A131666 -id:A131666 - cp's OEIS frontend

A131090 First differences of A131666.

0, 1, 0, 1, 1, 4, 7, 15, 28, 57, 113, 228, 455, 911, 1820, 3641, 7281, 14564, 29127, 58255, 116508, 233017, 466033, 932068, 1864135, 3728271, 7456540, 14913081, 29826161, 59652324, 119304647, 238609295, 477218588, 954437177, 1908874353

Offset: 0

Paul Curtz, Sep 24 2007

The first differences b(n)=a(n+1)-a(n) obey the recurrence b(n+1)-2b(n) = (-3,3,-2,3,-3,2), continued with period 6.
The 2nd differences c(n)=b(n+1)-b(n) obey the recurrence c(n+1)-2c(n) = (6,-5,5,-6,5,-5), periodically continued with period 6.
The hexaperiodic coefficients in these recurrences for A113405, A131666 and their higher order differences define a table,
0, 0, 1, 0, 0, -1 <- A113405
0, 1, -1, 0, -1, 1 <- A131666
1, -2, 1, -1, 2, -1 <- a(n)
-3, 3, -2, 3, -3, 2 <- b(n)
6, -5, 5, -6, 5, -5 <- c(n)
-11,10,-11, 11,-10, 11
21,-21,22,-21, 21,-22
...
in which the first three columns are A024495, A131708 and A024493, multiplied by a checkerboard pattern of signs.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

LinearRecurrence[{2,0,-1,2},{0,1,0,1},40] (* Harvey P. Dale, Jan 15 2016 *)

a(n) = A131666(n+1)-A131666(n).
a(n+1)-2a(n) = A131556(n), a sequence with period length 6.
G.f.: -(x-1)^2*x / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, Mar 04 2013

Edited by R. J. Mathar, Jun 28 2008

A135254 Binomial transform of A131666.

Original entry on oeis.org

0, 0, 1, 4, 12, 33, 90, 252, 729, 2160, 6480, 19521, 58806, 176904, 531441, 1595052, 4785156, 14353281, 43053282, 129146724, 387420489, 1162241784, 3486725352, 10460235105, 31380882462, 94143001680, 282429536481, 847289140884

Offset: 0

Paul Curtz, Nov 30 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,9).

Cf. A133474.

a:=[0,1,4];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; Concatenation([0], a); # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 21 2019

CoefficientList[Series[x^2(2x-1)/((3x^2-3x+1)(3x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-12,9},{0,0,1,4},30] (* Harvey P. Dale, May 26 2011 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)))) \\ G. C. Greubel, Nov 21 2019

def A135254_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x))).list()
A135254_list(30) # G. C. Greubel, Nov 21 2019

From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: x^2*(1-2*x)/((1 - 3*x + 3*x^2)*(1-3*x)).
a(n) = 6*a(n-1) - 12*a(n-2) + 9*a(n-3). (End)

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

A135258 Inverse binomial transform of A131666 after removing A131666(0) = 0.

Original entry on oeis.org

0, 1, -1, 2, -3, 7, -14, 29, -57, 114, -227, 455, -910, 1821, -3641, 7282, -14563, 29127, -58254, 116509, -233017, 466034, -932067, 1864135, -3728270, 7456541, -14913081, 29826162, -59652323, 119304647, -238609294, 477218589, -954437177, 1908874354

Offset: 0

Paul Curtz, Dec 01 2007

sign

The inverse binomial transform generally equals the sequence of first terms of the iterated differences (i.e., equals the diagonal of the arrangement in the standard hand-written display of the differences).

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

Cf. A113405.

LinearRecurrence[{-2, 0, 1, 2}, {0, 1, -1, 2}, 50] (* G. C. Greubel, Oct 04 2016 *)

concat(0, Vec(x*(1 + x)/((x^2 +x +1)*(1 +2*x)*(1-x)) + O(x^50))) \\ Michel Marcus, Oct 05 2016

O.g.f.: x*(1 + x)/((x^2 +x +1)*(1 +2*x)*(1-x)). - R. J. Mathar, Jul 22 2008

a(n) = -2*a(n-1) + a(n-3) + 2*a(n-4). - G. C. Greubel, Oct 04 2016

Edited and corrected by R. J. Mathar, Jul 22 2008

A135259 a(n) = 3*A131666(n) - A131666(n+1).

Original entry on oeis.org

0, -1, 2, 1, 3, 2, 7, 13, 30, 57, 115, 226, 455, 909, 1822, 3641, 7283, 14562, 29127, 58253, 116510, 233017, 466035, 932066, 1864135, 3728269, 7456542, 14913081, 29826163, 59652322, 119304647, 238609293, 477218590, 954437177, 1908874355, 3817748706

Offset: 0

Paul Curtz, Dec 01 2007

sign

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

Cf. A131666.

a:=[0,-1,2,1];; for n in [5..35] do a[n]:=2*a[n-1]-a[n-3]+2*a[n-4]; od; a; # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 35); [0] cat Coefficients(R!( x*(1-x)*(1-3*x)/((2*x-1)*(x+1)*(1-x+x^2)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series(x*(1-x)*(1-3*x)/((2*x-1)*(x+1)*(1-x+x^2)), x, n+1), x, n), n = 0 .. 35); # G. C. Greubel, Nov 21 2019

LinearRecurrence[{2,0,-1,2}, {0,-1,2,1}, 35] (* G. C. Greubel, Oct 05 2016 *)

concat(0, Vec(x*(1-x)*(1-3*x)/((2*x-1)*(x+1)*(1-x+x^2)) + O(x^35))) \\ Michel Marcus, Oct 05 2016

def A135259_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-x)*(1-3*x)/((2*x-1)*(x+1)*(1-x+x^2))).list()
A135259_list(35) # G. C. Greubel, Nov 21 2019

A131666(n) - a(n) = A092220(n).
O.g.f.: x*(1-x)*(1 -3*x)/( (2*x-1)*(x+1)*(1 -x +x^2) ). - R. J. Mathar, Jul 22 2008
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - G. C. Greubel, Oct 05 2016

Edited and extended by R. J. Mathar, Jul 22 2008

A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 0, 4, 3, 11, 17, 40, 74, 154, 301, 609, 1211, 2430, 4852, 9712, 19415, 38839, 77669, 155348, 310686, 621382, 1242753, 2485517, 4971023, 9942058, 19884104, 39768220, 79536427, 159072867, 318145721, 636291456, 1272582898, 2545165810

Offset: 0

Paul Curtz, Oct 17 2021

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,1,2).

Cf. A139797 (a(n) + a(n+1) = round(2^n/9) too, but a(0) = 0).
Cf. A001045, A104581, A113405, A131666.
Cf. A136298, A172481, A175656, A348407.

CoefficientList[ Series[(x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2), {x, 0, 40}], x] (* Thomas Scheuerle, Oct 17 2021 *)
nxt[{n_,a_}]:={n+1,Round[(2^n)/9]-a}; NestList[nxt,{0,1},40][[All,2]] (* or *) LinearRecurrence[{1,2,-1,1,2},{1,-1,1,-1,2},40] (* Harvey P. Dale, Apr 28 2022 *)

a(n+1) = 2*a(n) - A104581(n+6).
a(n) + a(n+1) = A113405(n).
a(n) + a(n+3) = A001045(n).
a(n+2) = a(n) + A131666(n).
From Thomas Scheuerle, Oct 18 2021: (Start)
G.f.: (x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2).
A172481(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k). With negative sign for ...*a(1+2*n-k) and ...*a(3+2*n-k) too.
A175656(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2+2*n-k).
A136298(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(4+2*n-k).
A348407(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(2+2*n-k) - 2*a(1+2*n-k) - a(2*n-k)).
(End)

a(22)-a(36) from Thomas Scheuerle, Oct 17 2021

cp's OEIS Frontend

A131090 First differences of A131666.

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A135254 Binomial transform of A131666.

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A135258 Inverse binomial transform of A131666 after removing A131666(0) = 0.

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A135259 a(n) = 3*A131666(n) - A131666(n+1).

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A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

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