A067358 -id:A067358 - cp's OEIS frontend

A067360 a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).

8, 240, 4888, 77280, 905768, 4839120, -116593352, -4896306240, -113193708472, -1980778750800, -26710380775592, -228866364286560, 853309115549288, 91741652745294480, 2505643247965090168, 48655959795562600320, 735547895204966951048

Offset: 1

Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Jan 17 2002

Note that A067360(n), A067361(n) and 17^n are primitive Pythagorean triples with hypotenuse 17^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (30, -289).

Cf. A067361 (17^n cos(2n arctan(1/4))).

Cf. A066770, A066771, A067358, A067359, A020888, A014498, A020892.

a[1] := 8/15; for n from 1 to 40 do a[n+1] := (8/15+a[n])/(1-8/15*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(1/4))

Table[Tan[2n ArcTan[1/4]] // TrigToExp // Simplify // Numerator, {n, 1, 17} ] (* Jean-François Alcover, Jul 25 2017 *)

a(n) = 17^n sin(2n arctan(1/4)). A recursive formula for T(n) = tan(2n arctan(1/4)) is T(n+1)=(8/15+T(n))/(1-8/15*T(n)). Unsigned a(n) is the absolute value of numerator of T(n).
Conjectures from Colin Barker, Jul 25 2017: (Start)
G.f.: 8*x / (1 - 30*x + 289*x^2).
a(n) = i*((15 - 8*i)^n - (15 + 8*i)^n)/2 where i=sqrt(-1).
a(n) = 30*a(n-1) - 289*a(n-2) for n>2.
(End)

A067361 a(n) = 17^ncos(2n*arctan(1/4)) or denominator of tan(2narctan(1/4)).

Original entry on oeis.org

15, 161, 495, -31679, -1093425, -23647519, -393425745, -4968639359, -35359140465, 375162560801, 21473668418415, 535788072480961, 9867752001506895, 141189807098209121, 1383913884510780975, 713562283940993281, -378544244105385903345

Offset: 1

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Note that A067360(n), A067361(n) and 17^n are primitive Pythagorean triples with hypotenuse 17^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (30, -289).

Cf. A067360 (17^n sin(2n arctan(1/4))).

Cf. A066770, A066771, A067358, A067359, A020888, A014498, A020892.

a[1] := 8/15; for n from 1 to 40 do a[n+1] := (8/15+a[n])/(1-8/15*a[n]):od: seq(abs(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(1/4))

Table[t = Tan[2 n ArcTan[1/4]] // TrigToExp // Simplify; Sign[t] * Denominator[t], {n, 1, 17}] (* Jean-François Alcover, Jul 25 2017 *)

a(n) = 17^n*cos(2*n*arctan(1/4)).
A recursive formula for T(n) = tan(2*n*arctan(1/4)) is T(n+1) = (8/15+T(n))/(1-8/15*T(n)). Unsigned a(n) is the absolute value of denominator of T(n). [And a(n) = 17^n*cos(n*arctan(8/15)). - Peter Luschny, Sep 29 2019]
From Colin Barker, Jul 25 2017: (Start)
G.f.: x*(15 - 289*x) / (1 - 30*x + 289*x^2).
a(n) = ((15 - 8*i)^n + (15 + 8*i)^n)/2 where i=sqrt(-1).
a(n) = 30*a(n-1) - 289*a(n-2) for n>2. (End)
a(n) = Re((8 + 15*i)^n) = Re((4 + i)^(2*n)) = (1/2)*V(2*n,P = 8,Q = 17), where V(n,P,Q) denotes the Lucas sequence of the second kind and i=sqrt(-1). - Peter Bala, Sep 24 2019

A067359 Real part of (5 + 12i)^n.

Original entry on oeis.org

1, 5, -119, -2035, -239, 341525, 3455641, -23161315, -815616479, -4241902555, 95420159401, 1671083125805, 584824319281, -276564805068235, -2864483360640839, 18094618450123325, 665043872449535041, 3592448206424508485, -76467932379726337079, -1371803070683005304755

Offset: 1

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Also 13^n*cos(2*n*arctan(2/3)) or denominator of tan(2*n*arctan(2/3)).

Note that A067358(n), a(n) and 13^n are primitive Pythagorean triples with hypotenuse 13^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (10, -169).

Cf. A067358 (13^n sin(2n arctan(2/3))).

Cf. A066770, A066771, A067360, A067361, A020888, A014498, A020892.

a[1] := 12/5; for n from 1 to 40 do a[n+1] := (12/5+a[n])/(1-12/5*a[n]):od: seq(abs(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

Table[Re[(5+12I)^n],{n,0,20}] (* Harvey P. Dale, Aug 24 2014 *)

PARI
```
a(n)=real((5+12*I)^n)
```

From Michael Somos, Jun 27 2002: (Start)
G.f.: (1-5*x)/(1-10*x+169*x^2).
a(n) = 10*a(n-1) - 169*a(n-2). (End)

Better description from Michael Somos, Jun 27 2002

cp's OEIS Frontend

A067360 a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A067361 a(n) = 17^ncos(2n*arctan(1/4)) or denominator of tan(2narctan(1/4)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A067359 Real part of (5 + 12i)^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

cp's OEIS Frontend

A067360 a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A067361 a(n) = 17^n*cos(2*n*arctan(1/4)) or denominator of tan(2*n*arctan(1/4)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A067359 Real part of (5 + 12i)^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A067361 a(n) = 17^ncos(2n*arctan(1/4)) or denominator of tan(2narctan(1/4)).