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.

A161482 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.

Original entry on oeis.org

0, 96, 189, 453, 969, 1496, 3020, 6020, 9089, 17969, 35453, 53340, 105096, 207000, 311253, 612909, 1206849, 1814480, 3572660, 7034396, 10575929, 20823353, 40999829, 61641396, 121367760, 238964880, 359272749, 707383509, 1392789753
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

Corresponding values y of solutions (x, y) are in A161483.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (187+78*sqrt(2))/151 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A161483, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,96,189,453,969,1496,3020},30] (* Harvey P. Dale, Jul 10 2023 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+302 for n > 6; a(1)=0, a(2)=96, a(3)=189, a(4)=453, a(5)=969, a(6)=1496.
G.f.: x*(96+93*x+264*x^2-60*x^3-31*x^4-60*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 151*A001652(k) for k >= 0.

A161485 Decimal expansion of (24723 + 6758*sqrt(2))/151^2.

Original entry on oeis.org

1, 5, 0, 3, 4, 5, 4, 0, 2, 6, 3, 3, 7, 3, 2, 6, 2, 7, 2, 5, 2, 3, 2, 1, 4, 4, 3, 7, 8, 8, 4, 7, 0, 4, 1, 6, 2, 7, 5, 4, 8, 7, 2, 2, 6, 5, 8, 5, 6, 6, 4, 7, 2, 9, 9, 6, 1, 5, 2, 8, 0, 1, 0, 4, 7, 9, 1, 0, 7, 8, 0, 2, 6, 8, 1, 6, 6, 7, 1, 1, 7, 0, 9, 3, 7, 7, 1, 8, 4, 7, 5, 3, 7, 3, 2, 8, 1, 9, 5, 0, 2, 8, 9, 1, 6
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 0, b = A161482.
lim_{n -> infinity} b(n)/b(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 1, b = A161483.

Examples

			(24723 + 6758*sqrt(2))/151^2 = 1.50345402633732627252...

Links

Crossrefs

Cf. A161482, A161483, A002193 (decimal expansion of sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151).

Programs

  • Magma
    (24723 + 6758*Sqrt(2))/151^2; // G. C. Greubel, Apr 07 2018
  • Mathematica
    RealDigits[(24723 + 6758*Sqrt[2])/151^2, 10, 100][[1]] (* G. C. Greubel, Apr 07 2018 *)
  • PARI
    (24723 + 6758*sqrt(2))/151^2 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (218 + 31*sqrt(2))/(218 - 31*sqrt(2)).
Equals (3 + 2*sqrt(2))*(13 - 3*sqrt(2))^2/(13 + 3*sqrt(2))^2.

A161483 Positive numbers y such that y^2 is of the form x^2+(x+151)^2 with integer x.

Original entry on oeis.org

109, 151, 265, 389, 755, 1481, 2225, 4379, 8621, 12961, 25519, 50245, 75541, 148735, 292849, 440285, 866891, 1706849, 2566169, 5052611, 9948245, 14956729, 29448775, 57982621, 87174205, 171640039, 337947481, 508088501, 1000391459, 1969702265
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

(-60, a(1)) and (A161482(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (187+78*sqrt(2))/151 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 1.

Examples

			(-60, a(1)) = (-60, 109) is a solution: (-60)^2+(-60+151)^2 = 3600+8281 = 11881 = 109^2.
(A161482(1), a(2)) = (0, 151) is a solution: 0^2+(0+151)^2 = 22801 = 151^2.
(A161482(3), a(4)) = (189, 389) is a solution: 189^2+(189+151)^2 = 35721+115600 = 151321 = 389^2.

Crossrefs

Cf. A161482, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).

Programs

  • PARI
    {forstep(n=-60, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=109, a(2)=151, a(3)=265, a(4)=389, a(5)=755, a(6)=1481.
G.f.: (1-x)*(109+260*x+525*x^2+260*x^3+109*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 151*A001653(k) for k >= 1.
Showing 1-3 of 3 results.

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