A099456 -id:A099456 - cp's OEIS frontend

A099457 A Chebyshev transform of A099456 associated to the knot 9_44.

1, 4, 10, 16, 9, -40, -169, -376, -490, 36, 2239, 7120, 13441, 12844, -16470, -109144, -283351, -448120, -229129, 1196064, 4879030, 10675276, 13561279, -2161760, -65753919, -204313516, -379184950, -347399104, 513198089

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 9_44. The g.f. is the image of the g.f. of A099456 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (4,-7,4,-1).

Cf. A099456, A099458.

LinearRecurrence[{4,-7,4,-1},{1,4,10,16},30] (* Harvey P. Dale, Jan 17 2024 *)

G.f.: (1+x^2)/(1-4*x+7*x^2-4*x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(Sum_{j=0..n-2*k} C(n-2*k-j, j)*(-5)^j*4^(n-2*k-2*j)).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099456(n-2*k).
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099456(k)/2.
a(n) = Sum_{k=0..n} A099458(n-k)*(1+(-1)^k)/2.

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882

Offset: 0

Gary W. Adamson, Apr 05 2008

Imaginary part of (2 + i)^n gives A099456.
Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.
Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).

Seiichi Manyama, Table of n, a(n) for n = 0..2862 (first 201 terms from Vincenzo Librandi)
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 (4,-5).

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

[ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011

restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009

Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)

a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */

Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

[lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Real part of (2 + i)^n, i^2 = -1.
Term (1,1) of matrix [2,-1; 1,2]^n.
(a(n))^2 + (A099456(n))^2 = 5^n.
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: (1-2x) /(1-4x+5x^2).
a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)
E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009
a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010
a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008
2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011
a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009
Added a(0)=1 by Michael Somos, Dec 26 2010
Edited by Franklin T. Adams-Watters, Apr 10 2011

A316658 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the imaginary part of f(n).

Original entry on oeis.org

0, 0, 1, 0, -1, 1, 1, 2, 1, 0, 2, 2, 3, 2, 1, -1, -1, 0, -1, -2, -2, -2, -1, -2, -3, 4, 4, 5, 4, 3, 5, 5, 6, 5, 4, 6, 6, 7, 6, 5, 3, 3, 4, 3, 2, 2, 2, 3, 2, 1, 3, 3, 4, 3, 2, 4, 4, 5, 4, 3, 5, 5, 6, 5, 4, 2, 2, 3, 2, 1, 1, 1, 2, 1, 0, -4, -4, -3, -4, -5, -3

Offset: 0

Rémy Sigrist, Jul 09 2018

See A316657 for the real part of f and additional comments.

Rémy Sigrist, Table of n, a(n) for n = 0..15624

Cf. A099456, A316657.

a[n_] := Module[{d, z}, d = IntegerDigits[n, 5] // Reverse; z = Sum[ If[d[[i]]>0, (2+I)^(i-1)*I^(d[[i]]-1), 0], {i, 1, Length[d]}]; Im[z]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2021, after PARI code *)

a(n) = my (d=Vecrev(digits(n, 5)), z=sum(i=1, #d, if (d[i], (2+I)^(i-1) * I^(d[i]-1), 0))); imag(z)

a(5^n) = A099456(n-1) for any n > 0.

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

Original entry on oeis.org

0, 1, 1, 0, -2, -4, -1, -1, 3, 11, 0, 2, 4, -2, -24, 1, 1, -3, -11, -7, 41, 0, -2, -4, 2, 24, 38, -44, -1, -1, 3, 11, 7, -41, -117, -29, 0, 2, 4, -2, -24, -38, 44, 278, 336, 1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 0, -2, -4, 2, 24, 38, -44, -278, -336, 718

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
   0;
   1,  1;
   0, -2, -4;
  -1, -1,  3,  11;
   0,  2,  4,  -2, -24;
   1,  1, -3, -11,  -7,  41;
   0, -2, -4,   2,  24,  38,  -44;
  -1, -1,  3,  11,   7, -41, -117, -29;
   0,  2,  4,  -2, -24, -38,   44, 278, 336.

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is related to A099456.
The next diagonal of the triangle is related to A139011.
Cf. A006495, A006496.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), this sequence (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).

T(n+1,n)^2 + T(n,n)^2 = 5^n.

A099458 An Alexander sequence for the knot 9_44.

Original entry on oeis.org

1, 4, 9, 12, -1, -56, -178, -336, -321, 412, 2729, 7084, 11202, 5724, -29911, -121988, -266881, -338976, 54222, 1644184, 5108159, 9479212, 8682249, -12837036, -79315198, -202151756, -313431031, -143085588, 892383039

Offset: 0

Paul Barry, Oct 16 2004

The denominator 1-4x+7x^2-4x^3+x^4 is a parameterization of the Alexander polynomial for the knot 9_44. 1/(1-4x+7x^2-4x^3+x^4) is the image of the g.f. of A099456 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Dror Bar-Natan, The knot 9_44 in the Rolfsen Knot Table at the Knot Atlas

G.f.: (1-x)*(1+x)*(1+x^2)/(1-4*x+7*x^2-4*x^3+x^4); a(n)=A099457(n)-A099457(n-2).

A185457 a(n) = abs( Im((2+i)^(2^n)) ).

Original entry on oeis.org

1, 4, 24, 336, 354144, 116749235904, 22940770664883067253376, 182503181432559739767250904458105698387204864

Offset: 0

Carmine Suriano, Feb 04 2011

The next term is too large to be displayed here.

Old name was: Leg of primitive Pythagorean triangle generated by repeated application of the basic formula y(n) = 2*x(n-1)*y(n-1), x(1)=2, y(1)=1.

			y(2)=24 since x(1)=3, y(1)=4 are the two legs of Pythagorean triangle obtained by p=2, q=1; second iteration p=3, q=4 gives 2*3*4=24.

Alois P. Heinz, Table of n, a(n) for n = 0..11

Cf. A176594, A120905.

Cf. A099456 ( imaginary part of (2+i)^n ).

a:= n-> abs(Im((2+I)^(2^n))):
seq(a(n), n=0..8);  # Alois P. Heinz, Apr 25 2013

Table[Abs[Im[(2 + I)^(2^n)]], {n, 0, 10}] (* G. C. Greubel, Jul 07 2017 *)

a(n) = abs(imag((2+I)^(2^n))); \\ Joerg Arndt, Apr 25 2013

from sympy import im, I
def a(n): return abs(im((2 + I)**(2**n)))
print([a(n) for n in range(11)]) # Indranil Ghosh, Jul 08 2017

a(n) = abs( Im((2+i)^(2^n)) ).

Better name from Joerg Arndt, Apr 25 2013

A102486 a(n) = 4a(n-1) - 5a(n-2).

Original entry on oeis.org

2, 6, 14, 26, 34, 6, -146, -614, -1726, -3834, -6706, -7654, 2914, 49926, 185134, 490906, 1037954, 1697286, 1599374, -2088934, -16352606, -54965754, -138099986, -277571174, -419784766, -291283194, 933791054, 5191580186, 16097365474, 38431560966, 73239416494, 100799861146

Offset: 0

N. J. A. Sloane, Feb 25 2005

Inverse binomial transform is 2,4,4,0,-8,-16,-16,.. essentially -A146559(n+3). - R. J. Mathar, Apr 07 2022

B. M. E. Moret and H. D. Shapiro, Algorithms from P to NP, Benjamin/Cummings, Vol. 1, 1991; p. 65.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5).

Cf. A099456.

I:=[2, 6]; [n le 2 select I[n] else 4*Self(n-1)-5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jan 15 2012

a := proc(n) option remember; if n = 0 then RETURN(2) end if; if n = 1 then RETURN(6) end if; 4*a(n - 1) - 5*a(n - 2); end proc;

Column[LinearRecurrence[{4,-5},{2,6},40]] (* Vincenzo Librandi, Jan 15 2012 *)

Vec(2*(1-x)/(1-4*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 15 2012

G.f.: 2*(1-x)/(1-4*x+5*x^2). [Colin Barker, Jan 14 2012]

A090133 Expansion of (1+4x)/(1+4x+5x^2).

Original entry on oeis.org

1, 0, -5, 20, -55, 120, -205, 220, 145, -1680, 5995, -15580, 32345, -51480, 44195, 80620, -543455, 1770720, -4365605, 8608820, -12607255, 7384920, 33496595, -170910980, 516160945, -1210088880, 2259550795, -2987758780, 653281145, 12325669320

Offset: 0

Paul Barry, Nov 21 2003

D. Cvijovic, J Klinowski, An Application for the Chebyshev Polynomials, Mat. Vesnik 50 (1998) 105-110

D. Cvijovic and J. Klinowski, An Application for the Chebyshev Polynomials, Mat. Vesnik 50 (1998) 105-110. [broken link]
Index entries for linear recurrences with constant coefficients, signature (-4,-5).

Cf. A099456.

a(n) = Sum_{k=0..n} C(n, k)*(-2)^(n-k)*(-1)^floor(k/2)*(1+(1-(-1)^k)/2).

a(n+1) = (-1)^n * (4*A099456(n) - A099456(n+1)). - Ralf Stephan, Jul 19 2013

Better name from Ralf Stephan, Jul 14 2013

cp's OEIS Frontend

A099457 A Chebyshev transform of A099456 associated to the knot 9_44.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A316658 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the imaginary part of f(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A292495 Triangle read by rows: T(n,k) = (-2)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-1)^m.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A099458 An Alexander sequence for the knot 9_44.

Views

Author

Keywords

Comments

Links

Formula

A185457 a(n) = abs( Im((2+i)^(2^n)) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

A102486 a(n) = 4a(n-1) - 5a(n-2).