A078069 -id:A078069 - cp's OEIS frontend

A090131 Expansion of (1+x)/(1 - 2x + 2x^2).

1, 3, 4, 2, -4, -12, -16, -8, 16, 48, 64, 32, -64, -192, -256, -128, 256, 768, 1024, 512, -1024, -3072, -4096, -2048, 4096, 12288, 16384, 8192, -16384, -49152, -65536, -32768, 65536, 196608, 262144, 131072, -262144, -786432, -1048576, -524288, 1048576, 3145728, 4194304, 2097152, -4194304

Offset: 0

Paul Barry, Nov 21 2003

Also first of two associated sequences a(n) and b(n) built from a(0)=1 and b(0)=2 by the formulas a(n) = a(n-1) + b(n-1) and b(n) = -a(n-1) + b(n-1). The initial terms of the second sequence b(n) are 2, 1, -2, -6, -8, -4, 8, 24, 32, 16, -32, -96, -128, -64, 128, 384, 512, 256, -1536, -2048, -1024, 2048, 6144, 8192, .... The formula for b(n) is the same as for a(n) but replacing cosines with sines. Indeed in the complex plane the points Mn=a(n)+b(n)*I are located where the logarithmic spiral Rho=A*(B^Theta) cuts the two pairs of orthogonal straight lines drawn from the origin with slopes 2, 1/3, -1/2 and -3. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007

Index entries for linear recurrences with constant coefficients, signature (2,-2)

Cf. A078069.

a[n_]:=(MatrixPower[{{1,-1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

a(n) = sum_{k=0..n} C(n, k)(-1)^floor(k/2)(1 + (1 - (-1)^k)/2).
a(n) = A*(B^Theta(n))*cos(Theta(n)) where A = 3.644691771.. = (5^0,5)*16^(arctan(2)/(2*Pi)) B = 0.64321824.. = 16^(-1/(2*Pi)) Theta(4p+1) = p*Pi + arctan(2) Theta(4*p+2) = p*Pi + arctan(1/3) Theta(4*p+3) = p*Pi + arctan(-1/2) Theta(4*p+4) = p*Pi + arctan(-3). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007
Also a(0)=1, a(1)=3, a(2)=4, a(3)=2 and for n>3 a(n) = -4 * a(n-4). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007
a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3). - Paul Curtz, Nov 20 2007
a(n) = 2a(n-1) - 2a(n-2) = A009545(n) + A009545(n+1) = (1/2)*((1+2*i)*(1-i)^n + (1-2*i)*(1+i)^n). - Ralf Stephan, Jul 21 2013

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

Original entry on oeis.org

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

Offset: 0

Philippe Deléham, Feb 24 2012

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
0, 3
0, 1, 5
0, 0, 5, 7
0, 0, 1, 14, 9
0, 0, 0, 7, 30, 11
0, 0, 0, 1, 27, 55, 13
0, 0, 0, 0, 9, 77, 91, 15
0, 0, 0, 0, 1, 44, 182, 140, 17
0, 0, 0, 0, 0, 11, 156, 378, 204, 19
0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # Peter Luschny, Aug 17 2016

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.
G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.
There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

New name using a formula of the author from Peter Luschny, Aug 17 2016

A134812 a(n) = 2a(n-2) + 4a(n-3), n >= 3.

Original entry on oeis.org

0, 1, -1, 1, 2, -2, 8, 4, 8, 40, 32, 112, 224, 352, 896, 1600, 3200, 6784, 12800, 26368, 52736, 103936, 210944, 418816, 837632, 1681408, 3350528, 6713344, 13426688, 26828800, 53706752, 107364352, 214728704, 429555712, 858914816, 1718026240, 3436052480, 6871711744, 13744209920

Offset: 0

Paul Curtz, Jan 28 2008

Joshua Zucker, Table of n, a(n) for n = 0..60
Index entries for linear recurrences with constant coefficients, signature (0,2,4).

Cf. A038521.

CoefficientList[Series[x*(-1 + x + x^2) / ( (2*x-1)*(2*x^2 + 2*x + 1) ),{x,0,38}],x] (* James C. McMahon, Apr 11 2025 *)

G.f.: x*(-1 + x + x^2) / ( (2*x-1)*(2*x^2 + 2*x + 1) ). - R. J. Mathar, Aug 11 2012

20*a(n) = 2^n - 6*A078069(n), n>0. - R. J. Mathar, Aug 11 2012

More terms from Joshua Zucker, Feb 23 2008

A176739 Expansion of 1/(1-2x^2-4x^3). (2,4)-Padovan sequence.

Original entry on oeis.org

1, 0, 2, 4, 4, 16, 24, 48, 112, 192, 416, 832, 1600, 3328, 6528, 13056, 26368, 52224, 104960, 209920, 418816, 839680, 1677312, 3354624, 6713344, 13418496, 26845184, 53690368, 107364352, 214761472, 429490176, 858980352, 1718026240, 3435921408, 6871973888

Offset: 0

Wolfdieter Lang, Jul 14 2010

See A000931 (Padovan), and the W. Lang link given there for a combinatorial interpretation and an explicit form.

a(n)/2^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 2's and 3's. The limiting ratio is 2/5. - Bob Selcoe, Jul 12 2013

			Combinatorics for (A,B)=(2,4) Padovan sequence with weighted (3,2)-Morse type code (see the W. Lang link under A000931): n=5, - -- and -- -, with weights 2^1*4^1 and 4^1*2^1, respectively, adding to 2*2*4=16=a(5).

Index entries for linear recurrences with constant coefficients, signature (0,2,4).

Cf. A000931, A108520.

seq(2^(n+1)/5 + Re((3-I)*(-1-I)^n)/5, n=0..100); # Robert Israel, Aug 26 2014

CoefficientList[Series[1/(1 - 2*x^2 - 4*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 26 2014 *)

O.g.f.: 1/((1+2*x+2*x^2)*(1-2*x)) = ((3+2*x)/(1+2*x+2*x^2) + 2/(1-2*x))/5.
a(n) = (3*b(n) + 2*b(n-1) + 2^(n+1))/5, with b(n):=A108520(n), and b(-1)=0.
a(n) = 2*a(n-2) + 4*a(n-3). - Bob Selcoe, Aug 26 2014
a(n) = 2^(n+1)/5 + Re((3-i)*(-1-i)^n)/5. - Robert Israel, Aug 26 2014
5*a(n) = 2^(n+1) -A078069(n+1). - R. J. Mathar, May 14 2024

A340670 Number of complex base i-1 points which can be represented within n bits and negated within those n bits.

Original entry on oeis.org

1, 1, 1, 3, 5, 15, 29, 47, 101, 199, 413, 847, 1621, 3255, 6541, 13087, 26373, 52423, 104637, 209711, 419253, 839511, 1678317, 3353919, 6710629, 13421287, 26845213, 53693007, 107366933, 214742391, 429498701, 858994271, 1718023109, 3435955975, 6871883645

Offset: 0

Kevin Ryde, Jan 15 2021

Complex base i-1 of Khmelnik and Penney uses an integer m to represent a complex integer point z(m) = A318438(m) + A318439(m)*i.  A340669(m) is the negation of z in this representation.  a(n) is how many n-bit m are negatable within those n bits, i.e., how many m in the range 0 <= m < 2^n have also 0 <= A340669(m) < 2^n.
A geometric interpretation of a(n) is to draw a unit square around each point z(0) to z(2^n-1), rotate a copy by 180 degrees about the origin, and measure its area of intersection with the original.
The bit-flip rule in A340669 gives the recurrence formula below.  A low 0-bit of m has a(n-1) negatables above it, or low 11 is one arbitrary bit then negatables above so 2*a(n-3), or low 01 is three arbitrary so 8*a(n-5).  This can be thought of the number of compositions of n (partitions with order) into parts 1,3,5 with 2 types of part 3 and 8 types of part 5.

			For n=3, the a(3)=3 points of n bits are m = 0,3,7 < 2^n, which negate to A340669(0,3,7) = 0,7,3 < 2^n.  These m are located at z = 0,i,-i,
               negate        intersection
  z(0..7)    (rotate 180)   a(3) = 3 points
                * *
   * *            * *             *
     o *        * o               o
   * *            * *             *
     * *

Kevin Ryde, Table of n, a(n) for n = 0..700
Kevin Ryde, Iterations of the Dragon Curve, see index MinusNegA.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,8).

Cf. A340669, A107920, A078069.

{ my(table=[4,-2,-2,6, -4,2,2,-6], p=Mod('x,2-'x+'x^2));
a(n) = (6<

a(n) = a(n-1) + 2*a(n-3) + 8*a(n-5).
a(n) = (2/5)*2^n + (h/15)*2^floor(n/2) + (2/3)*Im((1/2 + i*sqrt(7)/2)^(n+1)) where h = 4,-2,-2,6, -4,2,2,-6 according as n == 0 to 7 (mod 8) respectively.
a(n) = (2/5)*2^n + (1/3)*A107920(n+1) + (1/15)*A078069(n+2).
G.f.: 1/(1 - x - 2*x^3 - 8*x^5).
G.f.: (2/5)/(1-2*x) + (1/3)/(1-x+2*x^2) + (2/15)*(2+3*x)/(1+2*x+2*x^2).

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).

Original entry on oeis.org

-1, 1, 2, 5, 4, 1, -8, -15, -16, 1, 32, 65, 64, 1, -128, -255, -256, 1, 512, 1025, 1024, 1, -2048, -4095, -4096, 1, 8192, 16385, 16384, 1, -32768, -65535, -65536, 1, 131072, 262145, 262144, 1, -524288, -1048575, -1048576, 1, 2097152, 4194305, 4194304, 1, -8388608, -16777215, -16777216, 1, 33554432

Offset: 0

Creighton Dement, May 13 2005

Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,2).

Cf. A000035, A010673, A078069, A090131, A099087, A146559.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // G. C. Greubel, Sep 08 2021

CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* Harvey P. Dale, Mar 20 2013 *)

def A106664_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
A106664_list(50) # G. C. Greubel, Sep 08 2021

a(n) = (1/2)*(A010673(n) - A099087(n+2)).
a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - Ralf Stephan, Nov 16 2010
From G. C. Greubel, Sep 08 2021: (Start)
a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).
a(n) = A000035(n) - A146559(n).
E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)

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A090131 Expansion of (1+x)/(1 - 2*x + 2*x^2).

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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

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A134812 a(n) = 2a(n-2) + 4a(n-3), n >= 3.

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A176739 Expansion of 1/(1-2*x^2-4*x^3). (2,4)-Padovan sequence.

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A340670 Number of complex base i-1 points which can be represented within n bits and negated within those n bits.

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A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

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A090131 Expansion of (1+x)/(1 - 2x + 2x^2).

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

A176739 Expansion of 1/(1-2x^2-4x^3). (2,4)-Padovan sequence.

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).