A159668 -id:A159668 - cp's OEIS frontend

A238379 Expansion of (1 - x)/(1 - 36*x + x^2).

1, 35, 1259, 45289, 1629145, 58603931, 2108112371, 75833441425, 2727895778929, 98128414600019, 3529895029821755, 126978092658983161, 4567681440693572041, 164309553772309610315, 5910576254362452399299, 212616435603275976764449

Offset: 0

Bruno Berselli, Feb 25 2014

First bisection of A041611.

Bruno Berselli, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (36,-1).

Cf. similar sequences with g.f. (1-x)/(1-k*x+x^2): A122367 (k=3), A079935 (k=4), A004253 (k=5), A001653 (k=6), A049685 (k=7), A070997 (k=8), A070998 (k=9), A138288 (k=10), A078922 (k=11), A077417 (k=12), A085260 (k=13), A001570 (k=14), A160682 (k=15), A157456 (k=16), A161595 (k=17). From 18 to 38, even k only, except k=27 and k=31: A007805 (k=18), A075839 (k=20), A157014 (k=22), A159664 (k=24), A153111 (k=26), A097835 (k=27), A159668 (k=28), A157877 (k=30), A111216 (k=31), A159674 (k=32), A077420 (k=34), this sequence (k=36), A097315 (k=38).

[n le 2 select 35^(n-1) else 36*Self(n-1)-Self(n-2): n in [1..20]];

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1 - x)/(1 - 36*x + x^2))); // Marius A. Burtea, Jan 14 2020

CoefficientList[Series[(1 - x)/(1 - 36 x + x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{36, -1}, {1, 35}, 20]

a(n)=([0,1; -1,36]^n*[1;35])[1,1] \\ Charles R Greathouse IV, May 10 2016

m = 20; L. = PowerSeriesRing(ZZ, m); f = (1-x)/(1-36*x+x^2)
print(f.coefficients())

G.f.: (1 - x)/(1 - 36*x + x^2).
a(n) = a(-n-1) = 36*a(n-1) - a(n-2).
a(n) = ((19-sqrt(323))/38)*(1+(18+sqrt(323))^(2*n+1))/(18+sqrt(323))^n.
a(n+1) - a(n) = 34*A144128(n+1).
323*a(n+1)^2 - ((a(n+2)-a(n))/2)^2 = 34.
Sum_{n>0} 1/(a(n) - 1/a(n)) = 1/34.
See also Tanya Khovanova in Links field:
a(n) = 35*a(n-1) + 34*Sum_{i=0..n-2} a(i).
a(n+2)*a(n) - a(n+1)^2 = 36-2 = 34 = 34*1,
a(n+3)*a(n) - a(n+1)*a(n+2) = 36*(36-2) = 1224 = 34*36.
Generalizing:
a(n+4)*a(n) - a(n+1)*a(n+3) = 44030 = 34*1295,
a(n+5)*a(n) - a(n+1)*a(n+4) = 1583856 = 34*46584,
a(n+6)*a(n) - a(n+1)*a(n+5) = 56974786 = 34*1675729, etc.,
where 1, 36, 1295, 46584, 1675729, ... is the sequence A144128, which is the second bisection of A041611.
a(n)^2 - 36*a(n)*a(n+1) + a(n+1)^2 + 34 = 0 (see comments by Colin Barker in similar sequences).

A159669 Expansion of x(1 + x)/(1 - 28x + x^2).

Original entry on oeis.org

1, 29, 811, 22679, 634201, 17734949, 495944371, 13868707439, 387827863921, 10845311482349, 303280893641851, 8481019710489479, 237165271000063561, 6632146568291290229, 185462938641156062851, 5186330135384078469599, 145031780852113041085921

Offset: 1

Paul Weisenhorn, Apr 19 2009

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (28,-1).

Cf. A097311, A157456, A159668, A159673.

A097311:= func< n | Evaluate(ChebyshevSecond(n-1), 14) >;
[A097311(n+1) + A097311(n): n in [1..30]]; // G. C. Greubel, Sep 25 2022

for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq(simplify(ChebyshevU(n, 14) +ChebyshevU(n-1, 14)), n=1..30); # G. C. Greubel, Sep 25 2022

CoefficientList[Series[(1+x)/(1-28x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{28,-1},{1,29},20] (* Harvey P. Dale, Jul 01 2019 *)

Vec(x*(x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014

a(n) = round((14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26) \\ Colin Barker, Jul 25 2016

def A159669(n): return chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14)
[A159669(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).
The b(j) recurrence is b(1)=1, b(2)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (this sequence).
The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (A159673).
G.f.: x*(1+x)/(1-28*x+x^2). - Vincenzo Librandi, Feb 26 2014
a(n) = (14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26. - Colin Barker, Jul 25 2016
a(n) = chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14) = A097311(n) + A097311(n-1). - G. C. Greubel, Sep 25 2022

More terms and new name from Colin Barker, Feb 24 2014

A159673 Expansion of 56x^2/(1 - 783x + 783*x^2 - x^3).

Original entry on oeis.org

0, 56, 43848, 34289136, 26814060560, 20968561068840, 16397387941772376, 12822736401904929248, 10027363468901712899616, 7841385409944737582570520, 6131953363213315887857247080, 4795179688647403079566784646096, 3749824384568905994905337736000048

Offset: 1

Paul Weisenhorn, Apr 19 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (783,-783,1).

Cf. A157456, A159668, A159669.

b:= func< n | Evaluate(ChebyshevSecond(n),391) >;
[(14/195)*(-1 +b(n+1) -781*b(n)): n in [1..30]]; // G. C. Greubel, Sep 25 2022

for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq((14/195)*(simplify(ChebyshevU(n, 391) -781*ChebyshevU(n-1, 391)) -1), n=1..30); # G. C. Greubel, Sep 25 2022

CoefficientList[Series[56 x/(- x^3 + 783 x^2 - 783 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{783,-783,1},{0,56,43848},20] (* Harvey P. Dale, Jan 06 2019 *)

Vec(56*x^2/(-x^3+783*x^2-783*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014

a(n) = round(-((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390) \\ Colin Barker, Jul 25 2016

def A159673(n): return (14/195)*(-1 + chebyshev_U(n, 391) - 781*chebyshev_U(n-1, 391))
[A159673(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).
The b(j) recurrence is b(1)=1, b(2)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (this sequence).
G.f.: 56*x^2/((1-x)*(1 - 782*x + x^2)). - Vincenzo Librandi, Feb 26 2014
a(n) = -((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390. - Colin Barker, Jul 25 2016
a(n) = (14/195)*(-1 + ChebyshevU(n, 391) - 781*ChebyshevU(n-1, 391)). - G. C. Greubel, Sep 25 2022

More terms and new name from Colin Barker, Feb 24 2014

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Index entries for linear recurrences with constant coefficients, signature (40,-1).

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

[n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016

Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...

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A238379 Expansion of (1 - x)/(1 - 36*x + x^2).

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A159669 Expansion of x*(1 + x)/(1 - 28*x + x^2).

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A159673 Expansion of 56*x^2/(1 - 783*x + 783*x^2 - x^3).

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A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

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A159669 Expansion of x(1 + x)/(1 - 28x + x^2).

A159673 Expansion of 56x^2/(1 - 783x + 783*x^2 - x^3).