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.

A266506 a(n) = 2*a(n-4) + a(n-8) for n >= 8.

Original entry on oeis.org

2, -1, 2, 1, 1, 3, 3, 5, 4, 5, 8, 11, 9, 13, 19, 27, 22, 31, 46, 65, 53, 75, 111, 157, 128, 181, 268, 379, 309, 437, 647, 915, 746, 1055, 1562, 2209, 1801, 2547, 3771, 5333, 4348, 6149, 9104, 12875, 10497, 14845, 21979, 31083, 25342, 35839, 53062, 75041, 61181, 86523
Offset: 0

Views

Author

Raphie Frank, Dec 30 2015

Keywords

Comments

Previous name was: a(2n) = a(2n - 4) + a(2n - 3) and a(2n + 1) = 2*a(2n - 4) + a(2n - 3), with a(0) = 2, a(1) = -1, a(2) = 2, a(3) = 1. Alternatively, interleave denominators (A266504) and numerators (A266505) of convergents to sqrt(2).
a(2n) gives all x in N | 2*x^2 - 7(-1)^x = y^2. a(2n+1) gives associated y values.

Links

Crossrefs

Cf. A266504, A266505.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8+2*x^4-1))); // G. C. Greubel, Jul 27 2018
  • Mathematica
    CoefficientList[Series[(-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1), {x, 0, 50}], x] (* G. C. Greubel, Jul 27 2018 *)
  • PARI
    x='x+O('x^50); Vec((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8 + 2*x^4-1)) \\ G. C. Greubel, Jul 27 2018

Formula

From Chai Wah Wu, Sep 17 2016: (Start)
a(n) = 2*a(n-4) + a(n-8) for n > 7.
G.f.: (-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1).
(End)

Extensions

Edited, new name using given formula, Joerg Arndt, Jan 31 2024

A266505 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = -1, a(1) = 1, a(2) = 3, a(3) = 5.

Original entry on oeis.org

-1, 1, 3, 5, 5, 11, 13, 27, 31, 65, 75, 157, 181, 379, 437, 915, 1055, 2209, 2547, 5333, 6149, 12875, 14845, 31083, 35839, 75041, 86523, 181165, 208885, 437371, 504293, 1055907, 1217471, 2549185, 2939235, 6154277, 7095941, 14857739, 17131117, 35869755, 41358175, 86597249, 99847467
Offset: 0

Views

Author

Raphie Frank, Dec 30 2015

Keywords

Comments

a(n)/A266504(n) converges to sqrt(2).
Alternatively, bisection of A266506.
Alternatively, A135532(n) and A048655(n) interlaced.
Alternatively, A255236(n-1), A054490(n), A038762(n) and A101386(n) interlaced.
Let b(n) = (a(n) - (a(n) mod 2))/2, that is b(n) = {-1, 0, 1, 2, 2, 5, 6, 13, 15, 32, 37, 78, 90, ...}. Then:
A006451(n) = {b(4n+0) U b(4n+1)} gives n in N such that triangular(n) + 1 is square;
A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that triangular(n) follows form n^2 + n + 1 (twice a triangular number + 1).

Links

Crossrefs

Cf. A000129, A001333, A002203, A002965, A006451, A006452, A002965, A038761, A038762, A048654, A048655, A054490, A078343, A098586, A098790, A100525, A101386, A135532, A216134, A216162, A253811, A255236, A266504, A266505, A266507.

Programs

  • Magma
    I:=[-1,1,3,5]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
  • Maple
    a:=proc(n) option remember; if n=0 then -1 elif n=1 then 1 elif n=2 then 3 elif n=3 then 5 else 2*a(n-2)+a(n-4); fi; end:  seq(a(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016
  • Mathematica
    LinearRecurrence[{0, 2, 0, 1}, {-1, 1, 3, 5}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[(-1 + 3 x) (1 + x)^2/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 42}] (* Michael De Vlieger, Dec 31 2015 *)
  • PARI
    my(x='x+O('x^40)); Vec((-1+3*x)*(1+x)^2/(1-2*x^2-x^4)) \\ G. C. Greubel, Jul 26 2018

Formula

G.f.: (-1 + 3*x)*(1 + x)^2/(1 - 2*x^2 - x^4).
a(n) = (-(1+sqrt(2))^floor(n/2)*(-1)^n - sqrt(8)*(1-sqrt(2))^floor(n/2) - (1-sqrt(2))^floor(n/2)*(-1)^n + sqrt(8)*(1+sqrt(2))^floor(n/2))/2.
a(n) = 3*(((1+sqrt(2))^floor(n/2)-(1-sqrt(2))^floor(n/2))/sqrt(8)) - (-1)^n*(((1+sqrt(2))^(floor(n/2)-(-1)^n)-(1-sqrt(2))^(floor(n/2)-(-1)^n))/sqrt(8)).
a(n) = (3*A000129(floor(n/2)) - A000129(n-(-1)^n)), where A000129 gives the Pell numbers.
a(n) = sqrt(2*A266504(n)^2 - 7*(-1)^A266504(n))*sgn(2*n-1), where A266504 gives all x in N such that 2*x^2 - 7*(-1)^x = y^2. This sequence gives associated y values.
a(2n) = (-(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n - (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n) = A135532(n).
a(2n) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) - (((1+sqrt(2))^(n-1)-(1-sqrt(2))^(n-1))/sqrt(8)) = A135532(n).
a(2n+1) = (+(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n + (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n + 1) = A048655(n).
a(2n+1) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) + (((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/sqrt(8)) = A048655(n).
a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A255236(n-1).
a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A054490(n).
a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038762(n).
a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A101386(n).
(sqrt(2*(a(2n + 1) )^2 + 14*(-1)^floor(n/2)))/2 = A266504(n).
(a(2n + 1) + a(2n))/8 = A000129(n), where A000129 gives the Pell numbers.
a(2n + 1) - a(2n) = A002203(n), where A002203 gives the companion Pell numbers.
(a(2n + 2) + a(2n + 1))/2 = A000129(n+2).
(a(2n + 2) - a(2n + 1))/2 = A000129(n-1).

A266507 a(n) = 6*a(n - 1) - a(n - 2) with a(0) = 2, a(1) = 8.

Original entry on oeis.org

2, 8, 46, 268, 1562, 9104, 53062, 309268, 1802546, 10506008, 61233502, 356895004, 2080136522, 12123924128, 70663408246, 411856525348, 2400475743842, 13990997937704, 81545511882382, 475282073356588, 2770146928257146, 16145599496186288, 94103450048860582
Offset: 0

Views

Author

Raphie Frank, Dec 30 2015

Keywords

Comments

Bisection of A078343 = A078343(2*n + 1).
Quadrisection of A266504 = A266504(4*n + 1).
Octasection of A266506 = A266506(8*n + 2).

Links

Crossrefs

Bisection of A078343 = A078343(2n + 1).
Quadrisection of A266504 = A266504(4n + 1).
Octasection of A266506 = A266506(8n + 2).
Equals 2*A038723(n).

Programs

  • Magma
    I:=[2,8]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
  • Mathematica
    LinearRecurrence[{6, -1}, {2, 8}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[2 (1 - 2 x)/(1 - 6 x + x^2), {x, 0, n}], {n, 0, 22}] (* Michael De Vlieger, Dec 31 2015 *)
  • PARI
    Vec(2*(1-2*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Dec 31 2015

Formula

a(n) = (-sqrt(2)*(1+sqrt(2))^(2*n+1) - 3 *(1-sqrt(2))^(2*n+1) - sqrt(2)*(1-sqrt(2))^(2*n+1) + 3*(1+sqrt(2))^(2*n+1))/sqrt(8).
G.f.: 2*(1-2*x) / (1-6*x+x^2). - Colin Barker, Dec 31 2015
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