A128052 -id:A128052 - cp's OEIS frontend

A023039 a(n) = 18*a(n-1) - a(n-2).

1, 9, 161, 2889, 51841, 930249, 16692641, 299537289, 5374978561, 96450076809, 1730726404001, 31056625195209, 557288527109761, 10000136862780489, 179445175002939041, 3220013013190122249, 57780789062419261441

Offset: 0

David W. Wilson

The primitive Heronian triangle 3*a(n) +- 2, 4*a(n) has the latter side cut into 2*a(n) +- 3 by the corresponding altitude and has area 10*a(n)*A060645(n). - Lekraj Beedassy, Jun 25 2002
Chebyshev polynomials T(n,x) evaluated at x=9.
{a(n)} gives all (unsigned, integer) solutions of Pell equation a(n)^2 - 80*b(n)^2 = +1 with b(n) = A049660(n), n >= 0.
{a(n)} gives all possible solutions for x in Pell equation x^2 - D*y^2 = 1 for D=5, D=20 and D=80. The corresponding values for y are A060645 (D=5), A207832 (D=20) and A049660 (D=80). - Herbert Kociemba, Jun 05 2022
Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 14 2004
Appears to give all solutions > 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 24 2004
For all terms x of the sequence, 5*x^2 - 5 is a square, A004292(n)^2.
The a(n) are the x-values in the nonnegative integer solutions of x^2 - 5y^2 = 1, see A060645(n) for the corresponding y-values. - Sture Sjöstedt, Nov 29 2011
Rightmost digits alternate repeatedly: 1 and 9 in fact, a(2) = 18*9 - 1 == 1 (mod 10); a(3) = 18*1 - 9 == 9 (mod 10) therefore a(2n) == 1 (mod 10), a(2n+1) == 9 (mod 10). - Carmine Suriano, Oct 03 2013

			G.f. = 1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x4 + 930249*x^5 + 16692641*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..750 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A001077, A115032.
Row 2 of array A188645.
Row 4 of A322790.

I:=[1, 9]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 13 2012

a := n -> hypergeom([n, -n], [1/2], -4):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Jul 26 2020

LinearRecurrence[{18, -1}, {1, 9}, 50] (* Sture Sjöstedt, Nov 29 2011 *)
CoefficientList[Series[(1-9*x)/(1-18*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

{a(n) = fibonacci(6*n) / 2 + fibonacci(6*n - 1)}; /* Michael Somos, Aug 11 2009 */

x='x+O('x^30); Vec((1-9*x)/(1-18*x+x^2)) \\ G. C. Greubel, Dec 19 2017

a(n) ~ (1/2)*(sqrt(5) + 2)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
Limit_{n->infinity} a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 9) = (S(n, 18) - S(n-2, 18))/2, with S(n, x) := U(n, x/2) and T(n, x), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 18)=A049660(n+1).
a(n) = sqrt(80*A049660(n)^2 + 1) (cf. Richardson comment).
a(n) = ((9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/2.
G.f.: (1 - 9*x)/(1 - 18*x + x^2).
a(n) = cosh(2*n*arcsinh(2)). - Herbert Kociemba, Apr 24 2008
a(n) = A001077(2*n). - Michael Somos, Aug 11 2009
From Johannes W. Meijer, Jul 01 2010: (Start)
a(n) = 2*A167808(6*n+1) - A167808(6*n+3).
Limit_{k->infinity} a(n+k)/a(k) = a(n) + A060645(n)*sqrt(5).
Limit_{n->infinity} a(n)/A060645(n) = sqrt(5).
(End)
a(n) = (1/2)*A087215(n) = (1/2)*(sqrt(5) + 2)^(2*n) + (1/2)*(sqrt(5) - 2)^(2*n).
Sum_{n >= 1} 1/( a(n) - 5/a(n) ) = 1/8. Compare with A005248, A002878 and A075796. - Peter Bala, Nov 29 2013
a(n) = 2*A115032(n-1) - 1 = S(n, 18) - 9*S(n-1, 18), with A115032(-1) = 1, and see the above formula with S(n, 18) using its recurrence. - Wolfdieter Lang, Aug 22 2014
a(n) = A128052(3n). - A.H.M. Smeets, Oct 02 2017
a(n) = A049660(n+1) - 9*A049660(n). - R. J. Mathar, May 24 2018
a(n) = hypergeom([n, -n], [1/2], -4). - Peter Luschny, Jul 26 2020
a(n) = L(6*n)/2 for L(n) the Lucas sequence A000032(n). - Greg Dresden, Dec 07 2021
a(n) = cosh(6*n*arccsch(2)). - Peter Luschny, May 25 2022

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002

Sture Sjöstedt's comment corrected and reformulated by Wolfdieter Lang, Aug 24 2014

A178381 Number of paths of length n starting at initial node of the path graph P_9.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 125, 250, 450, 900, 1625, 3250, 5875, 11750, 21250, 42500, 76875, 153750, 278125, 556250, 1006250, 2012500, 3640625, 7281250, 13171875, 26343750, 47656250, 95312500, 172421875, 344843750

Offset: 0

Johannes W. Meijer, May 27 2010, May 29 2010

Counts all paths of length n, n>=0, starting at initial node on the path graph P_9, see the Maple program.
The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and g4; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3 and g5.
The path graphs P_(2*p) have as limit(a(n+1)/a(n), n =infinity) = 2 resp. hypergeom([(p-1)/(2*p+1),(p+2)/(2*p+1)],[1/2],3/4) and the path graphs P_(2*p+1) have as limit(a(n+1)/a(n), n =infinity) = 1+cos(Pi/(p+1)), p>=1; see the crossrefs. - Johannes W. Meijer, Jul 01 2010

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 70*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Trigonometric Identities.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5).

This is row 9 of A094718.
a(2*n) = A147748(n) and a(2*n+1) = A081567(n).
a(4*n+2) = A109106(n) and a(4*n+3) = A179135(n).
Cf. A000007 (P_1), A000012 (P_2), A016116 (P_3), A000045 (P_4), A038754 (P_5), A028495 (P_6), A030436 (P_7), A061551 (P_8), this sequence (P_9), A336675 (P_10), A336678 (P_11), and A001405 (P_infinity).
Cf. A216212 (P_9 starting in the middle).
Cf. A033191, A179131, A179132, A128052, A179133.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4))); // G. C. Greubel, Sep 18 2018

with(GraphTheory): P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=36; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P); od: seq(a(n),n=0..nmax);
r := j -> (-1)^(j/10) - (-1)^(1-j/10):
a := k -> add((2 + r(j))*r(j)^k, j in [1, 3, 5, 7, 9])/10:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 18 2020

CoefficientList[Series[(1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4), {x,0,50}], x] (* G. C. Greubel, Sep 18 2018 *)

x='x+O('x^50); Vec((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4)) \\ G. C. Greubel, Sep 18 2018

G.f.: (1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=5 with a(0)=1, a(1)=1, a(2)=2, a(3)=3 and a(4)=6.
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x / (1 - x / (1 + x / (1 + x)))))))). - Michael Somos, Feb 08 2015

A179133 Denominators of A178381(4n+3)/A178381(4n+2).

Original entry on oeis.org

2, 4, 5, 26, 68, 89, 466, 1220, 1597, 8362, 21892, 28657, 150050, 392836, 514229, 2692538, 7049156, 9227465, 48315634, 126491972, 165580141, 866988874, 2269806340, 2971215073, 15557484098, 40730022148, 53316291173, 279167724890

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the numerators see A128052.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A000045, A128052, A153727, A178381, A179131, A179132, A179133, A179134, A296182.

with(GraphTheory): nmax:=120; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+3)/A178381(4*n+2)) od: seq(a(n),n=0..nmax/4-1);

Flatten[Table[{2*Fibonacci[6 n + 1], 2*Fibonacci[6 n + 3],
Fibonacci[6 n + 5]}, {n, 0, 10}]] (* Greg Dresden, Oct 16 2021 *)
LinearRecurrence[{0,0,18,0,0,-1},{2,4,5,26,68,89},30] (* Harvey P. Dale, Oct 08 2024 *)

a(n) = A179134(n)*A153727(n+1)/2.
Lim_{n->infinity} A128052(n+1)/A179133(n) = 1+cos(Pi/5) = A296182.
From Colin Barker, Jun 27 2013: (Start)
G.f.: -(x^5+4*x^4+10*x^3-5*x^2-4*x-2)/((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)).
a(n) = 18*a(n-3)-a(n-6). (End)
From Greg Dresden, Oct 16 2021: (Start)
a(3*n) = 2*Fibonacci(6*n+1),
a(3*n+1) = 2*Fibonacci(6*n+3),
a(3*n+2) = Fibonacci(6*n+5). (End)

A179131 Numerators of A178381(4n+1)/A178381(4n).

Original entry on oeis.org

1, 5, 25, 65, 85, 445, 1165, 1525, 7985, 20905, 27365, 143285, 375125, 491045, 2571145, 6731345, 8811445, 46137325, 120789085, 158114965, 827900705, 2167472185, 2837257925, 14856075365, 38893710245, 50912527685, 266581455865

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the denominators see A179132.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052, A179133, A179132.

with(GraphTheory): nmax:=116; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= numer(A178381(4*n+1)/A178381(4*n)) od: seq(a(n),n=0..nmax/4-1);

a(n) = 5*A167808(2*n+1) for n>=1.
Limit(A179131(n)/A179132(n), n =infinity) = 1+cos(Pi/5) = A296182.
a(n) = 18*a(n-3)-a(n-6) for n>6. G.f.: -(4*x^6+5*x^5+5*x^4-47*x^3-25*x^2-5*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

A179132 Denominators of A178381(4n+1)/A178381(4n).

Original entry on oeis.org

1, 3, 14, 36, 47, 246, 644, 843, 4414, 11556, 15127, 79206, 207364, 271443, 1421294, 3720996, 4870847, 25504086, 66770564, 87403803, 457652254, 1198149156, 1568397607, 8212236486, 21499914244, 28143753123, 147362604494

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the numerators see A179131.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052 and A179133.

with(GraphTheory): nmax:=116; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+1)/A178381(4*n)) od: seq(a(n),n=0..nmax/4-1);

LinearRecurrence[{0,0,18,0,0,-1},{1,3,14,36,47,246,644},30] (* Harvey P. Dale, Jun 11 2022 *)

a(n) = A069705(n-1)*A128052(n) for n>=1.
Limit(A179131(n)/A179132(n), n =infinity) = 1+cos(Pi/5) = A296182.
a(n) = 18*a(n-3)-a(n-6) for n>6. G.f.: -(3*x^6+6*x^5+7*x^4-18*x^3-14*x^2-3*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

A179135 a(n) = (3-sqrt(5))((3+sqrt(5))/10)^(-n)/2+(3+sqrt(5))((3-sqrt(5))/10)^(-n)/2.

Original entry on oeis.org

3, 35, 450, 5875, 76875, 1006250, 13171875, 172421875, 2257031250, 29544921875, 386748046875, 5062597656250, 66270263671875, 867489013671875, 11355578613281250, 148646453857421875, 1945807342529296875

Offset: 0

Johannes W. Meijer, Jul 01 2010

Index entries for linear recurrences with constant coefficients, signature (15, -25).

Cf. A109106.

with(GraphTheory): nmax:=72; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax/4-1 do a(n):= A178381(4*n+3) od: seq(a(n),n=0..nmax/4-1);

a(n) = A178381(4*n+3).
G.f.: (3-10*z)/(1-15*z+25*z^2).
Limit(a(n+k)/a(k), k=infinity) = A000351(n)*A130196(n)/(A128052(n) - A167808(2*n)*sqrt(5)).
Limit(A128052(n)/A167808(2*n),n=infinity) = sqrt(5).
a(n) = 5^n*Lucas(2*(n+1)). - Ehren Metcalfe, Apr 22 2018

A128053 a(n)=A128056(n)/A128055(n).

Original entry on oeis.org

1, -3, -3, 7, 7, -9, -9, 47, 47, -123, -123, 161, 161, -843, -843, 2207, 2207, -2889, -2889, 15127, 15127, -39603, -39603, 51841, 51841, -271443, -271443, 710647, 710647, -930249, -930249, 4870847, 4870847, -12752043, -12752043

Offset: 0

Paul Barry, Feb 13 2007

Cf. A128052.

a(n)=(-1)^C(n+1,2)*(F(n-1)+F(n+1))(5/6-cos(pi*n/3)/3)(1+(-1)^n)/2+(F(n)+F(n+2))(5/6-cos(pi*(n+1)/3)/3)(1-(-1)^n)/2.

Empirical g.f.: -(x-1) * (x^10-2*x^9-5*x^8+2*x^7+9*x^6+9*x^4+2*x^3-5*x^2-2*x+1) / ((x^4+3*x^2+1)*(x^8-3*x^6+8*x^4-3*x^2+1)). - Colin Barker, Jun 27 2013

A179134 a(n) = (F(2n-1) + F(2n+2)) * (5/6 - cos(2Pin/3)/3), where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 4, 10, 13, 68, 178, 233, 1220, 3194, 4181, 21892, 57314, 75025, 392836, 1028458, 1346269, 7049156, 18454930, 24157817, 126491972, 331160282, 433494437, 2269806340, 5942430146, 7778742049, 40730022148, 106632582346

Offset: 0

Johannes W. Meijer, Jul 01 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052, A000045 (Fibonacci numbers).

Appears in A179133.

with(combinat): nmax:=28; for n from 0 to nmax do a(n):=(fibonacci(2*n-1)+fibonacci(2*n+2))*(5/6-cos(2*Pi*n/3)/3) od: seq(a(n),n=0..nmax);

a(n) = 18*a(n-3)-a(n-6). G.f.: -(2*x^5+4*x^4+5*x^3-10*x^2-4*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

cp's OEIS Frontend

A023039 a(n) = 18*a(n-1) - a(n-2).

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A178381 Number of paths of length n starting at initial node of the path graph P_9.

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A179133 Denominators of A178381(4*n+3)/A178381(4*n+2).

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Maple

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A179131 Numerators of A178381(4*n+1)/A178381(4*n).

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A179132 Denominators of A178381(4*n+1)/A178381(4*n).

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A179135 a(n) = (3-sqrt(5))*((3+sqrt(5))/10)^(-n)/2+(3+sqrt(5))*((3-sqrt(5))/10)^(-n)/2.

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A128053 a(n)=A128056(n)/A128055(n).

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A179134 a(n) = (F(2*n-1) + F(2*n+2)) * (5/6 - cos(2*Pi*n/3)/3), where F(n) = Fibonacci(n).

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A179133 Denominators of A178381(4n+3)/A178381(4n+2).

A179131 Numerators of A178381(4n+1)/A178381(4n).

A179132 Denominators of A178381(4n+1)/A178381(4n).

A179135 a(n) = (3-sqrt(5))((3+sqrt(5))/10)^(-n)/2+(3+sqrt(5))((3-sqrt(5))/10)^(-n)/2.

A179134 a(n) = (F(2n-1) + F(2n+2)) * (5/6 - cos(2Pin/3)/3), where F(n) = Fibonacci(n).