A066771 -id:A066771 - cp's OEIS frontend

A066770 a(n) = 5^nsin(2narctan(1/2)) or numerator of tan(2n*arctan(1/2)).

4, 24, 44, -336, -3116, -10296, 16124, 354144, 1721764, 1476984, -34182196, -242017776, -597551756, 2465133864, 29729597084, 116749235904, -42744511676, -3175197967656, -17982575014036, -28515500892816, 278471369994004, 2383715742284424, 7340510203856444

Offset: 1

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

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 (6,-25).

Cf. A066771, A000351 powers of 5 and also hypotenuse of right triangle with legs given by A066770 and A066771.

Note that A066770, A066771 and A000351 are primitive Pythagorean triples with hypotenuse 5^n. The offset of A000351 is zero, but the offset is 1 for A066770, A066771.

a[1] := 4/3; for n from 1 to 40 do a[n+1] := (4/3+a[n])/(1-4/3*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(1/2))

Table[ 5^n*Sin[2*n*ArcCot[2]] // Simplify, {n, 1, 23}] (* Jean-François Alcover, Mar 04 2013 *)

PARI
```
a(n)=imag((2+I)^(2*n))
```

G.f.: 4*x/(1-6*x+25*x^2). - Ralf Stephan, Jun 12 2003
a(n) = 5^n*sin(2*n*arctan(1/2)). A recursive formula for T(n) = tan(2*n*arctan(1/2)) is T(n+1) = (4/3+T(n))/(1-4/3*T(n)). Unsigned a(n) is the absolute value of numerator of T(n).
a(n) is the imaginary part of (2+I)^(2*n) = Sum_{k=0..n} 2^(2*n-2*k-1)*(-1)^k*binomial(2*n, 2*k+1). - Benoit Cloitre, Aug 03 2002
a(n) = 6*a(n-1)-25*a(n-2), n>2. - Gary Detlefs, Dec 11 2010
a(n) = 5^n*sin(n*x), where x = arcsin(4/5) = 0.927295218.. . - Gary Detlefs, Dec 11 2010

A139030 Real part of (4 + 3i)^n.

Original entry on oeis.org

1, 4, 7, -44, -527, -3116, -11753, -16124, 164833, 1721764, 9653287, 34182196, 32125393, -597551756, -5583548873, -29729597084, -98248054847, -42744511676, 2114245277767, 17982575014036, 91004468168113, 278471369994004, -47340744250793, -7340510203856444, -57540563024581727

Offset: 0

Gary W. Adamson, Apr 06 2008

sqrt (a(n)^2 + (A139031(n))^2) = 5^n. Example: a(3) = -44, A139031(3) = 117. Sqrt (-44^2 + 117^2) = 5^3.

If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 11 divides a(6n+3) for n >= 0; 31 divides a(8n+4) for n>= 0. See the Renault paper in Links. - Clark Kimberling, Oct 02 2024

			a(5) = -3116 since (4 + 3i)^5 = (-3116 - 237i) where -237 = A139031(5).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marc Renault, The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m, Math. Mag. 86 (2013) 372-380.
Index entries for linear recurrences with constant coefficients, signature (8,-25).

Cf. A139031, A066771, A066770, A374880, A376283, A376455.

a:= n-> Re((4+3*I)^n):
seq(a(n), n=0..24);  # Alois P. Heinz, Oct 15 2024

Re[(4+3I)^Range[40]] (* or *) LinearRecurrence[{8,-25},{4,7},40] (* Harvey P. Dale, Nov 09 2011 *)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
{a, b, c} = {3, 4, 5};
Table[TrigExpand[5^n Cos[n (A[b, c, a] - A[c, a, b])]], {n, 0, 50}] (* Clark Kimberling, Oct 02 2024 *)

Real part of (4 + 3i)^n. Term (1,1) of [4,-3; 3,4]^n. a(n), n>=2 = 8*a(n-1) - 25*a(n-2), given a(0) = 1, a(1) = 4. Odd-indexed terms of A066770 interleaved with even-indexed terms of A066771, irrespective of sign.
G.f.: (1-4*x) / ( 1-8*x+25*x^2 ). - R. J. Mathar, Feb 05 2011
a(n) = 5^n * cos(nB-nC), where B is the angle opposite side CA and C is the angle opposite side AB in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle. - Clark Kimberling, Oct 02 2024
E.g.f.: exp(4*x)*cos(3*x). - Stefano Spezia, Oct 03 2024

More terms from Harvey P. Dale, Nov 09 2011

a(0)=1 prepended by Alois P. Heinz, Oct 15 2024

A121622 Real part of (3 + 2i)^n.

Original entry on oeis.org

1, 3, 5, -9, -119, -597, -2035, -4449, -239, 56403, 341525, 1315911, 3455641, 3627003, -23161315, -186118929, -815616479, -2474152797, -4241902555, 6712571031, 95420159401, 485257533003, 1671083125805, 3718150825791, 584824319281

Offset: 0

Gary W. Adamson and Nick Williams, Aug 10 2006

Companion sequence A121621 is real((2 + 3i)^n).

			a(5) = -597 since (3 + 2i)^5 = (-597 + 122i).
a(5) = -597 = 6*(-119) - 13*(-9) = 6*a(5) -13*a(4).

Michael De Vlieger, Table of n, a(n) for n = 0..500
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (6,-13).

Cf. A121621.

Cf. A193410, A066771.

f[n_] := Re[(3 + 2I)^n]; Table[f[n], {n, 0, 24}] (* Robert G. Wilson v, Aug 17 2006 *)
LinearRecurrence[{6,-13},{1,3},30] (* Harvey P. Dale, Apr 24 2017 *)

a(n) = real((3 + 2*I)^n); \\ Michel Marcus, Jun 12 2021

a(n) = real((3 + 2i)^n).
a(n) = 6*a(n-1) - 13*a(n-2).
G.f.: ( 1-3*x ) / ( 1-6*x+13*x^2 ). - R. J. Mathar, Aug 12 2012
E.g.f.: exp(3*x)*cos(2*x). - Sergei N. Gladkovskii, Jan 20 2014

More terms from Robert G. Wilson v, Aug 17 2006

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

Original entry on oeis.org

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

A067358 Imaginary part of (5+12i)^n.

Original entry on oeis.org

0, 12, 120, -828, -28560, -145668, 3369960, 58317492, 13651680, -9719139348, -99498527400, 647549275812, 23290743888720, 123471611274972, -2701419604443960, -47880898349909868, -22269070348069440, 7869181117654073292, 82455284065364468280, -505338768229893703548

Offset: 0

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

Also 13^n sin(2n arctan(2/3)) or numerator of tan(2n arctan(2/3)).

Note that a(n), A067359(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. A067359 (13^n cos(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(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

Im[(5 + 12*I)^Range[0, 24]] (* or *)
LinearRecurrence[{10, -169}, {0, 12}, 25] (* Paolo Xausa, Apr 22 2024 *)

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

G.f.: 12*x/(1-10*x+169*x^2). a(n)=10*a(n-1)-169*a(n-2). - Michael Somos, Jun 27 2002

Better description from Michael Somos, Jun 27 2002

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

A094423 a(n) = A045873(n)^2.

Original entry on oeis.org

1, 4, 1, 144, 361, 484, 19321, 28224, 128881, 2427364, 1745041, 26501904, 285643801, 64995844, 4675961161, 31354493184, 149793121, 741117817924, 3178942795681, 545370434064, 107989070784841, 292105630845604

Offset: 1

Ralf Stephan, May 04 2004

The g.f. is an example of a rational function with nonnegative integer coefficients that is not N-rational.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Ira M. Gessel, Rational Functions With Nonnegative Integer Coefficients, slides, 50th Séminaire Lotharingien de Combinatoire, 2003.
Index entries for linear recurrences with constant coefficients, signature (-1,5,125).

Cf. A045873, A066771, A250102.

I:=[1,4,1]; [n le 3 select I[n] else -Self(n-1) +5*Self(n-2) +125*Self(n-3): n in [1..41]]; // G. C. Greubel, Jan 11 2024

LinearRecurrence[{-1,5,125}, {1,4,1}, 40] (* G. C. Greubel, Jan 11 2024 *)

Vec((x+5*x^2)/(1+x-5*x^2-125*x^3) + O(x^30)) \\ Michel Marcus, Aug 28 2015

@CachedFunction
def a(n): # a = A094423
    if (n<4): return (0,1,4,1)[n]
    else: return -a(n-1) + 5*a(n-2) + 125*a(n-3)
[a(n) for n in range(1,41)] # G. C. Greubel, Jan 11 2024

G.f.: x*(1+5*x)/(1+x-5*x^2-125*x^3).
a(n) = A250102(n)/16.
a(n) = (1/16)*( 2*5^n - (1+2*i)^(2*n) - (1-2*i)^(2*n) ) = (1/4)*( Im(1+2*i)^n )^2 = (1/4) * 5^n * sin(n*arctan(2))^2.
From G. C. Greubel, Jan 11 2024: (Start)
a(n) = (1/8)*5^n*(1 - ChebyshevU(n, -3/5) - (3/5)*ChebyshevU(n-1, -3/5)).
a(n) = (1/8)*( 5^n - (-1)^n*A066771(n) ).
E.g.f.: (1/8)*exp(-3*x)*(exp(8*x) - cos(4*x)). (End)

A193410 Expansion of (1-3x)/(1-6x+18*x^2).

Original entry on oeis.org

1, 3, 0, -54, -324, -972, 0, 17496, 104976, 314928, 0, -5668704, -34012224, -102036672, 0, 1836660096, 11019960576, 33059881728, 0, -595077871104, -3570467226624, -10711401679872, 0, 192805230237696, 1156831381426176, 3470494144278528

Offset: 0

Bruno Berselli, Aug 04 2011

Also real parts of 3^n*(1+i)^n, where i=sqrt(-1).

If |a(n)| > 0 then it is in A130505.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-18).

Cf. A066771, A121622, A130505.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)/(1-6*x+18*x^2))); /* or */ &cat[[r,3*r,0,-54*r] where r is (-324)^n: n in [0..6]];

I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)-18*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 26 2013

CoefficientList[Series[(1 - 3 x)/(1 - 6 x + 18 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{6,-18},{1,3},40] (* Harvey P. Dale, Jul 27 2021 *)

makelist(coeff(taylor((1-3*x)/(1-6*x+18*x^2), x, 0, n), x, n), n, 0, 25);

PARI
```
Vec((1-3*x)/(1-6*x+18*x^2) +O(x^26))
```

G.f.: (1-3*x)/(1-6*x+18*x^2).
a(n) = 3^n*A146559(n) = (1/2)*((3+3*i)^n+(3-3*i)^n), where i=sqrt(-1).
a(n) = 6*a(n-1)-18*a(n-2) for n>1.
a(n) = (3*sqrt(2))^n*cos(pi*n/4).
a(4k+2) = 0, a(4k+1) = 3*a(4k) = 18*a(4k-1) = 3*(-324)^k.
G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(3*k+3)/(x*(3*k+6) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

A093378 a(n) = numerator of any non-diagonal entry of the matrix A^n, where A is described in the Comments lines.

Original entry on oeis.org

2, 12, 22, 168, 1558, 5148, 8062, 177072, 860882, 738492, 121008888, 298775878, 1232566932, 14864798542, 58374617952

Offset: 1

Simone Severini, Apr 28 2004

A =
  [ -3/5 -(2/5)i,      -(2/5)i,      -(2/5)i,      -(2/5)i ]
  [      -(2/5)i, -3/5 +(2/5)i,      -(2/5)i,       (2/5)i ]
  [      -(2/5)i,      -(2/5)i, -3/5 +(2/5)i,       (2/5)i ]
  [      -(2/5)i,       (2/5)i,       (2/5)i, -3/5 -(2/5)i ]
is the Cayley transform of the matrix iH, where H =
  [1,  1,  1,  1]
  [1, -1,  1, -1]
  [1,  1, -1, -1]
  [1, -1, -1,  1]
is a Hadamard matrix of order 4 and i is the imaginary unit.

Cf. A066771.

cp's OEIS Frontend

A066770 a(n) = 5^n*sin(2n*arctan(1/2)) or numerator of tan(2n*arctan(1/2)).

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A139030 Real part of (4 + 3i)^n.

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A121622 Real part of (3 + 2i)^n.

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

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

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Maple

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A067358 Imaginary part of (5+12i)^n.

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Maple

Mathematica

PARI

Formula

Extensions

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

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Author

A066770 a(n) = 5^nsin(2narctan(1/2)) or numerator of tan(2n*arctan(1/2)).

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

A193410 Expansion of (1-3x)/(1-6x+18*x^2).