A108520 -id:A108520 - cp's OEIS frontend

A134142 List of quadruples: 2(-4)^n, -3(-4)^n, 2(-4^n), 2(-4)^n, n >= 0.

2, -3, 2, 2, -8, 12, -8, -8, 32, -48, 32, 32, -128, 192, -128, -128, 512, -768, 512, 512, -2048, 3072, -2048, -2048, 8192, -12288, 8192, 8192, -32768, 49152, -32768, -32768, 131072, -196608, 131072, 131072, -524288, 786432, -524288, -524288, 2097152, -3145728, 2097152, 2097152, -8388608

Offset: 0

Paul Curtz, Jan 29 2008

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

Cf. A081147, A038521.

A134142 := proc(n) (-4)^floor(n/4)*op(1+(n mod 4), [2,-3,2,2]) ; end: for n from 0 to 80 do printf("%d ",A134142(n)) ; od: # R. J. Mathar, Feb 05 2008

O.g.f.: (x+2)/(2*x^2+2*x+1). a(n) = 2*A108520(n)+A108520(n-1). - R. J. Mathar, Feb 05 2008
a(n) = (1 - I/2)*(-1 - I)^n + (1 + I/2)*(-1 + I)^n, n>=0. - Taras Goy, Apr 20 2019
a(n) = -2*a(n-1)-2*a(n-2) for n > 1. - Chai Wah Wu, May 19 2025
E.g.f.: exp(-x)*(2*cos(x) - sin(x)). - Stefano Spezia, May 19 2025

More terms from R. J. Mathar, Feb 05 2008

A137429 a(n) = -2a(n-1) - 2a(n-2), with a(0)=1 and a(1)=-4.

Original entry on oeis.org

1, -4, 6, -4, -4, 16, -24, 16, 16, -64, 96, -64, -64, 256, -384, 256, 256, -1024, 1536, -1024, -1024, 4096, -6144, 4096, 4096, -16384, 24576, -16384, -16384, 65536, -98304, 65536, 65536, -262144, 393216, -262144, -262144, 1048576, -1572864, 1048576, 1048576, -4194304, 6291456

Offset: 0

Paul Curtz, Apr 17 2008

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

Cf. A108520, A137444.

LinearRecurrence[{-2,-2},{1,-4},50] (* Harvey P. Dale, Mar 26 2013 *)

For n >= 4, a(n) = -4*a(n-4).
a(n) = (-1)^n*A137444(n) = A108520(n) - 2*A108520(n-1).
G.f.: (1-2*x)/(1 + 2*x + 2*x^2).
a(n) = (1/2 + 3*i/2)*(-1 + i)^n + (1/2 - 3*i/2)*(-1 - i)^n, n >= 0, where i=sqrt(-1). - Taras Goy, Apr 20 2019

A138377 a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2; thereafter a(n) = -4*a(n-4).

Original entry on oeis.org

0, 1, 3, 2, 0, -4, -12, -8, 0, 16, 48, 32, 0, -64, -192, -128, 0, 256, 768, 512, 0, -1024, -3072, -2048, 0, 4096, 12288, 8192, 0, -16384, -49152, -32768, 0, 65536, 196608, 131072, 0, -262144, -786432, -524288, 0, 1048576, 3145728, 2097152, 0, -4194304, -12582912, -8388608, 0, 16777216, 50331648

Offset: 0

Paul Curtz, May 08 2008

First and third differences have only 2^n's.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, -4).

Cf. A138380, A138382.

LinearRecurrence[{0,0,0,-4},{0,1,3,2},60] (* Harvey P. Dale, Mar 19 2012 *)

x='x+O('x^25); Vec(x*(1+x)*(2*x+1)/((1-2*x+2*x^2)*(1+2*x+2*x^2))) \\ G. C. Greubel, Feb 20 2017

From R. J. Mathar, May 09 2008: (Start)
O.g.f.: x*(1+x)*(2*x+1)/((1-2*x+2*x^2)*(1+2*x+2*x^2)).
a(n) = (5*A009545(n) - A108520(n))/4. (End)

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 1, -3, -3, 1, 1, 4, -6, -4, 1, -1, 5, 10, -10, -5, 1, -1, -6, 15, 20, -15, -6, 1, 1, -7, -21, 35, 35, -21, -7, 1, 1, 8, -28, -56, 70, 56, -28, -8, 1, -1, 9, 36, -84, -126, 126, 84, -36, -9, 1, -1, -10, 45, 120, -210, -252, 210, 120, -45, -10, 1, 1, -11, -55, 165, 330, -462, -462, 330, 165, -55, -11, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000984, A001700, A002054, A007318, A009116.

Cf. A090132, A108520, A260192, A333378.

A108086:= func< n,k | (-1)^Floor((n-k+1)/2)*Binomial(n,k) >;
[A108086(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2022

A108086[n_, k_]:= (-1)^(Floor[(n-k+1)/2])*Binomial[n, k];
Table[A108086[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2022 *)

def A108086(n,k): return (-1)^int((n-k+1)/2)*binomial(n,k)
flatten([[A108086(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1), with T(0, 0) = 1.
T(n, k) = (-1)^floor((n-k+1)/2) * A007318(n, k).
From G. C. Greubel, Dec 02 2022: (Start)
T(2*n, n) = (-1)^binomial(n+1,2) * A000984(n).
T(2*n, n+1) = (-1)^binomial(n,2) * A001791(n), n >= 1.
T(2*n, n-1) = (-1)^binomial(n+2,2) * A001791(n).
T(2*n+1, n-1) = (-1)^binomial(n-1,2) * A002054(n).
T(2*n+1, n+1) = (-1)^binomial(n+1,2) * A001700(n+1).
Sum_{k=0..n} T(n, k) = (-1)^n * A090132(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A108520(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A260192(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A333378(n+1). (End)

A130668 Diagonal of A129819.

Original entry on oeis.org

0, 0, 1, -2, 5, -11, 23, -48, 102, -220, 476, -1024, 2184, -4624, 9744, -20480, 42976, -90048, 188352, -393216, 819328, -1704192, 3539200, -7340032, 15203840, -31456256, 65010688, -134217728, 276826112, -570429440, 1174409216

Offset: 0

Paul Curtz, Jun 27 2007

sign

This sequence is connected to A124072. To see this, change the sign of every negative term and consider the differences of every line. Hence for the second line, and following lines, the four terms form periodic sequences:
 1   0   1   0
 0   0   1   1
 0   1   2   1
 1   3   3   1
 4   6   4   2
10  10   6   6
20  16  12  16
36  28  28  36
64  56  64  72
120 120 136 136
240 256 272 256.
The lines are connected as seen by the examples: (3rd line connected to 2nd, from right to left) 1+1=2, 1+0=1, 0+0=0, 0+1=1; (11th line connected to 10th) 136+136=272, 136+120=256, 120+120=240, 120+136=256.
The 4 columns are almost known (must the first line be suppressed?): A038503 (without the first 1), A000749 (without the first 0), A038505, A038504. Like the present sequence, every sequence of A124072 beginning with a negative number (-2, -11, ...) is a "twisted" sequence (see A129339 comments, A129961 and the present 4 columns). Periodic with period 2^n.
Inverse binomial transform of A129819. - R. J. Mathar, Feb 25 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-14,-16,-8).

a:=[-2,5,-11,23];; for n in [5..30] do a[n]:=-6*a[n-1]+-14*a[n-2] -16*a[n-3]-8*a[n-4]; od; Concatenation([0,0,1], a); # G. C. Greubel, Mar 24 2019

I:=[-2,5,-11,23]; [0,0,1] cat [n le 4 select I[n] else -6*Self(n-1) - 14*Self(n-2)-16*Self(n-3)-8*Self(n-4): n in [1..30]]; // G. C. Greubel, Mar 24 2019

gf = x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2); CoefficientList[Series[gf, {x, 0, 30}], x] (* Jean-François Alcover, Dec 16 2014, after R. J. Mathar *)
Join[{0, 0, 1}, LinearRecurrence[{-6,-14,-16,-8}, {-2,5,-11,23}, 30]] (* Jean-François Alcover, Feb 15 2016 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(1+3*x+4*x^2+3*x^3 )/((1+2*x +2*x^2)*(1+2*x)^2))) \\ G. C. Greubel, Mar 24 2019

(x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2 )).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Mar 24 2019

From R. J. Mathar, Feb 25 2009: (Start)
G.f.: x^2*(1+x)*(1 + 3*x + 4*x^2 + 3*x^3)/((1 + 2*x + 2*x^2)*(1+2*x)^2).
a(n) = ((-1)^n*A001787(n+1) - 4*A108520(n) + 4*A122803(n))/32, n > 2. (End)
a(n) = -6*a(n-1) - 14*a(n-2) - 16*a(n-3) - 8*a(n-4) for n >= 7. - G. C. Greubel, Mar 24 2019

Extended by R. J. Mathar, Feb 25 2009

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

A229893 Expansion of q^2 * f(-q) * f(-q^4) * f(-q^16) * f(-q^2, -q^14) in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -2, 1, 0, 2, 2, -1, -2, -1, 2, -1, -2, 0, 0, 0, 1, 3, 2, -2, 2, -6, -4, 3, 0, 4, 0, 3, 2, 0, -4, 0, -2, -2, 0, -3, 0, 2, 0, 0, 4, -5, -2, 1, 6, 0, 4, 0, -3, 2, 2, 5, -8, 2, 4, -6, 0, 3, -4, -9, -8, 0, 8, 0, -2, -5, 4, 6, 0, 10, -2, 4, 6, -3, -6, 2, -2

Offset: 2

Michael Somos, Oct 02 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q^2 - q^3 - 2*q^4 + q^5 + 2*q^7 + 2*q^8 - q^9 - 2*q^10 - q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A108520, A229502.

Basis( CuspForms( Gamma1(16), 2), 79)[2];

a[ n_] := SeriesCoefficient[ q^2 QPochhammer[ q^2, q^16] QPochhammer[ q^14, q^16] QPochhammer[ q^16]^2 QPochhammer[ q] QPochhammer[ q^4], {q, 0, n}]

{a(n) = local(A, m); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n\2, if( issquare( 16*k + 9, &m), (-1)^k * x^(2*k), 0), A), n))}

Sage
```
CuspForms( Gamma1(16), 2, prec=79).1
```

Euler transform of period 16 sequence [-1, -2, -1, -2, -1, -1, -1, -2, -1, -1, -1, -2, -1, -2, -1, -4, ...].

a(2^n) = A108520(n-1). a(16*n + 1) = a(16*n + 15) = 0. -2 * a(n) = A229502(2*n). a(8*n) = 2 * A229502(n).

A143462 Expansion of 1/(1 + 4x + 8x^2).

Original entry on oeis.org

1, -4, 8, 0, -64, 256, -512, 0, 4096, -16384, 32768, 0, -262144, 1048576, -2097152, 0, 16777216, -67108864, 134217728, 0, -1073741824, 4294967296, -8589934592, 0, 68719476736, -274877906944, 549755813888, 0, -4398046511104, 17592186044416

Offset: 0

Michael Somos, Aug 16 2008

			1 - 4*x + 8*x^2 - 64*x^4 + 256*x^5 - 512*x^6 + 4096*x^8 - 16384*x^9 + ...

Index entries for linear recurrences with constant coefficients, signature (-4,-8).

A030210(2^n) = 2^n * A108520(n) = a(n).

I:=[1,-4]; [n le 2 select I[n] else -4 * Self(n-1) - 8 * Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

A143462 := n -> `if`(n=0, 1, (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):
seq(simplify(A143462(n)), n=0..29); # Peter Luschny, Dec 17 2015

CoefficientList[Series[1/(1 + 4*x + 8*x^2), {x, 0, 30}], x] (* Jinyuan Wang, Mar 10 2020 *)

{a(n) = (-64)^(n \ 4) * [1, -4, 8, 0][n%4 + 1]}

{a(n) = n--; -2 * 2^n * ((-1 + I)^n + (-1 - I)^n)}

{a(n) = n--; simplify( -4 * (2 * quadgen(8))^n * polchebyshev(n, 1, -1 / quadgen(8)))}

G.f.: 1/(1 + 4*x + 8*x^2).
E.g.f.: (cos(2*x) - sin(2*x))/exp(2*x).
a(n) = -4*a(n-1) - 8*a(n-2).
a(n+4) = -64*a(n).
G.f.: 1/(1 + 4*x/(1 - 2*x/(1 + 2*x))) = 1 - 4*x/(1 + 2*x/(1 - 2*x/(1 + 4*x))). - Michael Somos, Jan 03 2013
a(n) = (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2) for n >= 1. - Peter Luschny, Dec 17 2015

A111806 Riordan array (1/(1+3x+2x^2),x/(1+3x+2x^2)).

Original entry on oeis.org

1, -3, 1, 7, -6, 1, -15, 23, -9, 1, 31, -72, 48, -12, 1, -63, 201, -198, 82, -15, 1, 127, -522, 699, -420, 125, -18, 1, -255, 1291, -2223, 1795, -765, 177, -21, 1, 511, -3084, 6562, -6768, 3840, -1260, 238, -24, 1, -1023, 7181, -18324, 23276, -16758, 7266, -1932, 308, -27, 1, 2047, -16398, 49029, -74616, 65870

Offset: 0

Paul Barry, Aug 18 2005

Signed version of A110441. Factors as (1/(1+x),x/(1+x))*((1-x)/(1+x),x(1-x)/(1+x)), or inverse binomial transform of A080246. Inverse of little Schroeder number array A110440. Row sums are A108520. Diagonal sums are (-1)^n*A001906(n+1).

			Triangle starts
1;
-3,1;
7,-6,1;
-15,23,-9,1;
31,-72,48,-12,1;

T(n,k)=-3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 04 2013

cp's OEIS Frontend

A134142 List of quadruples: 2*(-4)^n, -3*(-4)^n, 2*(-4^n), 2*(-4)^n, n >= 0.

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A137429 a(n) = -2*a(n-1) - 2*a(n-2), with a(0)=1 and a(1)=-4.

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A138377 a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2; thereafter a(n) = -4*a(n-4).

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A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.

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A130668 Diagonal of A129819.

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

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A229893 Expansion of q^2 * f(-q) * f(-q^4) * f(-q^16) * f(-q^2, -q^14) in powers of q where f() is a Ramanujan theta function.

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A134142 List of quadruples: 2(-4)^n, -3(-4)^n, 2(-4^n), 2(-4)^n, n >= 0.

A137429 a(n) = -2a(n-1) - 2a(n-2), with a(0)=1 and a(1)=-4.

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

A143462 Expansion of 1/(1 + 4x + 8x^2).