A009545 -id:A009545 - cp's OEIS frontend

A117355 Riordan array (1/(1-x^2),x(1-2x^2)/(1-x^2)).

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, -2, 0, 1, 1, 0, -2, 0, -3, 0, 1, 0, -2, 0, -2, 0, -4, 0, 1, 1, 0, -2, 0, -1, 0, -5, 0, 1, 0, -3, 0, 0, 0, 1, 0, -6, 0, 1, 1, 0, -1, 0, 3, 0, 4, 0, -7, 0, 1, 0, -4, 0, 3, 0, 6, 0, 8, 0, -8, 0, 1, 1, 0, 1, 0, 6, 0, 8, 0, 13, 0, -9, 0, 1

Offset: 0

Paul Barry, Mar 09 2006

Row sums are A077948.

Diagonal sums are the aerated version of A009545 with g.f. 1/(1-2x^2+2x^4).

			Triangle begins
1,
0, 1,
1, 0, 1,
0, 0, 0, 1,
1, 0, -1, 0, 1,
0, -1, 0, -2, 0, 1,
1, 0, -2, 0, -3, 0, 1,
0, -2, 0, -2, 0, -4, 0, 1

Number triangle T(n,k)=sum{j=0..n-k, C(j-(n-k)/2-1,j)C(k,j)(1+(-1)^(n-k))/2}

T(n,k)=T(n-1,k-1)+T(n-2,k)-2*T(n-3,k-1), T(0,0)=T(1,1)=T(2,0)=T(2,2)=1, T(1,1)=T(2,1)=0, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 16 2014

A136258 a(n) = 2a(n-1) - 2a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 8, 6, -4, -20, -32, -24, 16, 80, 128, 96, -64, -320, -512, -384, 256, 1280, 2048, 1536, -1024, -5120, -8192, -6144, 4096, 20480, 32768, 24576, -16384, -81920, -131072, -98304, 65536, 327680, 524288, 393216, -262144, -1310720, -2097152, -1572864, 1048576

Offset: 0

Paul Curtz, Mar 18 2008

Sequence opposite in sign to its second differences.
Binomial transform of 1, 4, -1, -4.
A bisection gives A135520.
This sequence with offset 0 is the binomial transform of (-1)^floor(n/2)*A010685(n). - R. J. Mathar, Feb 22 2009

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

Cf. A010685, A135520.

[n le 2 select 5^(n-1) else 2*(Self(n-1) - Self(n-2)): n in [1..41]]; // G. C. Greubel, Dec 02 2021

LinearRecurrence[{2,-2},{1,5},50] (* Harvey P. Dale, May 21 2014 *)

vector(100,n,t=if(n<3,[t1=1,5][n],-2*t1+2*t1=t)) \\ M. F. Hasler, May 01 2008

A136258=BinaryRecurrenceSequence(2,-2,1,5)
[A136258(n) for n in (0..40)] # G. C. Greubel, Dec 02 2021

a(4n+1) = 5*(-4)^n, a(4n+3) = 6*(-4)^n. - M. F. Hasler, May 01 2008
G.f.: x*(1+3*x)/(1-2*x+2*x^2). - R. J. Mathar, Feb 22 2009
From Paul Curtz, Apr 27 2011: (Start)
a(n)= -4 * a(n-4).
a(n)= 3*A009545(n) + A009545(n+1). (End)
E.g.f.: exp(x)*( cos(x) + 4*sin(x) ). - G. C. Greubel, Dec 02 2021

Edited and extended by M. F. Hasler, May 01 2008

Offset corrected Paul Curtz, Apr 27 2011

A137500 Binomial transform of b(n) = (0, 0, A007910).

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 149, 439, 1309, 3927, 11797, 35423, 106301, 318903, 956645, 2869807, 8609293, 25827879, 77483893, 232452191, 697357085, 2092071255, 6276212741, 18828636175, 56485906477, 169457719431, 508373162389, 1525119495359, 4575358494269, 13726075482807

Offset: 0

Paul Curtz, Apr 27 2008

b(n) is binomial transform of (0, 0, A077973).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,6).

Cf. A007910, A009545.

LinearRecurrence[{5,-8,6},{0,0,1},40] (* Harvey P. Dale, Sep 27 2020 *)

concat([0,0], Vec(1/((1 - 3*x)*(1 - 2*x + 2*x^2)) + O(x^40))) \\ Andrew Howroyd, Jan 03 2020

a(n) = 3*a(n-1) + A009545(n-1) for n > 0.
From Andrew Howroyd, Jan 03 2020: (Start)
a(n) = Sum_{k=0..n-2} binomial(n, k+2)*A007910(k).
a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) for n >= 3.
G.f.: x*2/((1 - 3*x)*(1 - 2*x + 2*x^2)). (End)

Terms a(11) and beyond from Andrew Howroyd, Jan 03 2020

A138232 First differences of A138231.

Original entry on oeis.org

1, 0, 0, 1, -2, 2, -4, 2, -4, 0, 0, -4, 8, -8, 16, -8, 16, 0, 0, 16, -32, 32, -64, 32, -64, 0, 0, -64, 128, -128, 256, -128, 256, 0, 0, 256, -512, 512, -1024, 512, -1024, 0, 0, -1024, 2048, -2048, 4096, -2048, 4096, 0, 0, 4096, -8192, 8192, -16384, 8192, -16384, 0, 0, -16384, 32768

Offset: 0

Paul Curtz, May 05 2008

sign

The sequence contains 2 copies of 1 and 3 copies of the higher powers 2^j (up to sign).

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

Cf. A000079, A131577.

LinearRecurrence[{0,2,0,-2},{1,0,0,1},70] (* Harvey P. Dale, Apr 23 2022 *)

a(n) = 2a(n-2)-2a(n-4). a(n) = -4a(n-8).
a(2n) = (-1)^(n+1)*A090132(n). a(2n+1) = A009545(n).
O.g.f.: (x-1)(x^2-x-1)/(1-2x^2+2x^4). - R. J. Mathar, Jul 08 2008

Edited by R. J. Mathar, Jul 08 2008

A309878 The real part of b(n) where b(n) = (n + b(n-1)) * (1 + i) with b(-1)=0; i = sqrt(-1).

Original entry on oeis.org

0, 1, 2, 1, -4, -13, -22, -23, -8, 23, 54, 53, -12, -141, -270, -271, -16, 495, 1006, 1005, -20, -2069, -4118, -4119, -24, 8167, 16358, 16357, -28, -32797, -65566, -65567, -32, 131039, 262110, 262109, -36, -524325, -1048614, -1048615, -40, 2097111, 4194262, 4194261, -44

Offset: 0

Jesse Fiedler, Aug 21 2019

Observe that (starting with n=1) the sequence has a pattern of a cluster of 3 positive numbers followed by a cluster of 5 negative numbers.
Observe also if the clusters of 3 positive numbers are represented by x, y, z; then y = (x * 2) + (8 * k) where k a positive integer ; when this happens, k = (n - 1) / 8  ; therefore y = x * 2 + n - 1; z = y - 1
Observe also that within each cluster of 5 negative numbers, the first and last are orders of magnitude less than the middle 3 values.  The first and last always differ by 4 and are always equal to -n.
Observe also if the clusters of 5 negative numbers are represented by c, d, e, f, g ; then d - c  = e - d; f = e - 1; g = c - 4

			For n = -1; b(n) = 0
For n =  0; b(n) = (0+0)*(1+i) = 0
For n =  1; b(n) = (1+0)*(1+i) = 1+i ; a(1) = Re(1+i) = 1
For n =  2; b(n) = (2+1+i)*(1+i) = (3+i)*(1+i) = 3+i+3i-1 = 2+4i ; a(2) = Re(2+4i) = 2
For n =  3; b(n) = 1+9i ; a(3) = 1
For n =  4; b(n) = -4+14i ; a(4) = -4
For n =  5; b(n) = -13+15i ; a(5) = -13
For n =  6; b(n) = -22+8i ; a(6) = -22
For n =  7; b(n) = -23-7i ; a(7) = -23
...
For n =  31; b(n) = -65567-65503i ; a(31) = -65567
For n =  32; b(n) = (32-65567-65503i)*(1+i) = (-65535-65503i)*(1+i) = -65535-65503i-65535i+65503 = -32-131038i ; a(32) = -32
For n =  33; b(n) = 131039-131037i ; a(33) = 131039

Sela Fried, Sequence A309878, 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 9.

b(n) = if (n==0, 0, (n + b(n-1)) * (1 + I));
for (n=0, 50, print1(real(b(n)), ", ")) \\ Michel Marcus, Aug 21 2019

Conjectures from Colin Barker, Aug 21 2019: (Start)
G.f.: x*(1 - 2*x) / ((1 - x)^2*(1 - 2*x + 2*x^2)).
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 2*a(n-4) for n>3.
a(n) = i*((1-i)^n - (1+i)^n + i*n) where i=sqrt(-1).
(End)
E.g.f.: exp(x)*(-x + 2*sin(x)). - Conjectured by Stefano Spezia, Aug 21 2019 after Colin Barker
Conjecture: a(n) = 2*A009545(n)-n. - R. J. Mathar, Mar 06 2022
All conjectures stated above hold true. See links. - Sela Fried, Jul 27 2024.

A316386 Binomial transform of [0, 1, 2, -3, -4, 5, 6, -7, -8, ...].

Original entry on oeis.org

0, 1, 4, 6, 0, -20, -48, -56, 0, 144, 320, 352, 0, -832, -1792, -1920, 0, 4352, 9216, 9728, 0, -21504, -45056, -47104, 0, 102400, 212992, 221184, 0, -475136, -983040, -1015808, 0, 2162688, 4456448, 4587520, 0, -9699328, -19922944, -20447232, 0, 42991616

Offset: 0

Paul Curtz, Jul 01 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-8,8,-4).

Cf. A009545, A140230.

seq(coeff(series(x*(1-2*x^2)/(1-2*x+2*x^2)^2, x,n+1),x,n),n=0..45); # Muniru A Asiru, Jul 01 2018

CoefficientList[Series[x (1 - 2 x^2)/(1 - 2 x + 2 x^2)^2, {x, 0, 41}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{4, -8, 8, -4}, {0, 1, 4, 6}, 42] (* Robert G. Wilson v, Jul 15 2018 *)

concat(0, Vec(x*(1 - 2*x^2) / (1 - 2*x + 2*x^2)^2 + O(x^40))) \\ Colin Barker, Jul 01 2018

a(n) = n * A009545(n).
a(n+1) = a(n) + A140230(n).
From Colin Barker, Jul 01 2018: (Start)
G.f.: x*(1 - 2*x^2) / (1 - 2*x + 2*x^2)^2.
a(n) = 4*a(n-1) - 8*a(n-2) + 8*a(n-3) - 4*a(n-4) for n>3.
a(n) = i*(((1-i)^n - (1+i)^n)*n) / 2 where i=sqrt(-1).
(End)

More terms from Colin Barker, Jul 01 2018

A323225 a(n) = ((2^nn + i(1 - i)^n - i*(1 + i)^n))/4, where i is the imaginary unit.

Original entry on oeis.org

0, 1, 3, 7, 16, 38, 92, 220, 512, 1160, 2576, 5648, 12288, 26592, 57280, 122816, 262144, 557184, 1179904, 2490624, 5242880, 11009536, 23067648, 48233472, 100663296, 209717248, 436211712, 905973760, 1879048192, 3892305920, 8053047296, 16642981888, 34359738368

Offset: 0

Peter Luschny, Mar 18 2019

Related to Clifford algebras (see A323100 and A323346).

Robert Israel, Table of n, a(n) for n = 0..3308
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Antidiagonal sums of A323346.

Cf. A001787, A009545, A321959, A323100.

a := n -> ((2^n*n + I*(1 - I)^n - I*(1 + I)^n))/4:
seq(a(n), n=0..32);

LinearRecurrence[{6, -14, 16, -8}, {0, 1, 3, 7}, 40] (* Jean-François Alcover, Mar 20 2019 *)
Table[((2^n n + I (1 - I)^n - I (1 + I)^n))/4, {n, 0, 29}] (* Alonso del Arte, Mar 27 2020 *)

a(n) = Sum_{k = 0..n} A323346(n - k, k - 1).
a(n) = (A001787(n) + A009545(n))/2.
a(n) = [x^n] (x*(3*x^2 - 3*x + 1))/((2*x - 1)^2*(2*x^2 - 2*x + 1)).
a(n) = n! [x^n] (exp(2*x)*x + exp(x)*sin(x))/2.
a(n) = (4*n*a(n-3) + (2 - 6*n)*a(n-2) + (4*n - 2)*a(n-1))/(n - 1) for n >= 3.

A370359 Imaginary part of (n + n*i)^n where i = sqrt(-1).

Original entry on oeis.org

0, 1, 8, 54, 0, -12500, -373248, -6588344, 0, 6198727824, 320000000000, 9129973459552, 0, -19384006821904192, -1422336873671426048, -56050417968750000000, 0, 211773507042902211629312, 20145360934551827238617088, 1012950863698080557631477248, 0, -5982809106827246101894271407104

Offset: 0

Chai Wah Wu, Feb 16 2024

sign

Cf. A009545, A121625, A146559.

a(n) = imag((n + n*I)^n); \\ Michel Marcus, Feb 16 2024

def A370359(n): return n**n*((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1)

a(n) = n^n*A009545(n) = n^n*Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*(-1)^j.
a(n) = 0 if and only if n == 0 mod 4.
a(4n) = 0.
a(4n+1) = (4n+1)^(4n+1)*(-4)^n.
a(4n+2) = 2*(4n+2)^(4n+2)*(-4)^n.
a(4n+3) = 2*(4n+3)^(4n+3)*(-4)^n.

A101892 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)J(k), where J = A001045.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 33, 77, 187, 459, 1121, 2717, 6555, 15795, 38081, 91893, 221867, 535755, 1293633, 3123277, 7540187, 18203139, 43945441, 106092997, 256131435, 618357915, 1492851361, 3604064733, 8700980827, 21006018195, 50713000833

Offset: 0

Paul Barry, Dec 22 2004

Transform of A001045 under the mapping g(x)-> (1/(1-x))*g(x^2/((1-x)^2)). Binomial transform of aerated Jacobsthal numbers 0,0,1,0,1,0,3,0,5,0,11,...

J(n) may be recovered as Sum_{k=0..2*n} Sum_{j=0..k} C(0,2*n-k)*C(k,j)*(-1)^(k-j)*a(j). - Paul Barry, Jun 10 2005

Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2).

Cf. A001045, A001333, A009545.

G.f.: x^2*(1 - x)/((1 - 2*x - x^2)*(1 - 2*x + 2*x^2)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + 2*a(n-4).
a(n) = Sum_{k=0..n} binomial(n, k)*A001045(k/2)*(1+(-1)^k)/2.
a(n) = (1/6)*( 2*A001333(n) - A009545(n+2) ). - Ralf Stephan, May 17 2007

A132402 Binomial transform of A004524 starting at 1.

Original entry on oeis.org

1, 3, 7, 15, 32, 70, 156, 348, 768, 1672, 3600, 7696, 16384, 34784, 73664, 155584, 327680, 688256, 1442048, 3014912, 6291456, 13106688, 27261952, 56622080, 117440512, 243271680, 503320576, 1040191488, 2147483648

Offset: 0

Paul Curtz, Nov 12 2007

Twisted numbers. b(n)=a(n)-2^n=0, 1, 3, 7, 16, 38, 92, 220, 512, 1160, 2576, twisted numbers. b(n+1)-2b(n)=1, 1, 1, 2, 6, 16, 36, 72, 136, 256.

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

LinearRecurrence[{6,-14,16,-8},{1,3,7,15},30] (* Harvey P. Dale, Mar 30 2022 *)

a(n+1)-2a(n) = 1, 1, 1, 2, 6, 16, 36, 72, 136, 256 = essentially A038503.
O.g.f.: (1-x)^3/[(1-2x+2x^2)(-1+2x)^2]. a(n)=6*a(n-1)-14*a(n-2)+16*a(n-3)-8*a(n-4). - R. J. Mathar, Apr 02 2008
4*a(n) = (n+4)*2^n+2*A009545(n). - R. J. Mathar, Nov 01 2021

More terms from R. J. Mathar, Apr 02 2008

cp's OEIS Frontend

A117355 Riordan array (1/(1-x^2),x(1-2x^2)/(1-x^2)).

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A136258 a(n) = 2*a(n-1) - 2*a(n-2), with a(0)=1, a(1)=5.

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A137500 Binomial transform of b(n) = (0, 0, A007910).

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A138232 First differences of A138231.

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A309878 The real part of b(n) where b(n) = (n + b(n-1)) * (1 + i) with b(-1)=0; i = sqrt(-1).

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A316386 Binomial transform of [0, 1, 2, -3, -4, 5, 6, -7, -8, ...].

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A323225 a(n) = ((2^n*n + i*(1 - i)^n - i*(1 + i)^n))/4, where i is the imaginary unit.

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A370359 Imaginary part of (n + n*i)^n where i = sqrt(-1).

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A136258 a(n) = 2a(n-1) - 2a(n-2), with a(0)=1, a(1)=5.

A323225 a(n) = ((2^nn + i(1 - i)^n - i*(1 + i)^n))/4, where i is the imaginary unit.

A101892 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)J(k), where J = A001045.