cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A188645 Array of ((k^n)+(k^(-n)))/2 where k=(sqrt(x^2+1)+x)^2 for integers x>=1.

Original entry on oeis.org

1, 3, 1, 17, 9, 1, 99, 161, 19, 1, 577, 2889, 721, 33, 1, 3363, 51841, 27379, 2177, 51, 1, 19601, 930249, 1039681, 143649, 5201, 73, 1, 114243, 16692641, 39480499, 9478657, 530451, 10657, 99, 1, 665857, 299537289, 1499219281, 625447713, 54100801, 1555849, 19601, 129, 1
Offset: 0

Views

Author

Charles L. Hohn, Apr 06 2011

Keywords

Comments

Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; and constant k=f(x, y); then for all integers x>=1 and y=[+-]1, k may be irrational, but ((k^n)+(k^(-n)))/2 always produces integer sequences; y=1 results shown here; y=-1 results are A188644.
Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{k}(x), evaluated at x=2*n^2+1. - Seiichi Manyama, Jan 01 2019

Examples

			Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   3,     17,        99,          577, ...
   2 | 1,   9,    161,      2889,        51841, ...
   3 | 1,  19,    721,     27379,      1039681, ...
   4 | 1,  33,   2177,    143649,      9478657, ...
   5 | 1,  51,   5201,    530451,     54100801, ...
   6 | 1,  73,  10657,   1555849,    227143297, ...
   7 | 1,  99,  19601,   3880899,    768398401, ...
   8 | 1, 129,  33281,   8586369,   2215249921, ...
   9 | 1, 163,  53137,  17322499,   5647081537, ...
  10 | 1, 201,  80801,  32481801,  13057603201, ...
  11 | 1, 243, 118097,  57394899,  27893802817, ...
  12 | 1, 289, 167041,  96549409,  55805391361, ...
  13 | 1, 339, 229841, 155831859, 105653770561, ...
  14 | 1, 393, 308897, 242792649, 190834713217, ...
  15 | 1, 451, 406801, 366934051, 330974107201, ...
  ...

Links

Crossrefs

Row 1 is A001541, row 2 is A023039, row 3 is A078986, row 4 is A099370, row 5 is A099397, row 6 is A174747, row 8 is A176368, (row 1)*2 is A003499, (row 2)*2 is A087215.
Column 1 is A058331, (column 1)*2 is A005899.
A188644 (f(x, y) as above with y=-1).
Diagonal gives A173128.
Cf. A188647.

Programs

  • Mathematica
    max = 9; y = 1; t = Table[k = ((x^2 + y)^(1/2) + x)^2; ((k^n) + (k^(-n)))/2 // FullSimplify, {n, 0, max - 1}, {x, 1, max}]; Table[ t[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

Formula

A(n,k) = (A188647(n,k-1) + A188647(n,k))/2.
A(n,k) = Sum_{j=0..k} binomial(2*k,2*j)*(n^2+1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

Extensions

Edited and extended by Seiichi Manyama, Jan 01 2019

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Original entry on oeis.org

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030
Offset: 0

Views

Author

Gary Detlefs, Feb 20 2012

Keywords

Comments

Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2.
.
k x y
- ----------------------- -----------------------
2 A001542 {0,2,6,-1} A001541 {1,3,6,-1}
3 A001353 {0,1,4,-1} A001075 {1,2,4,-1}
5 A060645 {0,4,18,-1} A023039 {1,9,18,-1}
6 A001078 {0,2,10,-1} A001079 {1,5,10,-1}
7 A001080 {0,3,16,-1} A001081 {1,8,16,-1}
8 A001109 {0,1,6,-1} A001541 {1,3,6,-1}
10 A084070 {0,1,38,-1} A078986 {1,19,38,-1}
11 A001084 {0,3,20,-1} A001085 {1,10,20,-1}
12 A011944 {0,2,14,-1} A011943 {1,7,14,-1}
13 A075871 {0,180,1298,-1} A114047 {1,649,1298,-1}
14 A068204 {0,4,30,-1} A069203 {1,15,30,-1}
15 A001090 {0,1,8,-1} A001091 {1,4,8,-1}
17 A121740 {0,8,66,-1} A099370 {1,33,66,-1}
18 A202299 {0,4,34,-1} A056771 {1,17,34,-1}
19 A174765 {0,39,340,-1} A114048 {1,179,340,-1}
20 a(n) {0,2,18,-1} A023039 {1,9,18,-1}
21 A174745 {0,12,110,-1} A114049 {1,55,110,-1}
22 A174766 {0,42,394,-1} A114050 {1,197,394,-1}
23 A174767 {0,5,48,-1} A114051 {1,24,48,-1}
24 A004189 {0,1,10,-1} A001079 {1,5,10,-1}
26 A174768 {0,10,102,-1} A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

Links

Crossrefs

Cf. A023039, A049660, A079962.

Programs

  • Magma
    m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019
  • Maple
    readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;
  • Mathematica
    LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)
  • Maxima
    makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

Formula

a(n) = 18*a(n-1) - a(n-2).
From Bruno Berselli, Feb 21 2012: (Start)
G.f.: 2*x/(1-18*x+x^2).
a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019
For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021

A174768 y-values in the solution to x^2 - 26*y^2 = 1.

Original entry on oeis.org

0, 10, 1020, 104030, 10610040, 1082120050, 110365635060, 11256212656070, 1148023325284080, 117087122966320090, 11941738519239365100, 1217940241839448920110, 124217962929104550486120, 12669014278526824700664130, 1292115238446807014917255140, 131783085307295788696859360150
Offset: 1

Views

Author

Vincenzo Librandi, Apr 14 2010

Keywords

Comments

The corresponding values of x of this Pell equation are in A099397.

Links

Crossrefs

Cf. A041041, A099397.

Programs

  • Magma
    I:=[0, 10]; [n le 2 select I[n] else 102*Self(n-1)-Self(n-2): n in [1..20]];
  • Mathematica
    LinearRecurrence[{102,-1}, {0,10}, 30]
a[ n_] := Fibonacci[2*n-2, 10]; (* Michael Somos, Oct 25 2022 *)

Formula

a(n) = 102*a(n-1)-a(n-2) with a(1)=0, a(2)=10.
G.f.: 10*x^2/(1-102*x+x^2).
a(n+1) = A041041(2*n-1). - Michael Somos, Oct 25 2022
Showing 1-3 of 3 results.

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