A038503 -id:A038503 - cp's OEIS frontend

A100212 Expansion of 4x^4(2 + x)/(1 - 2x + 2x^2 - 4x^4 + 8x^5 - 8*x^6).

0, 0, 0, 0, 8, 20, 24, 8, 0, 0, 0, 0, 128, 320, 384, 128, 0, 0, 0, 0, 2048, 5120, 6144, 2048, 0, 0, 0, 0, 32768, 81920, 98304, 32768, 0, 0, 0, 0, 524288, 1310720, 1572864, 524288, 0, 0, 0, 0, 8388608, 20971520, 25165824, 8388608, 0, 0, 0, 0, 134217728, 335544320

Offset: 0

Creighton Dement, Nov 08 2004

a(n) = 0 iff n == {0, 1, 2 or 3} (mod 8). - Robert G. Wilson v, Nov 12 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,0,4,-8,8).

Cf. A038503, A009116, A100213.

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) )); // G. C. Greubel, Apr 01 2024

CoefficientList[ Series[4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6), {x, 0, 55}], x] (* Robert G. Wilson v, Nov 12 2004 *)
LinearRecurrence[{2,-2,0,4,-8,8},{0,0,0,0,8,20},60] (* Harvey P. Dale, Oct 10 2012 *)

Vec(4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

def A100212_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) ).list()
A100212_list(60) # G. C. Greubel, Apr 01 2024

a(8n+4) = a(8n+7) = 2^(4n+3), a(8n+5) = (5/2)*2^(4n+3), a(8n+6) = 3*2^(4n+3), a(8n+8) = 0, a(8n+9) = 0, a(8n+10) = 0, a(8n+11) = 0.
(a(n)) = negseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=8, a(5)=20, a(n) = 2*a(n-1) - 2*a(n-2) + 4*a(n-4) - 8*a(n-5) + 8*a(n-6). - Harvey P. Dale, Oct 10 2012

More terms from Robert G. Wilson v, Nov 12 2004

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

A133212 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

Original entry on oeis.org

1, 4, 12, 32, 72, 144, 272, 512, 992, 1984, 4032, 8192, 16512, 33024, 65792, 131072, 261632, 523264, 1047552, 2097152, 4196352, 8392704, 16781312, 33554432, 67100672, 134201344, 268419072, 536870912, 1073774592, 2147549184

Offset: 0

Paul Curtz, Oct 11 2007

Conjecture: a(n) = 2*A038503(n+3) if n > 0. - R. J. Mathar, Oct 23 2007

Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Cf. A009116, A038503, A099087.

A133212 := proc(n) option remember ; if n <= 3 then op(n+1,[1,4,12,32]) ; else 4*A133212(n-1)-6*A133212(n-2)+4*A133212(n-3) ; fi ; end: seq(A133212(n),n=0..50) ; # R. J. Mathar, Oct 23 2007

Join[{1},LinearRecurrence[{4, -6, 4},{4, 12, 32},29]] (* Ray Chandler, Sep 23 2015 *)

Sequence is identical to its fourth differences.
From R. J. Mathar, Nov 18 2007: (Start)
G.f.: -(1 + 2*x^2 + 4*x^3)/((2*x - 1)*(2*x^2 - 2*x + 1)). - [Corrected by Georg Fischer, May 12 2019]
a(n) = -2*(-1)^n*A009116(n)+3*2^n. (End)
E.g.f.: exp(x)*(3*cosh(x) - 2*(cos(x) + sin(x)) + 5*sinh(x)). - Stefano Spezia, Jan 03 2023

More terms from R. J. Mathar, Oct 23 2007

A137171 Interleaved reading of A000749 and its first to third differences.

Original entry on oeis.org

0, 0, 0, 1, 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, 496, 528, 528, 496, 1024, 1056, 1024, 992, 2080, 2080, 2016, 2016, 4160, 4096, 4032, 4096, 8256, 8128, 8128

Offset: 0

Paul Curtz, May 11 2008

A000749 is identical to its fourth differences, which implies that the 2nd differences equal the 5th, the 3rd differences the 6th and so on and implies that each of the sequences of these differences obeys the recurrence a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3.
The table containing A000749 and its first differences (essentially A038505), 2nd differences (A038504) and 3rd differences (A038503) as the 4 rows is
O, 0, 0, 1, 4, 10, 20, 36, 64, ...
0, 0, 1, 3, 6, 10, 16, 28, 56, ...
0, 1, 2, 3, 4, 6, 12, 28, 64, ...
1, 1, 1, 1, 2, 6, 16, 36, 72, ...
Columns sums are 1, 2, 4, 8, 16, 32 ... = 2^n =A000079. The sequence reads this table column by column.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4).

Cf. A137172, A137173.

Join[{0, 0, 0, 1},LinearRecurrence[{0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4},{0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 1},59]] (* Ray Chandler, Sep 23 2015 *)

Edited by R. J. Mathar, Jun 28 2008

A098179 Expansion of (1-3x+3x^2)/(1-5x+10x^2-10x^3+4x^4).

Original entry on oeis.org

1, 2, 3, 5, 11, 27, 63, 135, 271, 527, 1023, 2015, 4031, 8127, 16383, 32895, 65791, 131327, 262143, 523775, 1047551, 2096127, 4194303, 8390655, 16781311, 33558527, 67108863, 134209535, 268419071, 536854527, 1073741823, 2147516415

Offset: 0

Paul Barry, Aug 30 2004

Partial sums of A038503. Binomial transform of A098178.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4).

LinearRecurrence[{5,-10,10,-4},{1,2,3,5},40] (* or *) CoefficientList[ Series[(1-3 x+3 x^2)/(1-5 x+10 x^2-10 x^3+4 x^4),{x,0,40}],x] (* Harvey P. Dale, Oct 06 2011 *)

a(n) = 2^n+2^(n/2)cos(pi*n/4)-1; a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-4a(n-4).

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

A100212 Expansion of 4*x^4*(2 + x)/(1 - 2*x + 2*x^2 - 4*x^4 + 8*x^5 - 8*x^6).

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A130668 Diagonal of A129819.

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A133212 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

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A137171 Interleaved reading of A000749 and its first to third differences.

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A098179 Expansion of (1-3*x+3*x^2)/(1-5*x+10*x^2-10*x^3+4*x^4).

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A132402 Binomial transform of A004524 starting at 1.

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A100212 Expansion of 4x^4(2 + x)/(1 - 2x + 2x^2 - 4x^4 + 8x^5 - 8*x^6).

A133212 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

A098179 Expansion of (1-3x+3x^2)/(1-5x+10x^2-10x^3+4x^4).