A255232 -id:A255232 - cp's OEIS frontend

A054490 Expansion of (1+5x)/(1-6x+x^2).

1, 11, 65, 379, 2209, 12875, 75041, 437371, 2549185, 14857739, 86597249, 504725755, 2941757281, 17145817931, 99933150305, 582453083899, 3394785353089, 19786259034635, 115322768854721, 672150354093691, 3917579355707425, 22833325780150859

Offset: 0

Barry E. Williams, May 04 2000

A Pellian-related second-order recursive sequence.
Third binomial transform of 1,8,8,64,64,512. - Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
Binomial transform of A164607. - R. J. Mathar, Oct 26 2011
Pisano period lengths: 1, 1, 4, 2, 6, 4, 3, 2, 12, 6, 12, 4, 14, 3, 12, 2, 8, 12, 20, 6, ... - R. J. Mathar, Aug 10 2012
From Wolfdieter Lang, Feb 26 2015: (Start)
This sequence gives all positive solutions x = x1 = a(n) of the first class of the (generalized) Pell equation x^2 - 2*y^2 = -7. For the corresponding y1 terms see 2*A038723(n). All positive solutions of the second class are given by (x2(n), y2(n)) = (A255236(n), A038725(n+1)), n >= 0. See (A254938(1), 2*A255232(1)) for the fundamental solution (1, 2) of the first class. See the Nagell reference, Theorem 111, p. 210, Theorem 110, p. 208, Theorem 108a, pp. 206-207.
This sequence also gives all positive solutions y = y1 of the first class of the Pell equation x^2 - 2*y^2 = 14. The corresponding solutions x1 are given in 4*A038723. This follows from the preceding comment. (End)
From Wolfdieter Lang, Mar 19 2015: (Start)
a(0) = -(2*A038761(0) - A038762(0)), a(n) = 2*A253811(n-1) + A101386(n-1), for n >= 1.
This follows from the general trivial fact that if X^2 - D*Y^2 = N (X, Y positive integers, D > 1, not a square, and N a non-vanishing integer) then x:= D*Y +/- X and y:= Y +/- X (correlated signs) satisfy x^2 - D*y^2 = -(D-1)*N. with integers x and y. Here D = 2 and N = 7. (End)

			n = 2: sqrt(8*23^2-7) = 65.
2*19 + 27  = 65. - _Wolfdieter Lang_, Mar 19 2015

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See y(n) in Lemma 7.9 p. 21.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (6,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000129, A001109, A038723, A038725, A054488, A054489, A101386, A253811, A255236.

a:=[1,11];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 20 2020

I:=[1,11]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

a[0]:=1: a[1]:=11: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..30); # Zerinvary Lajos, Jul 26 2006

CoefficientList[Series[(1+5x)/(1-6x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{6, -1}, {1, 11}, 30] (* G. C. Greubel, Jul 26 2018 *)

my(x='x+O('x^30)); Vec((1+5*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jul 26 2018

[lucas_number1(2*n+1,2,-1) + 3*lucas_number1(2*n,2,-1) for n in (0..30)] # G. C. Greubel, Jan 20 2020

a(n) = 6*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=11.
a(n) = sqrt(8*A038723(n)^2 - 7).
a(n) = (11*((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) - ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).
a(n) = 11*S(n, 6) + 5*S(n-1, 6), n >= 0, with Chebyshev's polynomials S(n, x) (A049310) evaluated at x=6: S(n, 6) = A001109(n-1). See the g.f. and the Pell equation comments above. - Wolfdieter Lang, Feb 26 2015
a(n) = 2*A253811(n-1) + A101386(n-1), for n >= 1. See the Mar 19 2015 comment above. - Wolfdieter Lang, Mar 19 2015
From G. C. Greubel, Jan 20 2020: (Start)
a(n) = Pell(2*n+1) + 3*Pell(2*n).
a(n) = ChebyshevU(n,3) + 5*ChebyshevU(n-1,3).
E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x) ). (End)

More terms from James Sellers, May 05 2000

More terms from Vincenzo Librandi, Mar 20 2015

A254938 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

Original entry on oeis.org

1, 3, 1, 5, 1, 7, 5, 1, 7, 11, 3, 1, 13, 7, 5, 11, 9, 17, 5, 3, 9, 19, 7, 13, 5, 3, 7, 19, 13, 1, 9, 25, 15, 7, 23, 27, 17, 9, 21, 7, 1, 13, 19, 11, 23, 17, 31, 7, 1, 33, 11, 17, 7, 27, 5, 35, 13, 25, 19, 11, 29, 9, 17, 5, 3, 1, 27, 21, 35, 17, 23, 15, 37, 41, 21, 13, 19, 7, 3, 23, 15, 33, 13

Offset: 1

Wolfdieter Lang, Feb 18 2015

nonn

For the corresponding term y1(n) see 2*A255232(n).
For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A255233(n) and A255234(n).
The present solutions of this first class are the smallest positive ones.
See the Nagell reference Theorem 111, p. 210, for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208, for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable and each prime from A007522 does not divide 4.
The present fundamental solutions are found according to the Nagell reference Theorem 108a, p. 206-207, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) (because odd y is out in this Pell equation). The interval to be scanned for X1(n) is [0, floor((sqrt(p(n))-1)/2)] and for Y1(n) it is [0, floor(sqrt(p(n))/2)], with p(n) = A007522(n).
The general positive proper solutions are for both classes obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental positive column vectors (x(n),y(n))^T. The n-th power is M^n = S(n-1, 6)*M - S(n-2, 6) 1_2, where 1_2 is the 2 X 2 identity matrix and S(n, 6), with S(-2, 6) = -1 and S(-1, 6) = 0, is the Chebyshev S-polynomial evaluated at x = 6, given in A001109(n).
The least positive x solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255235.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A007522(n) is listed as first entry):
  [7, [1, 2]], [23, [3, 4]], [31, [1, 4]],
  [47, [5, 6]], [71, [1, 6]], [79, [7, 8]],
  [103, [5, 8]], [127, [1, 8]], [151, [7, 10]],
  [167, [11, 12]], [191, [3, 10]], [199, [1, 10]], [223, [13, 14]], [239, [7, 12]], [263, [5, 12]], [271, [11, 14]], [311, [9, 14]], [359, [17, 18]], [367, [5, 14]], [383, [3, 14]], [431, [9, 16]], [439, [19, 20]], [463, [7, 16]], [479, [13, 18]], [487, [5, 16]], [503, [3, 16]], ...
n=1: 1^2 - 2*(2*1)^2 = 1 - 8 = -7 = -A007522(1), ...

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522 (primes == 7 mod 8).
Cf. A255232 (corresponding y2 values, halved), A255233 (x2 values), A255234 (y2 values), A255235.
Cf. A254934 - A254937 (similar for primes == 1 mod 8).

apply( {A254938(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][1]}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*(2*A255232(n))^2 = -A007522(n), n >= 1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 23 2015

A254934 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

Original entry on oeis.org

1, 3, 5, 3, 1, 7, 5, 7, 3, 1, 9, 13, 5, 1, 15, 7, 13, 17, 1, 11, 19, 3, 1, 17, 11, 21, 9, 7, 11, 17, 21, 5, 1, 23, 11, 27, 9, 7, 5, 3, 19, 11, 23, 27, 7, 31, 1, 13, 25, 19, 33, 7, 5, 21, 31, 25, 9, 29, 5, 15, 27, 13, 31, 11, 7, 17, 3, 37, 41, 31, 19, 25, 7, 35, 5, 17, 33, 13, 21, 19, 45, 25, 3

Offset: 1

Wolfdieter Lang, Feb 18 2015

For the corresponding term y1(n) see A254935(n).
For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254936(n) and A254937(n).
The present solutions of this first class are the smallest positive ones.
See the Nagell reference Theorem 111, p. 210, for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208, for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable and each prime from A007519 does not divide 4.
The present fundamental solutions are found according to the Nagell reference Theorem 108a, p. 206-207, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) + 1 (because even y is out in this Pell equation). The intervals to be scanned are identical for X1(n) and Y1(n), namely [0, floor((sqrt(p(n) - 1)/2)] with p(n) = A007519(n).
The general positive proper solutions are for both classes obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental positive column vectors (x(n),y(n))^T. The n-th power M^n = S(n-1, 6)*M - S(n-2, 6) 1_2 , where 1_2 is the 2 X 2 identity matrix and S(n, 6), with S(-2, 6) = -1 and S(-1, 6) = 0 is the Chebyshev S-polynomial evaluated at x = 6, given in A001109(n).
The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255235.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A007519(n) is listed as first entry):
[17, [1, 3]], [41, [3, 5]], [73, [5, 7]],
[89, [3, 7]], [97, [1, 7]], [113, [7, 9]],
[137, [5, 9]], [193, [7, 11]], [233, [3, 11]],
[241, [1, 11]], [257, [9, 13]], [281, [13, 15]],
[313, [5, 13]], [337, [1, 13]], [353, [15, 17]],
[401, [7, 15]], [409, [13, 17]], [433, [17, 19]],
[449, [1, 15]], [457, [11, 17]], [521, [19, 21]],
[569, [3, 17]], [577, [1, 17]], [593, [17, 21]],
[601, [11, 19]], [617, [21, 23]], [641, [9, 19]],
[673, [7, 19]], [761, [11, 21]], [769, [17, 23]],
...
n=1: 1^2 - 2*3^2 = 1 - 18 = -17, ...

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).

Cf. A254935 (corresponding y1 values), A254936 (x2 values), A254937 (y2 values), A254938, A255232, A255235, A254760.

apply( {A254934(n, p=A007519(n))=Set(abs(qfbsolve(Qfb(-1,0,2), p,1)))[1][1]}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*A254935(n)^2 = -A007519(n), n >=1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from M. F. Hasler, May 22 2025

A254935 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

Original entry on oeis.org

3, 5, 7, 7, 7, 9, 9, 11, 11, 11, 13, 15, 13, 13, 17, 15, 17, 19, 15, 17, 21, 17, 17, 21, 19, 23, 19, 19, 21, 23, 25, 21, 21, 27, 23, 29, 23, 23, 23, 23, 27, 25, 29, 31, 25, 33, 25, 27, 31, 29, 35, 27, 27, 31, 35, 33, 29, 35, 29, 31, 35, 31, 37, 31, 31, 33, 31, 41, 43, 39, 35, 37, 33, 41, 33, 35, 41

Offset: 1

Wolfdieter Lang, Feb 18 2015

For the corresponding term x1(n) see A254934(n).
See A254934 also for the Nagell reference.
The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255246.

			See A254934.
n = 3: 5^2 - 2*7^2 = 25 - 98 = -73.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).

Cf. A254934 (corresponding x1 values), A254936 (x2 values), A254937 (y2 values), A254938 (same for primes == 7 mod 8), A255232 (y2 values, halved).

apply( {A254935(n, p=A007519(n))=sqrtint((A254934(,p)^2+p)\2)}, [1..77]) \\ M. F. Hasler, May 22 2025

A254934(n)^2 - 2*a(n)^2 = -A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from M. F. Hasler, May 22 2025

A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

Original entry on oeis.org

5, 7, 13, 9, 21, 11, 17, 29, 19, 15, 31, 37, 17, 27, 33, 23, 29, 21, 41, 47, 37, 23, 43, 33, 49, 55, 51, 31, 41, 69, 53, 29, 43, 59, 35, 31, 45, 61, 41, 67, 85, 57, 47, 63, 43, 53, 35, 75, 93, 37, 71, 61, 83, 47, 89, 39, 73, 53, 63, 79, 49, 85, 69, 97, 103, 109, 55, 65, 47, 77, 67, 83, 49

Offset: 1

Wolfdieter Lang, Feb 18 2015

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n).

For comments and the Nagell reference see A254938.

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry):
  [7, [5, 4]], [23, [7, 6]], [31, [13, 10]],
  [47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]],
  [151, [19, 16]], [167, [15, 14]],
  [191, [31, 24]], [199, [37, 28]],
  [223, [17, 16]], [239, [27, 22]],
  [263, [33, 26]], [271, [23, 20]],
  [311, [29, 24]], [359, [21, 20]],
  [367, [41, 32]], [383, [47, 36]],
  [431, [37, 30]], [439, [23, 22]],
  [463, [43, 34]], [479, [33, 28]], ...
n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4).
a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522 (primes == 7 mod 8), A139487 (8k+7 is prime).

Cf. 2*A255234 (corresponding y2 values), A254938 (x1 values), 2*A255232 (y2 values), A255247, A254936.

apply( {A255233(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1.

More terms from Colin Barker, Feb 23 2015

Double-checked and extended by M. F. Hasler, May 22 2025

A255234 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n>=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

2, 3, 5, 4, 8, 5, 7, 11, 8, 7, 12, 14, 8, 11, 13, 10, 12, 10, 16, 18, 15, 11, 17, 14, 19, 21, 20, 14, 17, 26, 21, 14, 18, 23, 16, 15, 19, 24, 18, 26, 32, 23, 20, 25, 19, 22, 17, 29, 35, 18, 28, 25, 32, 21, 34, 19, 29, 23, 26, 31, 22, 33, 28, 37, 39, 41, 24, 27, 22, 31, 28, 33, 23, 22, 30

Offset: 1

Wolfdieter Lang, Feb 19 2015

The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A255233(n).

See the comments and the Nagell reference in A254938.

			n = 2: 7^2 - 2*(2*3)^2 = 49 - 72  = -23 = - A007522(2).
a(3) = -(1 - 3*2) = 5.
See also A255233.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522, A255233, A254938, A255232, A254937.

apply( {A255234(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-1,3/2]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

A255233(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(A254938(n) - 3*A255232(n)), n >= 1.

More terms from Colin Barker, Feb 24 2015

Double-checked and extended by M. F. Hasler, May 22 2025

A255235 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

Original entry on oeis.org

4, 1, 1, 3, 1, 3, 5, 1, 5, 7, 3, 1, 5, 7, 1, 5, 7, 11, 3, 7, 1, 13, 3, 7, 1, 9, 5, 11, 13, 9, 5, 1, 15, 17, 5, 3, 7, 13, 9, 17, 19, 1, 11, 7, 13, 5, 3, 19, 3, 1, 17, 7, 11, 19, 21, 13, 9, 1, 7, 9, 25, 15, 7, 11, 17, 21, 23, 27, 5

Offset: 1

Wolfdieter Lang, Feb 25 2015

For the corresponding term y1(n) see A255246(n).
The present solutions of this first class are the smallest positive ones.
For the positive fundamental proper (sometimes called primitive) solutions x2 and y2 of the second class of this (generalized) Pell equation see A255247 and A255248. There is no second class for prime 2.
For the first class solutions of this Pell equation with primes 1 (mod 8) see A254934 and A254935. For those with primes 7 (mod 8) see A254938 and 2*A255232. For the derivation of these solutions see A254934 and A254938, also for the Nagell reference.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are
  (the prime A038873(n) is listed as first entry):
  [2,[4, 3]], [7, [1, 2]], [17, [1, 3]],
  [23, [3, 4]], [31, [1, 4]], [41, [3, 5]],
  [47, [5, 6]], [71, [1, 6]], [73, [5, 7]],
  [79, [7, 8]], [89, [3, 7]], [97, [1, 7]],
  [103, [5, 8]], [113, [7, 9]], [127, [1, 8]],
  [137, [5, 9]], [151, [7, 10]], [167, [11, 12]], [191, [3, 10]], [193, [7, 11]], [199, [1, 10]], [223, [13, 14]], [233, [3, 11]], [239, [7, 12]], [241, [1, 11]], [257, [9, 13]], [263, [5, 12]], ...
n=1: 4^2 - 2*3^2 = -2 = -A038873(1),
n=2: 1^2 - 2*2^2 = 1 - 8 = -7 = -A038873(2).

Cf. A038873, A255246, A255247, A255248, A254934, A254935, A254938, 2*A255232, A002334.

a(n)^2 - A255246(n)^2 = - A038873(n), n >= 1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 26 2015

A255246 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

Original entry on oeis.org

3, 2, 3, 4, 4, 5, 6, 6, 7, 8, 7, 7, 8, 9, 8, 9, 10, 12, 10, 11, 10, 14, 11, 12, 11, 13, 12, 14, 15, 14, 13, 13, 17, 18, 14, 14, 15, 17, 16, 19, 20, 15, 17, 16, 18, 16, 16, 21, 17, 17, 21, 18, 19, 22, 23, 20, 19, 18, 19, 20, 26, 22, 20, 21, 23, 25, 26, 28, 21

Offset: 1

Wolfdieter Lang, Feb 25 2015

For the corresponding term x1(n) see A255235(n).
For the primes 1 (mod 8) see A154935, and for the primes 7 (mod 8) see 2*A255232.
See A254934 and A254938 also for the derivation based on the Nagell reference given there.

			See A255235.
n = 1: 4^2 - 2*3^2 = -2 = -A038873(1),
n = 3: 1^2 - 2*3^2 = 1 - 18 = -17 = -A038873(3).

Cf. A038873, A255235, A255247, A255248, A254935, 2*A255232, A002335.

A255235(n)^2 - 2*a(n)^2 = -A038873(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 26 2015

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A054490 Expansion of (1+5*x)/(1-6*x+x^2).

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A254938 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

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A254934 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

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A254935 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

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A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

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A255234 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n>=1 (primes congruent to 7 mod 8).

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A054490 Expansion of (1+5x)/(1-6x+x^2).