrt(13)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x) = 0 = U(-1, x).
a(n) = (-2/3)*i*((-1)^n)*T(2*n+1, 3*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-11*x+x^2).
a(n) = L(n,-11)*(-1)^n, where L is defined as in A108299; see also A078922 for L(n,+11). - Reinhard Zumkeller, Jun 01 2005
a(n) = 11*a(n-1) - a(n-2) with a(0)=1 and a(1)=12. - Philippe Deléham, Nov 17 2008
From Peter Bala, Mar 22 2015: (Start)
The aerated sequence (b(n))n>=1 = [1, 0, 12, 0, 131, 0, 1429, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -9, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials.
b(n) = 1/2*( (-1)^n - 1 )*F(n,3) + 1/3*( 1 + (-1)^(n+1) )*F(n+1,3), where F(n,x) is the n-th Fibonacci polynomial. The o.g.f. is x*(1 + x^2)/(1 - 11*x^2 + x^4).
Exp( Sum_{n >= 1} 6*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 6*A006190(n)*x^n.
Exp( Sum_{n >= 1} (-6)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 6*A006190(n)*(-x)^n. Cf. A002315, A004146, A113224 and A192425. (End)
a(n) = A006497(2n+1)/3. - Adam Mohamed, Aug 22 2024

A077416 Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 13, 155, 1847, 22009, 262261, 3125123, 37239215, 443745457, 5287706269, 63008729771, 750817050983, 8946795882025, 106610733533317, 1270382006517779, 15137973344680031, 180385298129642593

Offset: 0

Wolfdieter Lang, Nov 29 2002

7*b(n)^2 - 5*a(n)^2 = 2 with companion sequence b(n) = A077417(n), n>=0.
a(n) = L(n,-12)*(-1)^n, where L is defined as in A108299; see also A077417 for L(n,+12). - Reinhard Zumkeller, Jun 01 2005
The aerated sequence (b(n))n>=1 = [1, 0, 13, 0, 155, 0, 1857, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -10, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, May 12 2025

Ivan Panchenko, Table of n, a(n) for n = 0..200
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
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 (12,-1).

Cf. A054320(n-1) with companion A072256(n), n>=1.

I:=[1, 13]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

LinearRecurrence[{12,-1},{1,13},30] (* Harvey P. Dale, Apr 03 2013 *)

x='x+O('x^30); Vec((1+x)/(1-12*x+x^2)) \\ G. C. Greubel, Jan 18 2018

[(lucas_number2(n,12,1)-lucas_number2(n-1,12,1))/10 for n in range(1, 18)] # Zerinvary Lajos, Nov 10 2009

a(n) = 12*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
a(n) = S(n, 12) + S(n-1, 12) = S(2*n, sqrt(14)) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 12) = A004191(n).
G.f.: (1+x)/(1-12*x+x^2).
a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).
a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n-k,k) * 14^(n-k).
a(n) = sqrt((7*A077417(n)^2 - 2)/5).
From Peter Bala, May 09 2025: (Start)
a(n) = Dir(n, 6), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.
a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 = 14.
More generally, for real x, a(n+x)^2 - 12*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 14, where a(n) := (ap^(2*n+1) - am^(2*n+1))/(ap - am), ap := sqrt(7/2) + sqrt(5/2) and am := sqrt(7/2) - sqrt(5/2), as given above.
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/14 (telescoping series).
Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(7/5) (telescoping product). (End)

A006238 Complexity of (or spanning trees in) a 3 X n grid.

Original entry on oeis.org

1, 15, 192, 2415, 30305, 380160, 4768673, 59817135, 750331584, 9411975375, 118061508289, 1480934568960, 18576479568193, 233018797965135, 2922930580320960, 36664523428884015, 459910778352898337, 5769007865476035840, 72365017995700730081, 907729015392142395375

Offset: 1

Frans J. Faase

nonn

a(n) is a divisibility sequence - m divides n implies that a(m) divides a(n). - Paul Raff, Mar 06 2009
Also number of domino tilings of the 5 X (2n-1) rectangle with upper left corner removed.  For n=2 the 15 domino tilings of the 5 X 3 rectangle with upper left corner removed are:
. ._. . ._. . ._. . ._. . ._. . ._. . ._. . ._.
.|__| .| | | .|___| .|__| .|__| .| | | .| | | .|__|
| |_| | ||| | | | | | |_| | |_| | ||| | ||| | | | |
||__| ||__| |||_| || | | ||___| || | | ||___| |||_|
| |_| | |_| | |_| | ||| | | | | | ||| | | | | | | | |
||__| ||__| ||__| ||__| |||_| ||__| |||_| |||_|
. ._. . ._. . ._. . ._. . ._. . ._. . ._.
.|__| .|__| .|__| .|__| .|__| .|__| ._| | |
|_| | | | | | |_| | |_| | |_| | | |_| | |||
|_|_| |||_| | | || |__|_| |_|_| ||__| ||__|
|_| | |_| | ||| | | | | | | |_| |_| | |_| |
|_|_| |_|_| |_|_| |||_| ||__| |_|_| |_|_|
- Alois P. Heinz, Apr 14 2011

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..200
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
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.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of P_3 x P_n. Contains sequence, recurrence, generating function, and more.
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index to divisibility sequences
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (15,-32,15,-1).

Row 3 of A116469. A100047.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|15|-32|15>>^n. <<1, 0, 1, 15>>)[2, 1]: seq(a(n), n=1..30);  # Alois P. Heinz, Apr 14 2011

LinearRecurrence[{15,-32,15,-1},{1,15,192,2415},30] (* Harvey P. Dale, May 14 2013 *)

a(n) = 15a(n-1) - 32a(n-2) + 15a(n-3) - a(n-4), n>4.
G.f.: -x(x^2-1)/(x^4-15x^3+32x^2-15x+1). - Paul Raff, Mar 06 2009
a(n) = A001906(n)*A004254(n). - R. J. Mathar, Jun 03 2009
From Peter Bala, Mar 25 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (15 + sqrt(105))/4 and beta = (15 - sqrt(105))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2X2 matrix T(n, M), where M is the 2X2 matrix [0, -15/2; 1, 15/2].
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) = (A003775(n+1)+A003775(n-2))/24-(A003775(n)+A003775(n-1))/3, n>1. - Sergey Perepechko, Apr 26 2016

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

A003690 Number of spanning trees in K_3 X P_n.

Original entry on oeis.org

3, 75, 1728, 39675, 910803, 20908800, 479991603, 11018898075, 252954664128, 5806938376875, 133306628004003, 3060245505715200, 70252340003445603, 1612743574573533675, 37022849875187828928, 849912803554746531675, 19510971631883982399603

Offset: 1

Frans J. Faase

Column 3 of A173958. The sequence a(n)/3 is linear divisibility sequence of the fourth order; it is the case P1 = 25, P2 = 46, Q = 1 of the three parameter family of divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008. [From _Paul Raff_, Mar 06 2009]
P. Raff, Analysis of the Number of Spanning Trees of K_3 x P_n. Contains sequence, recurrence, generating function, and more. [From _Paul Raff_, Mar 06 2009] [broken link]
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 trees
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A100047, A173958 (column 3).

I:=[3,75,1728]; [n le 3 select I[n] else 24*Self(n-1)-24*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 28 2014

CoefficientList[Series[3 (1 + x)/((1 - x) (1 - 23 x + x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 28 2014 *)

Vec(3*x*(1+x)/((1-x)*(1-23*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 06 2016

a(n) = (A090731(n)-2)/7.
a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3), n>3.
G.f.: 3*x*(1+x)/((1-x)*(1-23*x+x^2)). - R. J. Mathar, Dec 16 2008
a(n) = 3*(A004254(n))^2. - R. K. Guy, seqfan list, Mar 28 2009, - R. J. Mathar, Jun 03 2009
From Peter Bala, Apr 27 2014: (Start)
Product {n >= 2} (1 - 3/a(n)) = 1/2 + sqrt(21)/10.
a(n) = (2/7)*( T(n,23/2) - 1), where T(n,x) is the Chebyshev polynomial of the first kind.
a(n) = 3 * the bottom left entry of the 2 X 2 matrix T(n,M), where M is the 2 X 2 matrix [0, -23/2; 1, 25/2].
a(n) = 3*U(n-1,5/2)^2, where U(n,x) is the Chebyshev polynomial of the second 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+(2/(23+5*sqrt(21)))^n+(1/2*(23+5*sqrt(21)))^n)/7. - Colin Barker, Mar 06 2016

A003757 Number of perfect matchings (or domino tilings) in D_4 X P_(n-1).

Original entry on oeis.org

0, 1, 1, 6, 13, 49, 132, 433, 1261, 3942, 11809, 36289, 109824, 335425, 1018849, 3104934, 9443629, 28756657, 87504516, 266383153, 810723277, 2467770054, 7510988353, 22861948801, 69584925696, 211799836801, 644660351425

Offset: 0

Frans J. Faase

nonn

Here D_4 is the graph on 4 vertices with edges (1,2), (1,3), (2,3), (1.4): a triangular kite with a tail.
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 22 2008
This is the case P1 = 1, P2 = -8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Vincenzo Librandi, Table of n, a(n) for n = 0..160
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.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
F. J. Faase, Results from the counting program
Paul Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
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 to divisibility sequences
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (1,6,1,-1).

I:=[0,1,1,6]; [n le 4 select I[n] else Self(n-1)+6*Self(n-2)+Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 24 2011

CoefficientList[Series[x(1-x^2)/(1-x-6x^2-x^3+x^4), {x,0,30}], x] (* T. D. Noe, Dec 22 2008 *)
LinearRecurrence[{1,6,1,-1},{0,1,1,6},40] (* Harvey P. Dale, Sep 23 2011 *)

a(n) = a(n-1) + 6a(n-2) + a(n-3) - a(n-4), n>4.
G.f.: x(1-x^2)/(1-x-6x^2-x^3+x^4). [T. D. Noe, Dec 22 2008]
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(33))/4 and beta = (1 - sqrt(33))/4 and T(n,x) denotes the Chebyshev polynomial of the first 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, 2; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(3))/sqrt(8))*U(n-1,i*(1 - sqrt(3))/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

Offset and name changed by T. D. Noe, Dec 22 2008

0 and 1 prepended by T. D. Noe, Dec 22 2008

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

cp's OEIS Frontend

A100051 A Chebyshev transform of 1,1,1,...

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A152090 a(n) = 2^n*Product_{k=1..floor((n-1)/2)} (1 + 2*cos(k*Pi/n)^2 + 4*cos(k*Pi/n)^4).

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A075796 Numbers k such that 5*k^2 + 5 is a square.

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A097783 Chebyshev polynomials S(n,11) + S(n-1,11) with Diophantine property.

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A077416 Chebyshev S-sequence with Diophantine property.

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A006238 Complexity of (or spanning trees in) a 3 X n grid.

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A054493 A Pellian-related recursive sequence.

A152090 a(n) = 2^nProduct_{k=1..floor((n-1)/2)} (1 + 2cos(kPi/n)^2 + 4cos(k*Pi/n)^4).