A065113 -id:A065113 - cp's OEIS frontend

A075870 Numbers k such that 2*k^2 - 4 is a square.

2, 10, 58, 338, 1970, 11482, 66922, 390050, 2273378, 13250218, 77227930, 450117362, 2623476242, 15290740090, 89120964298, 519435045698, 3027489309890, 17645500813642, 102845515571962, 599427592618130, 3493720040136818, 20362892648202778, 118683635849079850

Offset: 1

Gregory V. Richardson, Oct 16 2002

Lim_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2).
Also gives solutions to the equation x^2-2 = floor(x*r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 14 2004
The upper intermediate convergents to 2^(1/2) beginning with 10/7, 58/41, 338/239, 1970/1393 form a strictly decreasing sequence; essentially, numerators = A075870, denominators = A002315. - Clark Kimberling, Aug 27 2008
Numbers n such that sqrt(floor(n^2/2 - 1)) is an integer. The integer square roots are given by A002315. - Richard R. Forberg, Aug 01 2013
a(n) are the integer square roots of m^2 + (m+2)^2. The values of m are given by A065113 (except for m = 0). The values of this expression are given by A165518. - Richard R. Forberg, Aug 15 2013
Values of x (or y) in the solutions to x^2 - 6*x*y + y^2 + 16 = 0. - Colin Barker, Feb 04 2014
Also integers k such that k^2 is equal to the sum of four consecutive triangular numbers. - Colin Barker, Dec 20 2014
Equivalently, numbers x such that (x-1)*x/2 + x*(x+1)/2 = (y-1)^2 + (y+1)^2. y-values are listed in A002315. Example: for x=58 and y=41, 57*58/2 + 58*59/2 = 40^2 + 42^2. - Bruno Berselli, Mar 19 2018

			From _Muniru A Asiru_, Mar 19 2018: (Start)
For k=2, 2*2^2 - 4 = 8 - 4 = 4 = 2^2.
For k=10, 2*10^2 - 4 = 200 - 4 =  196 = 14^2.
For k=58, 2*58^2 - 4 = 6728 - 4 =  6724 = 82^2.
... (End)

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238. - N. J. A. Sloane, Mar 03 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A000217, A000290, A002315.

Twice A001653.

a:=[2,10];; for n in [3..25] do a[n]:=6*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Mar 19 2018

a:= proc(n) option remember: if n = 1 then 2 elif n = 2 then 10 elif  n >= 3 then 6*procname(n-1) - procname(n-2) fi; end: seq(a(n), n = 0..25); # Muniru A Asiru, Mar 19 2018

LinearRecurrence[{6,-1},{2,10},30] (* Harvey P. Dale, Sep 27 2018 *)

Vec(2*x*(1-x)/(1-6*x+x^2) + O(x^100)) \\ Colin Barker, Dec 20 2014

a(n) = 2 * A001653(n).
a(n) = (1/sqrt(2))*((1+sqrt(2))^(2*n-1) - (1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2).
G.f.: 2*x*(1-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008
a(n) = round(((2+sqrt(2))*(3+2*sqrt(2))^(n-1))/2). - Paul Weisenhorn, Jun 11 2020
From Peter Bala, Aug 21 2022: (Start)
a(n) = 2*Pell(2*n-1).
1/a(n) - 1/a(n+1) = 1/(Pell(2*n) + 1/Pell(2*n)), where Pell(n) = A000129(n). (End)

More terms from Colin Barker, Dec 20 2014

A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 20, 27, 40, 49, 54, 58, 63, 68, 88, 93, 119, 136, 150, 167, 221, 238, 288, 300, 310, 322, 334, 338, 360, 372, 382, 394, 406, 508, 530, 542, 696, 737, 771, 812, 833, 867, 908, 942, 983, 1242, 1276, 1317, 1392, 1681, 1710, 1734, 1763, 1792, 1802

Offset: 1

Amiram Eldar, Mar 12 2022

A000129(n) - 2 is a term for n > 1. The maximal Pell representations of these numbers are 0, 11, 121, 1221, 12221, ... (0 and A132583).
A048739 is a subsequence since these are the repunit numbers in the maximal Pell representation.
A065113 is a subsequence since the maximal Pell representation of A065113(n) is 2*n 2's.

			The first 10 terms are:
   n  a(n)  A352339(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    6              22
   5    8             111
   6   10             121
   7   20            1111
   8   27            1221
   9   40            2222
  10   49           11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A048739, A065113, A132583, A352339.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazy[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Select[Range[0, 2000], PalindromeQ[lazy[#]] &]

A114696 Expansion of (1+4x+x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 6, 15, 40, 97, 238, 575, 1392, 3361, 8118, 19599, 47320, 114241, 275806, 665855, 1607520, 3880897, 9369318, 22619535, 54608392, 131836321, 318281038, 768398399, 1855077840, 4478554081, 10812186006, 26102926095, 63018038200, 152139002497, 367296043198

Offset: 0

Creighton Dement, Feb 18 2006

Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.

Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

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

Cf. A000129, A002203, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114689, A114695, A114697.

Q:= proc(n) option remember; # Q=A002203
    if n<2 then 2
  else 2*Q(n-1) + Q(n-2)
    fi; end:
seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # G. C. Greubel, May 24 2021

CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x,0,30}], x] (* or *) LinearRecurrence[{2,2,-2,-1}, {1,6,15,40}, 30] (* Harvey P. Dale, Jan 23 2014 *)

Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ Colin Barker, May 26 2016

[(lucas_number2(n+2,2,-1) -3 -(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 24 2021

G.f.: (1 +4*x +x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Harvey P. Dale, Jan 23 2014
a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - Colin Barker, May 26 2016
From G. C. Greubel, May 24 2021: (Start)
a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.
a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)

A241976 Values of k such that k^2 + (k+3)^2 is a square.

Original entry on oeis.org

0, 9, 60, 357, 2088, 12177, 70980, 413709, 2411280, 14053977, 81912588, 477421557, 2782616760, 16218279009, 94527057300, 550944064797, 3211137331488, 18715879924137, 109084142213340, 635788973355909, 3705649697922120, 21598109214176817, 125883005587138788

Offset: 1

Colin Barker, Aug 10 2014

A075841 gives the corresponding values of sqrt(n^2 + (n+3)^2).

			9 is in the sequence because 9^2 + 12^2 = 225 = 15^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A001652, A065113, A075841.

CoefficientList[Series[3 x (x - 3)/((x - 1) (x^2 - 6 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 11 2014 *)

concat(0, Vec(3*x^2*(x-3)/((x-1)*(x^2-6*x+1)) + O(x^100)))

G.f.: 3*x^2*(x-3) / ((x-1)*(x^2-6*x+1)).
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = 3*A001652(n-1).
a(n) = -3*(2 + (3-2*sqrt(2))^n*(1+sqrt(2)) - (-1+sqrt(2))*(3+2*sqrt(2))^n) / 4. - Colin Barker, Apr 13 2017

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A075870 Numbers k such that 2*k^2 - 4 is a square.

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A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

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A114696 Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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A241976 Values of k such that k^2 + (k+3)^2 is a square.

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A114696 Expansion of (1+4x+x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.