A041085 -id:A041085 - cp's OEIS frontend

A091761 a(n) = Pell(4n) / Pell(4).

0, 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775, 92665403695822344828176, 3147895910861898495432209

Offset: 0

Paul Barry, Feb 06 2004

A000129(k*n)/A000129(k) = ((sqrt(2)-1)^k(-1)^k-(sqrt(2)+1)^k)((sqrt(2)-1)^(k*n)(-1)^(k*n)-(sqrt(2)+1)^(k*n))/((sqrt(2)-1)^(2k)+(sqrt(2)+1)^(2k)-2(-1)^k).
All squares of the form (3m-1)^3 + (3m)^3 + (3m+1)^3 (cf. A116108) are given for m = 24 b, where b is a square of this sequence. From Ribenboim & McDaniel, it follows there are no squares > 1 in this sequence. - M. F. Hasler, Jun 05 2007
A divisibility sequence, cf. R. K. Guy's post to the SeqFan list. - M. F. Hasler, Feb 05 2013
a(n) gives all nonnegative solutions of the Pell equation b(n)^2 - 32*(3*a(n))^2 = +1, together with b(n) = A056771(n). - Wolfdieter Lang, Mar 09 2019

M. F. Hasler, Table of n, a(n) for n = 0..99
R. K. Guy, A new sequence, post to the SeqFan list, Feb 05 2013.
Tanya Khovanova, Recursive Sequences
Paulo Ribenboim and Wayne L. McDaniel, The Square Terms in Lucas Sequences, Journal of Number Theory 58, 104-123 (1996).
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (34,-1).

A029547 is an essentially identical sequence, cf. formula.

Cf. A000129, A001109, A002965, A029547, A041085, A056771, A116108.

I:=[0,1]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // G. C. Greubel, Mar 11 2019

with (combinat):seq(fibonacci(4*n,2)/12, n=0..17); # Zerinvary Lajos, Apr 21 2008

LinearRecurrence[{34,-1}, {0,1}, 20] (* G. C. Greubel, Mar 11 2019 *)

A091761(n, x=[ -1,17],A=[17,72*4;1,17]) = vector(n,i,(x*=A)[1]) \\ M. F. Hasler, May 26 2007

A091761(n)=([34,1;-1,0]^(n-1))[1,1] \\ M. F. Hasler, Jun 05 2007

[lucas_number1(n,34,1) for n in range(0, 16)]# Zerinvary Lajos, Nov 07 2009

G.f.: x/(1-34*x+x^2).
a(n) = A000129(4n)/A000129(4).
a(n) = ((1+sqrt(2))^(4n) - (1-sqrt(2))^(4n))*sqrt(2)/48.
From M. F. Hasler, Jun 05 2007: (Start)
a(n) = n (mod 2^m) for any m >= 0.
a(n) = sinh(4*n*log(sqrt(2)+1))/(12*sqrt(2)).
a(n) = A[1,1], first element of the 2 X 2 matrix A = (34,1;-1,0)^(n-1). (End)
a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=1. - Philippe Deléham, Nov 03 2008
A029547(n) = a(n+1). - M. F. Hasler, Feb 05 2013
a(n) = sqrt((A056771(n)^2 - 1)/(32*9)), n >= 0. See the Pell remark above. - Wolfdieter Lang, Mar 11 2019
E.g.f.: exp(17*x)*sinh(12*sqrt(2)*x)/(12*sqrt(2)). - Stefano Spezia, Apr 16 2023
a(n) = A002965(8*n)/12. - Gerry Martens, Jul 14 2023

A041084 Numerators of continued fraction convergents to sqrt(50).

Original entry on oeis.org

7, 99, 1393, 19601, 275807, 3880899, 54608393, 768398401, 10812186007, 152139002499, 2140758220993, 30122754096401, 423859315570607, 5964153172084899, 83922003724759193, 1180872205318713601, 16616132878186749607, 233806732499933208099, 3289910387877251662993

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (14,1).

Cf. A010503, A041085.

Numerator[Convergents[Sqrt[50],30]] (* or *) LinearRecurrence[{14,1},{7,99},30] (* Harvey P. Dale, Aug 18 2013 *)
CoefficientList[Series[(7 + x)/(1 - 14 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)

a(n) = 14*a(n-1)+a(n-2), n>1 ; a(0)=7, a(1)=99 . G.f.: (7+x)/(1-14*x-x^2). - Philippe Deléham, Nov 21 2008

More terms from Colin Barker, Nov 04 2013

A142588 A trisection of A000129, the Pell numbers.

Original entry on oeis.org

0, 5, 70, 985, 13860, 195025, 2744210, 38613965, 543339720, 7645370045, 107578520350, 1513744654945, 21300003689580, 299713796309065, 4217293152016490, 59341817924539925, 835002744095575440, 11749380235262596085, 165326326037771920630, 2326317944764069484905

Offset: 0

Paul Curtz, Sep 22 2008

Colin Barker, Table of n, a(n) for n = 0..850
Index entries for linear recurrences with constant coefficients, signature (14,1).

Cf. A000129, A041085.

[n le 2 select 5*(n-1) else 14*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Apr 13 2021

LinearRecurrence[{14,1},{0,5},20] (* Harvey P. Dale, Jul 05 2019 *)
Fibonacci[3*Range[0, 30], 2] (* G. C. Greubel, Apr 13 2021 *)

concat(0, Vec(5*x/(1-14*x-x^2) + O(x^20))) \\ Colin Barker, Jan 25 2016

[lucas_number1(3*n,2,-1) for n in (0..30)] # G. C. Greubel, Apr 13 2021

a(n) = A000129(3n).
From R. J. Mathar, Sep 22 2008: (Start)
G.f.: 5*x/(1-14*x-x^2).
a(n) = 5*A041085(n-1). (End)
a(n) = ( (7+5*sqrt(2))^n - (7-5*sqrt(2))^n )/( 2*sqrt(2) ). - Colin Barker, Jan 25 2016

Changed offset and extended by R. J. Mathar, Sep 22 2008

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 6, 5, 4, 0, 1, 14, 35, 12, 5, 0, 1, 34, 197, 204, 29, 6, 0, 1, 82, 1155, 2772, 1189, 70, 7, 0, 1, 198, 6725, 39236, 39005, 6930, 169, 8, 0, 1, 478, 39203, 551532, 1332869, 548842, 40391, 408, 9, 0, 1, 1154, 228485, 7761996, 45232349, 45278310, 7722793, 235416, 985, 10

Offset: 0

Seiichi Manyama, May 07 2025

			Square array begins:
  0,  0,    0,     0,       0,        0, ...
  1,  1,    1,     1,       1,        1, ...
  2,  2,    6,    14,      34,       82, ...
  3,  5,   35,   197,    1155,     6725, ...
  4, 12,  204,  2772,   39236,   551532, ...
  5, 29, 1189, 39005, 1332869, 45232349, ...

Columns k=0..6 give A001477, A000129, A001109, A041085(n-1), A091761, A292423, A097731(n-1).
Rows n=0..5 give A000004, A000012, A002203, A383720, A383740, A383741.
Main diagonal gives A380083.
Cf. A028412.

A[n_, k_] := Fibonacci[k*n, 2]/Fibonacci[k, 2]; A[n_, 0] := n; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 08 2025 *)

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n, k) = if(k==0, n, pell(k*n)/pell(k));

A(0,k) = 0, A(1,k) = 1; A(n,k) = A002203(k) * A(n-1,k) - (-1)^k * A(n-2,k) for n > 1.

A(n,k) = Pell(k*n)/Pell(k) for k > 0.

A093144 Third binomial transform of Pell(3*n)/Pell(3).

Original entry on oeis.org

0, 1, 20, 350, 6000, 102500, 1750000, 29875000, 510000000, 8706250000, 148625000000, 2537187500000, 43312500000000, 739390625000000, 12622187500000000, 215474218750000000, 3678375000000000000

Offset: 0

Paul Barry, Mar 26 2004

Index entries for linear recurrences with constant coefficients, signature (20,-50).

Cf. A000129, A041085.

LinearRecurrence[{20,-50},{0,1},30] (* Harvey P. Dale, Aug 04 2021 *)

G.f.: x/(1-20*x+50*x^2);

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

cp's OEIS Frontend

A091761 a(n) = Pell(4n) / Pell(4).

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A041084 Numerators of continued fraction convergents to sqrt(50).

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A142588 A trisection of A000129, the Pell numbers.

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A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)*x + (-1)^k*x^2).

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A093144 Third binomial transform of Pell(3*n)/Pell(3).

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A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).