A077925 -id:A077925 - cp's OEIS frontend

A036827 a(n) = 26 + 2^(n+1)(-13 +9n -3*n^2 +n^3).

0, 2, 34, 250, 1274, 5274, 19098, 63002, 194074, 567322, 1591322, 4317210, 11395098, 29392922, 74350618, 184942618, 453378074, 1097334810, 2626158618, 6222250010, 14610858010, 34032582682, 78693531674, 180757725210, 412685959194

Offset: 0

N. J. A. Sloane

			a(3) = 2^0*0^3 + 2^1*1^3 + 2^2*2^3 + 2^3*3^3 = 250.

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), this sequence (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

a036827 n = 2^(n+1) * (n^3 - 3*n^2 + 9*n - 13) + 26
-- Reinhard Zumkeller, May 24 2012

[2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)): n in [0..35]]; // G. C. Greubel, Mar 31 2021

A036827:= n-> 2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)); seq(A026827(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[26 +2^(n+1)(-13 +9n -3n^2 +n^3), {n, 0, 30}] (* or *) LinearRecurrence[ {9, -32, 56, -48, 16}, {0, 2, 34, 250, 1274}, 31] (* Harvey P. Dale, Dec 15 2011 *)

a(n)=26+2^(n+1)*(-13+9*n-3*n^2+n^3) \\ Charles R Greathouse IV, Oct 07 2015

[2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)) for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = Sum_{k=0..n} 2^k*k^3. - Benoit Cloitre, Jun 11 2003
G.f.: 2*x*(1 +8*x +4*x^2)/((1-x)*(1-2*x)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
a(n) = 9*a(n-1) -32*a(n-2) +56*a(n-3) -48*a(n-4) +16*a(n-5) for n>4 with a(0)=0, a(1)=2, a(2)=34, a(3)=250, a(4)=1274. - Harvey P. Dale, Dec 15 2011
a(n) = Sum_{k=0..n} Sum_{i=0..n} k^3 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2 (13*exp(x) + (-13 +14*x +8*x^3)*exp(2*x)). - G. C. Greubel, Mar 31 2021

A014992 a(n) = (1 - (-10)^n)/11.

Original entry on oeis.org

1, -9, 91, -909, 9091, -90909, 909091, -9090909, 90909091, -909090909, 9090909091, -90909090909, 909090909091, -9090909090909, 90909090909091, -909090909090909, 9090909090909091, -90909090909090909

Offset: 1

Olivier Gérard

q-integers for q = -10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-9,10).

Cf. A077925, A014983, A014985, A014986, A014987, A014989, A014990, A014991, A014993, A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1, -9]; [n le 2 select I[n] else -9*Self(n-1) +10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-10)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

CoefficientList[Series[1/((1 - x)*(1 + 10*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1, 30, print1((1-(-10)^n)/11, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-10) for n in range(1,19)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
G.f.: x/((1 - x)*(1 + 10*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -9*a(n-1) + 10*a(n-2). - Vincenzo Librandi, Oct 22 2012
a(n) = (-1)^(n+1)*A015585(n). - R. J. Mathar, Oct 26 2015
E.g.f.: (exp(x) - exp(-10*x))/11. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A077953 Expansion of 1/(1-x+2x^2-2x^3).

Original entry on oeis.org

1, 1, -1, -1, 3, 3, -5, -5, 11, 11, -21, -21, 43, 43, -85, -85, 171, 171, -341, -341, 683, 683, -1365, -1365, 2731, 2731, -5461, -5461, 10923, 10923, -21845, -21845, 43691, 43691, -87381, -87381, 174763, 174763, -349525, -349525, 699051, 699051, -1398101, -1398101, 2796203, 2796203, -5592405

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 17 2007

Essentially the same as A077980.

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

Cf. A077980.

Cf. A007420, A077925. - Reinhard Zumkeller, Oct 07 2008

a:=[1,1,-1];; for n in [4..50] do a[n]:=a[n-1]-2*a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^2-2*x^3) )); // G. C. Greubel, Aug 07 2019

seq(coeff(series(1/(1-x+2*x^2-2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Aug 07 2019

CoefficientList[Series[1/(1-x+2x^2-2x^3),{x,0,50}],x] (* or *) LinearRecurrence[{1,-2,2},{1,1,-1},50] (* Harvey P. Dale, Aug 27 2014 *)

Vec(1/(1-x+2*x^2-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 25 2012

(1/(1-x+2*x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019

From Reinhard Zumkeller, Oct 07 2008: (Start)
a(n+1) = a(n) - 2*a(n-1) + 2*a(n-2).
a(n) = A077925(floor(n/2)-1) for n>1. (End)

A014991 a(n) = (1 - (-9)^n)/10.

Original entry on oeis.org

1, -8, 73, -656, 5905, -53144, 478297, -4304672, 38742049, -348678440, 3138105961, -28242953648, 254186582833, -2287679245496, 20589113209465, -185302018885184, 1667718169966657, -15009463529699912

Offset: 1

Olivier Gérard

q-integers for q = -9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-8,9).

Cf. A077925, A014983, A014985-A014987, A014989-A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1,-8]; [n le 2 select I[n] else -8*Self(n-1)+9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-9)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

((-9)^Range[30]-1)/-10 (* or *) LinearRecurrence[{-8,9},{1,-8},30] (* Harvey P. Dale, Aug 08 2011 *)
CoefficientList[Series[1/((1 - x)*(1 + 9*x)), {x, 0, 30}], x]; (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1,30, print1((1-(-9)^n)/10, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-9) for n in range(1,19)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
a(0)=1, a(1)=-8, a(n) = -8*a(n-1) + 9*a(n-2). - Harvey P. Dale, Aug 08 2011
G.f.: x/((1 - x)*(1 + 9*x)). - Vincenzo Librandi, Oct 22 2012
E.g.f.: (exp(x) - exp(-9*x))/10. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A014993 a(n) = (1 - (-11)^n)/12.

Original entry on oeis.org

1, -10, 111, -1220, 13421, -147630, 1623931, -17863240, 196495641, -2161452050, 23775972551, -261535698060, 2876892678661, -31645819465270, 348104014117971, -3829144155297680, 42120585708274481

Offset: 1

Olivier Gérard

q-integers for q = -11.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (-10,11).

Cf. A077925, A014983, A014985, A014986, A014987, A014989, A014990, A014991, A014992, A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1, -10]; [n le 2 select I[n] else -10*Self(n-1) +11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-11)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

LinearRecurrence[{-10, 11}, {1, -10}, 40] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1,30, print1((1-(-11)^n)/12, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-11) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^{(n-1)} = {(q^n - 1) / (q - 1)}.
G.f.: x/((1 - x)*(1 + 11*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -10*a(n-1) + 11*a(n-2). - Vincenzo Librandi, Oct 22 2012
E.g.f.: (exp(x) - exp(-11*x))/12. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A140431 2*A094555(n).

Original entry on oeis.org

0, 2, 2, 12, 22, 92, 222, 772, 2102, 6732, 19342, 59732, 175782, 534172, 1588862, 4793892, 14327062, 43090412, 129052782, 387595252, 1161911942, 3487483452, 10458955102, 31383855812, 94137586422, 282440721292, 847266239822

Offset: 0

Paul Curtz, Jun 19 2008

Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

b(n) = A091002(n-1); b(n+1)-3b(n)= A077925(n-2), where b(n)=floor(a(n)/10).

a(n) = (1-(-2)^n+3^n)/3 for n>0. a(n) = 2*a(n-1)+5*a(n-2)-6*a(n-3) for n>3. G.f.: 2*x*(1-x-x^2)/((1-x)*(1+2*x)*(1-3*x)). [Colin Barker, Sep 21 2012]

Edited and extended by R. J. Mathar, Aug 02 2008

A166035 a(n) = (3^n+6*(-4)^n)/7.

Original entry on oeis.org

1, -3, 15, -51, 231, -843, 3615, -13731, 57111, -221883, 907215, -3569811, 14456391, -57294123, 230770815, -918300291, 3687550071, -14707153563, 58957754415, -235443597171, 942936650151, -3768259816203, 15083499618015

Offset: 0

Philippe Deléham, Oct 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,12).

Cf. A077925, A091004.

LinearRecurrence[{-1, 12}, {1, -3}, 100] (* G. C. Greubel, Apr 24 2016 *)

a(n)= (3^n+6*(-4)^n)/7;
for(n=0,33,print1(a(n),", "));

a(n) = -a(n-1) + 12*a(n-2), a(0) = 1, a(1) = -3, for n>1.
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-4)^k.
G.f.: (1-2x)/(1+x-12x^2).
E.g.f.: (1/7)*(exp(3*x) + 6*exp(-4*x)). - G. C. Greubel, Apr 24 2016

a(5) corrected by Tilman Neumann, Dec 31 2010

A166036 a(n) = (4^n+8*(-5)^n)/9.

Original entry on oeis.org

1, -4, 24, -104, 584, -2664, 14344, -67624, 354504, -1706984, 8797064, -42936744, 218878024, -1077612904, 5455173384, -27007431464, 136110899144, -676259528424, 3398477511304, -16923668079784

Offset: 0

Philippe Deléham, Oct 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,20).

Cf. A077925, A091004, A166035.

Table[(4^n+8(-5)^n)/9,{n,0,30}] (* or *) LinearRecurrence[{-1,20},{1,-4},30] (* Harvey P. Dale, Mar 05 2016 *)

vector(100, n, n--; (4^n+8*(-5)^n)/9) \\ Altug Alkan, Apr 24 2016

a(n) = -a(n-1) + 20*a(n-2), a(0) = 1, a(1) = -4, for n>1.
a(n) = Sum_{k=0..n} A112555(n,k)*(-5)^k.
G.f.: (1-3x)/(1+x-20x^2).
E.g.f.: (1/9)*(exp(4*x) + 8*exp(-5*x)). - G. C. Greubel, Apr 24 2016

A077898 Expansion of (1 - x)^(-1)/(1 + x - 2*x^2).

Original entry on oeis.org

1, 0, 3, -2, 9, -12, 31, -54, 117, -224, 459, -906, 1825, -3636, 7287, -14558, 29133, -58248, 116515, -233010, 466041, -932060, 1864143, -3728262, 7456549, -14913072, 29826171, -59652314, 119304657, -238609284, 477218599, -954437166, 1908874365, -3817748696, 7635497427

Offset: 0

N. J. A. Sloane, Nov 17 2002

Partial sums of A077925 (signed Jacobsthal numbers). - Paul Barry, Aug 26 2003

The generalized (3,-2)-Padovan sequence p(3,-2;n). See the W. Lang link under A000931 with (A,B)=(3,-2). - Wolfdieter Lang, Jun 28 2010

			(3,-2)-Padovan combinatorics from the (3,2)-Morse code with weights -2 and 3 for 3-lines -- and 2-lines -, respectively (see the W. Lang link under A000931). n=5: two codes - -- and -- - with the weights (3^1)*(-2)^1 and (-2)^1*3^1, respectively, adding up to 2*(3)(-2) = -12 = a(5). - _Wolfdieter Lang_, Jun 28 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,-2).

Cf. A000931, A053088, A077925.

[(5+(-1)^n*2^(2+n)+3*n)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 21 2016

A077898:=n->(5+(-1)^n*2^(2+n)+3*n)/9: seq(A077898(n), n=0..50); # Wesley Ivan Hurt, Apr 21 2016

CoefficientList[Series[(1 - x)^(-1)/(1 + x - 2*x^2), {x, 0, 40}], x] (* Wesley Ivan Hurt, Apr 21 2016 *)

Vec(1/(1-x)/(1+x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: (1-x)^(-1)/(1+x-2*x^2).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(-3)^i. - Paul Barry, Aug 26 2003
a(n) = (-1)^n * A053088(n). - R. J. Mathar, Aug 30 2008
From Colin Barker, Apr 21 2016: (Start)
a(n) = 3*a(n-2) - 2*a(n-3) for n>2.
a(n) = (5+(-1)^n*2^(2+n)+3*n)/9. (End)
E.g.f.: (4*exp(-2*x) + (5 + 3*x)*exp(x))/9. - Ilya Gutkovskiy, Apr 21 2016
a(n) = Sum_{k=0..n} (n+1-k)*(-2)^k. - Bruno Berselli, May 15 2018

A015000 q-integers for q=-13.

Original entry on oeis.org

1, -12, 157, -2040, 26521, -344772, 4482037, -58266480, 757464241, -9847035132, 128011456717, -1664148937320, 21633936185161, -281241170407092, 3656135215292197, -47529757798798560, 617886851384381281

Offset: 1

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (-12,13).

Cf. A077925, A014983, A014985-A014987, A014989-A014994.

I:=[1,-12]; [n le 2 select I[n] else -12*Self(n-1)+13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-13)^j, j=0..n-1): seq(a(n), n=0..20); # Zerinvary Lajos, Dec 16 2008

QBinomial[Range[20],1,-13] (* Harvey P. Dale, May 02 2012 *)
LinearRecurrence[{-12, 13}, {1, -12}, 30] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1, 30, print1((1-(-13)^n)/14, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-13) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1), with q=-13.
a(n) = Sum_{j=0..n-1} (-13)^j. - Zerinvary Lajos, Dec 16 2008
G.f.: x/((1 - x)*(1 + 13*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -12*a(n-1) + 13*a(n-2). - Vincenzo Librandi, Oct 22 2012
From G. C. Greubel, May 26 2018: (Start)
a(n) = (1 - (-13)^n)/14.
E.g.f.: (exp(x) - exp(-13*x))/14. (End)

Edited by N. J. A. Sloane, Jan 13 2009 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A036827 a(n) = 26 + 2^(n+1)*(-13 +9*n -3*n^2 +n^3).

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A014992 a(n) = (1 - (-10)^n)/11.

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A077953 Expansion of 1/(1-x+2*x^2-2*x^3).

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A014991 a(n) = (1 - (-9)^n)/10.

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A014993 a(n) = (1 - (-11)^n)/12.

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A140431 2*A094555(n).

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A036827 a(n) = 26 + 2^(n+1)(-13 +9n -3*n^2 +n^3).

A077953 Expansion of 1/(1-x+2x^2-2x^3).