A090390 -id:A090390 - cp's OEIS frontend

A098212 Expansion of (5-x^2)/((1+x)(1-6x+x^2)).

5, 25, 149, 865, 5045, 29401, 171365, 998785, 5821349, 33929305, 197754485, 1152597601, 6717831125, 39154389145, 228208503749, 1330096633345, 7752371296325, 45184131144601, 263352415571285, 1534930362283105, 8946229758127349, 52142448186480985

Offset: 0

Creighton Dement, Oct 25 2004

Old name was: Relates the squares of Pell numbers with the squares of the numerators of continued fraction convergents to sqrt(2).

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[(j' + k' + 'ii')*('j + 'k + 'ii')] - Creighton Dement, Aug 16 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A000129, A001333, A001541, A079291, A075848, A090390.

I:=[5,25,149]; [n le 3 select I[n] else 5*Self(n-1)+5*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 26 2015

a[0]= 5; a[1]= 25; a[2]= 149; a[n_]:= a[n]= 5 a[n-1] + 5 a[n-2] - a[n-3]; Table[ a[n], {n,0,40}] (* Robert G. Wilson v, Nov 05 2004 *)
CoefficientList[Series[(5-x^2)/((1+x)(1-6x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{5,5,-1},{5,25,149},40] (* Harvey P. Dale, Jun 09 2011 *)

Vec((5-x^2)/((1+x)*(1-6*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

def Pell(n): return lucas_number1(n,2,-1)
[4*Pell(n+1)^2 +(Pell(n+1) +Pell(n))^2  for n in (0..40)] # G. C. Greubel, Aug 20 2022

G.f.: (5-x^2)/((1+x)*(1-6*x+x^2)).
a(n) = 4*A079291(n+1) + A090390(n+1) = 4(A000129(n+1))^2 + (A001333(n+1))^2.
a(n) + a(n+1) = A075848(n+2) - A075848(n+1).
a(n) = A001541(n+1) + 2*A079291(n+1). - Creighton Dement, Oct 26 2004
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3), a(0) = 5, a(1) = 25, a(2) = 149. - Robert G. Wilson v, Nov 05 2004
2*a(n) = (-1)^n + 3*A001541(n+1). - R. J. Mathar, Sep 11 2019

A114620 2*A084158 (twice Pell triangles).

Original entry on oeis.org

0, 2, 10, 60, 348, 2030, 11830, 68952, 401880, 2342330, 13652098, 79570260, 463769460, 2703046502, 15754509550, 91824010800, 535189555248, 3119313320690, 18180690368890, 105964828892652, 617608282987020

Offset: 0

Creighton Dement, Feb 17 2006

Cross-referenced sequences A116484, A001109, A108475, A090390 are also generated by A*B given in the following FAMP code.
Floretion Algebra Multiplication Program, FAMP Code: 1jesleftseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'
Related to the reciprocals of the differences between successive convergents of the continued fraction of sqrt(2) (i.e., 1, 2, -10, 60, -348, 2030, -11830, 68952, ...). 1/1 + 1/2 - 1/10 + 1/60 - 1/348 + 1/2030 + ... = sqrt(2). 2, 10, 60, ... are products of the denominators of two successive convergents of sqrt(2) (e.g., 11830 = 70*169, cf. A000129 (Pell numbers)). - Gerald McGarvey, Feb 28 2006
a(n) is half of the even leg (b(n)) of the ordered Pythagorean triple (x(n), y(n)=x(n)+1, z(n)). In fact b(n) = x(n) + (1-(-1)^n)/2: x(0)=0, b(0)=0, a(0)=0; x(1)=3, b(1)=4, a(1)=2. - George F. Johnson, Aug 13 2012
Given a square shape composed of A001110(n+1) elements, thinking of it graphically as a sum of layers, each layer having an odd number of elements (all layers together being a sum of consecutive odd numbers), a(n) is the number of last layers that we have to subtract from the square to get a square of squares that is made of A002965(2*(n+1))^4 elements. - Daniel Poveda Parrilla, Jul 17 2016
Also numbers m such that 8*m^2 - 4*m + 1 or 8*m^2 + 4*m + 1 is a perfect square (square roots are then A001653). - Lamine Ngom, Jul 25 2023

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A116484, A001109, A108475, A090390.

Cf. A000129.

Table[Fibonacci[n, 2] Fibonacci[n + 1, 2], {n, 0, 20}] (* or *)
LinearRecurrence[{5, 5, -1}, {0, 2, 10}, 21] (* or *)
CoefficientList[Series[2 x/((x + 1) (x^2 - 6 x + 1)), {x, 0, 20}], x] (* Michael De Vlieger, Jul 17 2016 *)

G.f.: 2*x/((x+1)*(x^2-6*x+1)).
From George F. Johnson, Aug 13 2012: (Start)
a(n) = ((sqrt(2) + 1)^(2*n+1) - (sqrt(2) - 1)^(2*n+1) - 2*(-1)^n)/8. - corrected by Ilya Gutkovskiy, Jul 18 2016
4*a(n)*(2*a(n) + (-1)^n) + 1 = A000129(2*n+1)^2 is a perfect square.
For n >= 0, a(n+1) = 3*a(n) + (-1)^n + sqrt(4*a(n)*(2*a(n) + (-1)^n) + 1).
For n >  0, a(n-1) = 3*a(n) + (-1)^n - sqrt(4*a(n)*(2*a(n) + (-1)^n) + 1).
a(n+1) = 6*a(n) - a(n-1) + 2*(-1)^n.
a(n+1) = 5*a(n) + 5*a(n-1) - a(n-2).
For n > 0, a(n+1)*a(n-1) = a(n)*(a(n) + 2*(-1)^n).
a(n) = A046729(n)/2. (End)
a(n) = A000129(n)*A000129(n+1). - Philippe Deléham, Apr 10 2013
a(n) = A002965(2*(n+1))*(A002965(2*(n+1)+1) - A002965(2*(n+1))). - Daniel Poveda Parrilla, Jul 17 2016

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).

Original entry on oeis.org

1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673

Offset: 0

Creighton Dement, Apr 04 2005

A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).
Floretion Algebra Multiplication Program, FAMP Code:
1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1)

Cf. A000129, A001109, A001333, A046729, A077444.

Cf. A077445, A079291, A082639, A084158, A090390.

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

CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x,0,30}], x] (* Robert G. Wilson v, Apr 06 2005 *)
LinearRecurrence[{5,5,-1}, {1,13,69}, 30] (* Harvey P. Dale, Jun 03 2017 *)

[lucas_number1(2*n+2,2,-1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022

a(n) = 2 * A001109(n+1) - (-1)^n.
G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013
From G. C. Greubel, Aug 21 2022: (Start)
a(n) = A000129(2*n+2) - (-1)^n.
E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)

A111587 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.

Original entry on oeis.org

1, 4, 9, 20, 49, 120, 289, 696, 1681, 4060, 9801, 23660, 57121, 137904, 332929, 803760, 1940449, 4684660, 11309769, 27304196, 65918161, 159140520, 384199201, 927538920, 2239277041, 5406093004, 13051463049, 31509019100, 76069501249

Offset: 0

Creighton Dement, Aug 08 2005

Let (b(n)) be the p-INVERT of (1,2,2,2,2,2,...) using p(S) = 1 - S^2; then
b(0) = 0 and b(n) = a(n-1) for n >= 1; see A292400. - Clark Kimberling, Sep 30 2017
Floretion Algebra Multiplication Program, FAMP Code: 2kbasekseq[J+G] with J = + j' + k' + 'ii' and G = + .5'ii' + .5'jj' + .5'kk' + .5e

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).

Cf. A019898, A046729, A090390, A111588.

I:=[1,4,9,20]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-3)+Self(n-4): n in [1..35]]; // Vincenzo Librandi, Oct 01 2017

LinearRecurrence[{2,0,2,1},{1,4,9,20},30] (* Harvey P. Dale, Jul 26 2011 *)
CoefficientList[Series[(x + 1)^2 / ((x^2 + 1) (1 - 2 x - x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 01 2017 *)

a(2n) = A090390(n+1), a(2n+1) = A046729(n+1);
G.f.: (x+1)^2/((x^2+1)*(1-2*x-x^2)). [sign flipped by R. J. Mathar, Nov 10 2009]
a(n) = A057077(n+1)/2 - A001333(n+2)/2. - R. J. Mathar, Nov 10 2009

A123219 Expansion of -x(x^4 + 52x^3 - 122x^2 - 28x + 1) / ((x-1)(x^2 - 34x + 1)(x^2 + 6x + 1)).

Original entry on oeis.org

1, 1, 81, 2401, 83521, 2825761, 96059601, 3262808641, 110841719041, 3765342321601, 127910874833361, 4345203949621921, 147609026049038401, 5014361666349715681, 170340687719412376401, 5786569020271612560001

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 05 2006

G. C. Greubel, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (29,174,-174,-29,1).

Cf. A001333, A090390, A120717, A121801.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-x*(x^4+52*x^3-122*x^2-28*x+1)/((x-1)*(x^2-34*x+1)*(x^2+6*x+1)))); // G. C. Greubel, Oct 12 2018

seq(coeff(series(-x*(x^4+52*x^3-122*x^2-28*x+1)/((x-1)*(x^2-34*x+1)*(x^2+6*x+1)),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 13 2018

LinearRecurrence[{29,174,-174,-29,1},{1,1,81,2401,83521},20] (* Harvey P. Dale, Jun 01 2018 *)

x='x+O('x^30); Vec(-x*(x^4+52*x^3-122*x^2-28*x+1)/((x-1)*(x^2-34*x+1)*(x^2+6*x+1))) \\ G. C. Greubel, Oct 12 2018

G.f.: -x*(x^4 + 52*x^3 - 122*x^2 - 28*x + 1) / ((x-1)*(x^2 - 34*x + 1)*(x^2 + 6*x + 1)). - Colin Barker, Jan 04 2013

New name from Colin Barker, Jan 04 2013

Edited by Joerg Arndt, Oct 13 2018

A184327 a(1)=1, a(2)=17; thereafter a(n) = 6*a(n-1)-a(n-2)+c, where c=-4 if n is odd, c=12 if n is even.

Original entry on oeis.org

1, 17, 97, 577, 3361, 19601, 114241, 665857, 3880897, 22619537, 131836321, 768398401, 4478554081, 26102926097, 152139002497, 886731088897, 5168247530881, 30122754096401, 175568277047521, 1023286908188737, 5964153172084897, 34761632124320657

Offset: 1

N. J. A. Sloane, Dec 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. V. Leyendekkers and A. G. Shannon, Pellian sequence relationships among pi, e, sqrt(2), Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See Table 3, {y_n}.
Index entries for linear recurrences with constant coefficients, signature (6,0,-6,1).

/* By definition: */ a:=[1,17]; c:=func; [n le 2 select a[n] else 6*Self(n-1)-Self(n-2)+c(n): n in [1..22]]; // Bruno Berselli, Dec 26 2012

CoefficientList[Series[(1 + 11 x - 5 x^2 + x^3)/((1 - x) (1 + x) (1 - 6 x + x^2)), {x, 0, 24}], x] (* Bruno Berselli, Dec 26 2012 *)

From Bruno Berselli, Dec 26 2012: (Start)
G.f.: x*(1+11*x-5*x^2+x^3)/((1-x)*(1+x)*(1-6*x+x^2)).
a(n) = a(-n) = 6*a(n-1)-6*a(n-3)+a(n-4).
a(n) = ((1+sqrt(2))^(2n)+(1-sqrt(2))^(2n))/2+(-1)^n-1.
a(n) = 2*A090390(n)-1. (End)

Edited from Bruno Berselli, Dec 26 2012

cp's OEIS Frontend

A098212 Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)).

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A114620 2*A084158 (twice Pell triangles).

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A105058 Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).

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A111587 a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.

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A123219 Expansion of -x*(x^4 + 52*x^3 - 122*x^2 - 28*x + 1) / ((x-1)*(x^2 - 34*x + 1)*(x^2 + 6*x + 1)).

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A184327 a(1)=1, a(2)=17; thereafter a(n) = 6*a(n-1)-a(n-2)+c, where c=-4 if n is odd, c=12 if n is even.

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A098212 Expansion of (5-x^2)/((1+x)(1-6x+x^2)).

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).

A111587 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.

A123219 Expansion of -x(x^4 + 52x^3 - 122x^2 - 28x + 1) / ((x-1)(x^2 - 34x + 1)(x^2 + 6x + 1)).