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.

User: Raphie Frank

Raphie Frank's wiki page.

Raphie Frank has authored 24 sequences. Here are the ten most recent ones:

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

A266504 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = a(1) = 2, a(2) = 1, a(3) = 3.

Original entry on oeis.org

2, 2, 1, 3, 4, 8, 9, 19, 22, 46, 53, 111, 128, 268, 309, 647, 746, 1562, 1801, 3771, 4348, 9104, 10497, 21979, 25342, 53062, 61181, 128103, 147704, 309268, 356589, 746639, 860882, 1802546, 2078353, 4351731, 5017588, 10506008, 12113529, 25363747, 29244646, 61233502
Offset: 0

Views

Author

Raphie Frank, Dec 30 2015

Keywords

Comments

This sequence gives all x in N | 2*x^2 - 7(-1)^x = y^2. The companion sequence to this sequence, giving y values, is A266505.
A266505(n)/a(n) converges to sqrt(2).
Alternatively, 1/4*(3*A002203(floor[n/2]) - A002203(n-(-1)^n)), where A002203 gives the Companion Pell numbers, or, in Lucas sequence notation, V_n(2, -1).
Alternatively, bisection of A266506.
Alternatively, A048654(n -1) and A078343(n + 1) interlaced.
Alternatively, A100525(n-1), A266507(n), A038761(n) and A253811(n) interlaced.
Let b(n) = (a(n) - a(n)(mod 2))/2, that is b(n) = {1, 1, 0, 1, 2, 4, 4, 9, 11, 23, 26, 55, 64, ...}. Then:
A006452(n) = {b(4n+0) U b(4n+1)} gives n in N such that n^2 - 1 is triangular;
A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that n^2 + n + 1 is triangular (indices of Sophie Germain triangular numbers);
A216162(n) = {b(4n+0) U b(4n+2) U b(4n+1) U b(4n+3)}, sequences A006452 and A216134 interlaced.

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:=[2,2,1,3]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
  • Mathematica
    LinearRecurrence[{0, 2, 0, 1}, {2, 2, 1, 3}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[(1 - x) (2 + 4 x + x^2)/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 41}] (* Michael De Vlieger, Dec 31 2015 *)
  • PARI
    Vec((1-x)*(2+4*x+x^2)/(1-2*x^2-x^4) + O(x^50)) \\ Colin Barker, Dec 31 2015

Formula

a(n) = 1/sqrt(8)*(+sqrt(2)*(1+sqrt(2))^(floor(n/2)-(-1)^n)*(-1)^n - 3*(1-sqrt(2))^(floor(n/2)-(-1)^n) + sqrt(2)*(1-sqrt(2))^(floor(n/2)-(-1)^n)*(-1)^n + 3*(1+sqrt(2))^(floor(n/2)-(-1)^n)).
a(n) = 1/4*((3*((1+sqrt(2))^floor(n/2)+(1-sqrt(2))^floor(n/2))) - (-1)^n*((1+sqrt(2))^(floor(n/2)-(-1)^n)+(1-sqrt(2))^(floor(n/2)-(-1)^n))).
a(2n) = (+sqrt(2)*(1+sqrt(2))^(n-1) - 3 *(1-sqrt(2))^(n-1) + sqrt(2)*(1-sqrt(2))^(n-1) + 3*(1 + sqrt(2))^(n-1))/sqrt(8) = A048654(n -1).
a(2n) = 1/4*((3*((1+sqrt(2))^n+(1-sqrt(2))^n)) - ((1+sqrt(2))^(n-1)+(1-sqrt(2))^(n-1))) = A048654(n -1).
a(2n + 1) = (-sqrt(2)*(1+sqrt(2))^(n+1) - 3 *(1-sqrt(2))^(n+1) - sqrt(2)*(1-sqrt(2))^(n+1) + 3*(1+sqrt(2))^(n+1))/sqrt(8) = A078343(n + 1).
a(2n + 1) =1/4*((3*((1+sqrt(2))^n+(1-sqrt(2))^n)) + ((1+sqrt(2))^(n+1)+(1-sqrt(2))^(n+1))) = A078343(n + 1).
a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A100525(n-1).
a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A266507(n).
a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038761(n).
a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A253811(n).
sqrt(2*a(n)^2 - 7(-1)^a(n))*sgn(2*n - 1) = A266505(n).
(a(2n + 1) + a(2n))/2 = A002203(n), where A002203 gives the companion Pell numbers.
(a(2n + 1) - a(2n))/2 = A000129(n), where A000129 gives the Pell numbers.
(a(2n+2) + a(2n+1))*2 = A002203(n+2)
(a(2n+2) - a(2n+1))*2 = A002203(n-1).
G.f.: (1-x)*(2+4*x+x^2) / (1-2*x^2-x^4). - Colin Barker, Dec 31 2015

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

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).

A227079 Solutions n to the Diophantine equation: n = (2*x^2 - 1)^2 = (6*y^2 - 5).

Original entry on oeis.org

1, 49, 289, 5041, 274266721
Offset: 0

Views

Author

Raphie Frank, Aug 08 2013

Keywords

Comments

x = {1, 2, 3, 6, 91} = A180445(n).
y = {1, 3, 7, 29, 6761} = A227077(n).
(sqrt(2*sqrt(n) + 2) - 1)^2 is a Ramanujan-Nagell Square = {1, 9, 25, 121, 32761} = A227078(n).

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
  • L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.

Links

Crossrefs

Cf. A060728, A180445, A227077, A227078.

A227078 The Ramanujan-Nagell squares: A038198(n)^2.

Original entry on oeis.org

1, 9, 25, 121, 32761
Offset: 0

Views

Author

Raphie Frank, Jun 30 2013

Keywords

Comments

a(n) = (2*x - 1)^2 = (sqrt(2)*sqrt(sqrt(6*y^2 - 5) + 1) - 1)^2 = 2^(z + 3) - 7 for x, y, z are the solutions to two Diophantine equations noted by R. K. Guy: 2*x^2*(x^2 - 1) = 3*(y^2 - 1) & x*(x - 1)/2 = 2^z - 1 (see A180445). x = {1, 2, 3, 6, 91} = A180445(n), y = {1, 3, 7, 29, 6761} = A227077(n), and z = {0, 1, 2, 4, 12} = A215795(n).

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
  • L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.

Links

Crossrefs

Cf. A060728, A076046, A180445, A227077, A215795.

Formula

a(n) + 7 = 2^A060728(n).
(a(n) - 1)/8 = A076046(n).

A227077 y solutions to the Diophantine equation 2*x^2*(x^2 - 1) = 3*(y^2 - 1).

Original entry on oeis.org

1, 3, 7, 29, 6761
Offset: 0

Views

Author

Raphie Frank, Jun 30 2013

Keywords

Comments

Also solutions to (2*x^2 - 1)^2 = 6*y^2 - 5 as outlined in A180445, which gives the x solutions to this equation {1, 2, 3, 6, 91}.
(sqrt(2)*sqrt(sqrt(6*a(n)^2 - 5) + 1) - 1)^2 = A038198(n)^2 gives the Ramanujan-Nagell squares listed in A227078.

Links

Crossrefs

Cf. A180445, A038198, A227078.

Programs

  • Mathematica
    Select[Table[Sqrt[3-2x^2+2x^4]/Sqrt[3],{x,0,100}],IntegerQ]//Union (* Harvey P. Dale, Aug 11 2019 *)

A222786 a(n) = A222785(n)/A211202(n).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 18, 30, 54, 156, 270, 411, 870, 1320, 2268, 4050, 8190
Offset: 1

Views

Author

Raphie Frank, Mar 05 2013

Keywords

Comments

a(n) = A222785(n)/A211202(n).

Links

Crossrefs

Cf. A211202, A222785.

A222785 Terms A002336(k) that are divisible by k.

Original entry on oeis.org

2, 6, 12, 24, 40, 72, 126, 240, 648, 2340, 4320, 7398, 17400, 27720, 49896, 93150, 196560
Offset: 1

Views

Author

Raphie Frank, Mar 05 2013

Keywords

Comments

Average number of spheres/dimension are given in A222786.
Associated dimensions are given in A211202.

Crossrefs

Cf. A222786, A211202, A002336.

A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 16, 18, 20, 21, 22, 23, 24
Offset: 1

Views

Author

Raphie Frank, Feb 18 2013

Keywords

Comments

Observations:
For all n in this sequence to n = 24, then y = Lambda_n/n follows form: y = (x^2 + x^k) - (floor[z^2/4]) or y = (x^2 + x^k) + (floor[z^2/4]); k = 1 or 2 and z = 0, 1, 3, 6 or 7. y (= A222786) gives the average number of spheres/dimension of the laminated lattice Kissing numbers in A222785.
e.g. Where T_x is the x-th triangular number = (1/2*(x^2 + x)), 2*T_x is the x-th pronic number = (x^2 + x) = floor[(2*x + 1)^2/4], and S_x is the x-th square = (x^2) = floor[(2*x)^2/4]:
For k = 1, z = 0 or 1, then n = {1, 4, 6, 8, 15, 20, 24}, x = {1, 2, 3, 5, 12, 29, 90}, and y = 2*T_x = {2, 6, 12, 30, 156, 870, 8190}.
For k = 2, z = 0 or 1, then n = {1, 5, 7, 23}, x = {1, 2, 3, 45}, and y = 2*T_x + 2*T_(-x) = 2*S_x = {2, 8, 18, 4050}.
For k = 1, z = 3, then n = {3, 7, 12, 16}, x = {2, 4, 7, 16}, and y = 2*T_x - 2*T_1 = {4, 18, 54, 270}.
For k = 1, z = 6, then n = {2, 18}, x = {3, 20}, and y = 2*T_x - S_3 = {3, 411}.
For k = 1, z = 7, then n = {5, 7, 8, 21}, x = {4, 5, 6, 36}, and y = 2*T_x - 2*T_3 = {8, 18, 30, 1320}.
For k = 1, z = 7, then n = {6, 7, 12, 22}, x = {0, 2, 6, 47}, and y = 2*T_x + 2*T_3 = {12, 18, 54, 2268}.
For the special case where k = 1 and z = 0 or 1, then all associated x values follow form (A216162(n) - A216162(n - 2)) [type 1] or (A216162(n) - A216162(n - 1)) [type II] for some n in N. Type II x values = {1, 2, 5, 90} (= A215797(n+1)) are associated with the positive Ramanujan-Nagell triangular numbers = {1, 3, 15, 4095} (= A076046(n+1)) by the formula 1/2*(x^2 + x) = T_x.

Examples

			Lambda_6/6 = 72/6 = 12, so 6 is in this sequence.
Lambda_12/12 = 648/12 = 54, so 12 is in this sequence.
Lambda_18/18 = 7398/18 = 411, so 18 is in this sequence.
Lambda_24/24 = 196560/24 = 8190, so 24 is in this sequence.
But...
Lambda_19/19 = 10668/19 = 561.47368..., so 19 is not in this sequence.

Crossrefs

Cf. A222785, A222786, A002336, A216162

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