A232600 -id:A232600 - cp's OEIS frontend

A077925 Expansion of 1/((1-x)(1+2x)).

1, -1, 3, -5, 11, -21, 43, -85, 171, -341, 683, -1365, 2731, -5461, 10923, -21845, 43691, -87381, 174763, -349525, 699051, -1398101, 2796203, -5592405, 11184811, -22369621, 44739243, -89478485, 178956971, -357913941, 715827883, -1431655765, 2863311531, -5726623061

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n+1) is the reflection of a(n) through a(n-1) on the numberline. - Floor van Lamoen, Aug 31 2004
If a zero is added as the (new) a(0) in front, the sequence represents the inverse binomial transform of A001045. Partial sums are in A077898. - R. J. Mathar, Aug 30 2008
a(n) = A077953(2*n+3). - Reinhard Zumkeller, Oct 07 2008
Related to the Fibonacci sequence by an INVERT transform: if A(x) = 1+x^2*g(x) is the generating function of the a(n) prefixed with 1, 0, then 1/A(x) = 2+(x+1)/(x^2-x+1) is the generating function of 1, 0, -1, 1, -2, 3, ..., the signed Fibonacci sequence A000045 prefixed with 1. - Gary W. Adamson, Jan 07 2011
Also: Gaussian binomial coefficients [n+1,1], or q-integers, for q=-2, diagonal k=1 in the triangular (or column r=1 in the square) array A015109. - M. F. Hasler, Nov 04 2012
With a leading zero, 0, 1, -1, 3, -5, 11, -21, 43, -85, 171, -341, 683, ... we obtain the Lucas U(-1,-2) sequence. - R. J. Mathar, Jan 08 2013
Let m = a(n). Then 18*m^2 - 12*m + 1 = A000225(2n+3). - Roderick MacPhee, Jan 17 2013

			G.f. = 1 - x + 3*x^2 - 5*x^3 + 11*x^4 - 21*x^5 + 43*x^6 - 85*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (-1,2)
Index entries related to Gaussian binomial coefficients.
Index entries for Lucas sequences

Cf. A001045 (unsigned version).
Cf. A014983, A014985, A014986. - Zerinvary Lajos, Dec 16 2008
Cf. A232600, A232601, A232602.

[(1-(-2)^(n+1))/3: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

a:=n->sum ((-2)^j, j=0..n): seq(a(n), n=0..35); # Zerinvary Lajos, Dec 16 2008

CoefficientList[Series[(1 - x)^(-1)/(1 + 2 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

a(n)=(1+(-2)^n*2)/3 \\ Charles R Greathouse IV, Jun 21 2011

[gaussian_binomial(n,1,-2) for n in range(1,35)] # Zerinvary Lajos, May 28 2009

G.f.: 1/(1+x-2*x^2).
a(n) = (1-(-2)^(n+1))/3. - Vladeta Jovovic, Apr 17 2003
a(n) = Sum_{k=0..n} (-2)^k. - Paul Barry, May 26 2003
a(n+1) - a(n) = A122803(n). - R. J. Mathar, Aug 30 2008
a(n) = Sum_{k=0..n} A112555(n,k)*(-2)^k. - Philippe Deléham, Sep 11 2009
a(n) = A082247(n+1) - 1. - Philippe Deléham, Oct 07 2009
G.f.: Q(0)/(3*x), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k - 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k-1 + 2*x)/( x*(4*k+1 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
E.g.f.: (2*exp(-2*x) + exp(x))/3. - Ilya Gutkovskiy, Nov 12 2016
a(n) = A086893(n+2) - A061547(n+3), n >= 0. - Yosu Yurramendi, Jan 16 2017
a(n) = (-1)^n*A001045(n+1). - M. F. Hasler, Feb 13 2020
a(n) - a(n-1) = a(n-1) - a(n+1) = (-2)^n, a(n+1) = - a(n) + 2*a(n-1) = 1 - 2*a(n). - Michael Somos, Feb 22 2023

A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

Original entry on oeis.org

0, -1, 1, -2, 2, -3, 3, -4, 4, -5, 5, -6, 6, -7, 7, -8, 8, -9, 9, -10, 10, -11, 11, -12, 12, -13, 13, -14, 14, -15, 15, -16, 16, -17, 17, -18, 18, -19, 19, -20, 20, -21, 21, -22, 22, -23, 23, -24, 24, -25, 25, -26, 26, -27, 27, -28, 28, -29, 29, -30, 30, -31, 31, -32, 32

Offset: 0

Clark Kimberling, May 28 2007

Pisano period lengths: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ... - R. J. Mathar, Aug 10 2012

Partial sums of A038608. - Stanislav Sykora, Nov 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
Index entries for sequences that are permutations of the integers

Sums of the form Sum_{k=0..n} k^p * q^k: A059841 (p=0,q=-1), this sequence (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), A036827 (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).

Cf. A001057, A038608.

[((-1)^n*(2*n+1)-1)/4: n in [0..80]]; // Vincenzo Librandi, Mar 29 2015

A130472:=n->ceil(n/2)*(-1)^n; seq(A130472(k), k=0..50); # Wesley Ivan Hurt, Oct 22 2013

CoefficientList[Series[(x^2-x)/(1-x^2)^2, {x, 0, 64}], x] (* Geoffrey Critzer, Sep 29 2013 *)
LinearRecurrence[{-1,1,1},{0,-1,1},70] (* Harvey P. Dale, Mar 02 2018 *)

a(n)=(-1)^n*n\2 \\ Charles R Greathouse IV, Sep 02 2015

[((-1)^n*(2*n+1) - 1)/4 for n in (0..70)] # G. C. Greubel, Mar 31 2021

a(n) = -A001057(n).
a(2n) = n, a(2n+1) = -(n+1).
a(n) = Sum_{k=0..n} k*(-1)^k.
a(n) = -a(n-1) +a(n-2) +a(n-3).
G.f.: -x/( (1-x)*(1+x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = floor( (n/2)*(-1)^n ). - Wesley Ivan Hurt, Jun 14 2013
a(n) = ceiling( n/2 )*(-1)^n. - Wesley Ivan Hurt, Oct 22 2013
a(n) = ((-1)^n*(2*n+1) - 1)/4. - Adriano Caroli, Mar 28 2015
E.g.f.: (1/4)*(-exp(x) + (1-2*x)*exp(-x) ). - G. C. Greubel, Mar 31 2021

A036799 a(n) = 2 + 2^(n+1)*(n-1).

Original entry on oeis.org

0, 2, 10, 34, 98, 258, 642, 1538, 3586, 8194, 18434, 40962, 90114, 196610, 425986, 917506, 1966082, 4194306, 8912898, 18874370, 39845890, 83886082, 176160770, 369098754, 771751938, 1610612738, 3355443202, 6979321858, 14495514626, 30064771074, 62277025794

Offset: 0

N. J. A. Sloane

This sequence is a part of a class of sequences of the type: a(n) = Sum_{i=0..n} (C^i)*(i^k). This sequence has C=2, k=1. Sequence A036800 has C=2, k=2. Suppose C >= 2, k >= 1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008
Partial sums of A036289. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011
a(n) is the number of swaps needed in the worst case, when successively inserting 2^(n+1) - 1 keys into an initially empty binary heap (thus creating a tree with n+1 full levels). - Rudy van Vliet, Nov 09 2015
a(n) is also the total path length of the complete binary tree of height n, with nodes at depths 0,...,n. Total path length is defined to be the sum of depths over all nodes. - F. Skerman, Jul 02 2017
For n >= 1, every number greater than or equal to a(n-1) can be written as a sum of (not necessarily distinct) numbers of the form 2^n - 2^k with 0 <= k < n. However, a(n-1) - 1 cannot be written in this way. See problem N1 from the 2014 International Mathematics Olympiad Shortlist. - Dylan Nelson, Jun 02 2023

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 3, 4, 9.
Problem Selection Committee for the 2014 IMO, Shortlisted Problems with Solutions for the 55th International Mathematical Olympiad. See pp. 9, 68-70.
Stanislav 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 (5,-8,4).

Cf. A000337, A036289, A036800, A232599, A232600, A232601, A232602.

a036799 n = (n-1)*2^(n+1) + 2  -- Reinhard Zumkeller, May 24 2012

[2+2^(n+1)*(n-1) : n in [0..40]]; // Wesley Ivan Hurt, Nov 12 2015

A036799:=n->2+2^(n+1)*(n-1): seq(A036799(n), n=0..40); # Wesley Ivan Hurt, Nov 12 2015

Accumulate[Table[n*2^n, {n, 0, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 09 2011 *)

a(n)=2+(n-1)<<(n+1) \\ Charles R Greathouse IV, Sep 28 2015

concat(0, Vec(2*x/((1-x)*(1-2*x)^2) + O(x^40))) \\ Altug Alkan, Nov 09 2015

[2^(n+1)*(n-1) +2 for n in (0..40)] # G. C. Greubel, Mar 29 2021

a(n) = (n-1) * 2^(n+1) + 2.
a(n) = 2 * A000337(n).
a(n) = Sum_{k=1..n} k*2^k. - Benoit Cloitre, Oct 25 2002
G.f.: 2*x/((1-x)*(1-2*x)^2). - Colin Barker, Apr 30 2012
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 12 2015
a(n) = Sum_{k=0..n} Sum_{i=0..n} k * binomial(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2*exp(x) - 2*(1-2*x)*exp(2*x). - G. C. Greubel, Mar 29 2021

A232599 Alternating sum of cubes, i.e., Sum_{k=0..n} k^p*q^k for p=3, q=-1.

Original entry on oeis.org

0, -1, 7, -20, 44, -81, 135, -208, 304, -425, 575, -756, 972, -1225, 1519, -1856, 2240, -2673, 3159, -3700, 4300, -4961, 5687, -6480, 7344, -8281, 9295, -10388, 11564, -12825, 14175, -15616, 17152, -18785, 20519

Offset: 0

Stanislav Sykora, Nov 26 2013

			a(3) = 0^3 - 1^3 + 2^3 - 3^3 = -20.

Stanislav Sykora, 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 (-3,-2,2,3,1).

Cf. A000578 (cubes), A011934 (absolute values), A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (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).

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232599:= n-> (1 -(-1)^n*(1 -6*n^2 -4*n^3))/8; seq(A232599(n), n=0..30); # G. C. Greubel, Mar 31 2021

Accumulate[Times@@@Partition[Riffle[Range[0,40]^3,{1,-1},{2,-1,2}],2]] (* Harvey P. Dale, Jul 22 2016 *)

S3M1(n)=((-1)^n*(4*n^3+6*n^2-1)+1)/8;
v = vector(10001);for(k=1,#v,v[k]=S3M1(k-1))

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(4*n^3+6*n^2-1) +1)/8.
G.f.: (-x)*(1-4*x+x^2) / ( (1-x)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (exp(x) - (1 +10*x -18*x^2 +4*x^3)*exp(-x))/8. - G. C. Greubel, Mar 31 2021
a(n) = - 3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A089594 Alternating sum of squares to n.

Original entry on oeis.org

-1, 3, -6, 10, -15, 21, -28, 36, -45, 55, -66, 78, -91, 105, -120, 136, -153, 171, -190, 210, -231, 253, -276, 300, -325, 351, -378, 406, -435, 465, -496, 528, -561, 595, -630, 666, -703, 741, -780, 820, -861, 903, -946, 990, -1035, 1081, -1128, 1176, -1225, 1275

Offset: 1

Jon Perry, Dec 30 2003

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 24 2010

Also triangular numbers with alternating signs. - Stanislav Sykora, Nov 26 2013

			a(6) = 1 + 4 - 9 + 16 - 25 + 36 = 3 + 7 + 11 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), this sequence (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (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).
Cf. A000217.
Cf. A225144. [Bruno Berselli, Jun 06 2013]

[(-1)^n*n*(n+1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011

seq(sum(binomial(n,m), m=1..2)-n^2,n=2..51); # Zerinvary Lajos, Jun 19 2008
A089594 := n -> (-1)^n*n*(n+1)/2; # Peter Luschny, Jul 08 2011

nn = Range[50]; Accumulate[(-1)^nn*nn^2] (* Jayanta Basu, Jun 06 2013 *)

for(i=1,50, print1(","sum(j=1,i,(-1)^j*j^2)))

a(n)=(-1)^n*n*(n+1)/2 \\ Charles R Greathouse IV, Jul 08 2011

[(-1)^n*binomial(n+1,2) for n in (1..50)] # G. C. Greubel, Mar 31 2021

From R. J. Mathar, Nov 05 2011: (Start)
a(n) = Sum_{i=1..n} (-1)^i*i^2 = (-1)^n*n*(n+1)/2.
G.f.: -x / (1+x)^3. (End)
a(n) = (-1)^n*det(binomial(i+2,j+1), 1 <= i,j <= n-1). - Mircea Merca, Apr 06 2013
G.f.: -W(0)/(2+2*x), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) - (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
E.g.f.: (1/2)*x*(x-2)*exp(-x). - G. C. Greubel, Mar 31 2021
Sum_{n>=1} 1/a(n) = 2 - 4*log(2). - Amiram Eldar, Jan 31 2023

A036800 a(n) = -6 + 2^(n+1)(3 - 2n + n^2).

Original entry on oeis.org

0, 2, 18, 90, 346, 1146, 3450, 9722, 26106, 67578, 169978, 417786, 1007610, 2392058, 5603322, 12976122, 29753338, 67633146, 152567802, 341835770, 761266170, 1686110202, 3716153338, 8153726970, 17817403386, 38788923386

Offset: 0

N. J. A. Sloane

This sequence is a part of a class of sequences of the type: a(n) = sum(i=0,n,(C^i)*(i^k)). This sequence has C=2, k=2. Sequence A036799 has C=2, k=1. Suppose C>=2, k>=1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008

M. Petkovsek et al., A=B, Peters, 1996, p. 97.
Jolley, Summation of Series, Dover (1961), p. 6.

Vincenzo Librandi, 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 (7,-18,20,-8).

Cf. A232599, A232600, A232601, A232602.

[-6+2^(n+1)*(3-2*n+n^2): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

A036800:= n-> 2^(n+1)*(3-2*n+n^2) -6; seq(A036800(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[ -6+2^(n+1)*(3-2*n+n^2),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
LinearRecurrence[{7,-18,20,-8},{0,2,18,90},30] (* Harvey P. Dale, Jun 13 2015 *)

a(n)=2^(n+1)*(3 - 2*n + n^2) - 6 \\ Charles R Greathouse IV, Jun 11 2015

[2^(n+1)*(3-2*n+n^2) -6 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = Sum_{k=0..n} 2^k * k^2. - Benoit Cloitre, Jun 11 2003
From R. J. Mathar, Oct 03 2011: (Start)
G.f.: 2*x*(1+2*x) / ( (1-x)*(1-2*x)^3 ).
a(n) = 2*A036826(n). (End)
a(0)=0, a(1)=2, a(2)=18, a(3)=90, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Jun 13 2015
a(n) = Sum_{k=0..n} Sum_{i=0..n} k^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2*(3 -2*x +4*x^2)*exp(2*x) -6*exp(x). - G. C. Greubel, Mar 31 2021

A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

Original entry on oeis.org

0, -2, 14, -58, 198, -602, 1702, -4570, 11814, -29658, 72742, -175066, 414758, -969690, 2241574, -5131226, 11645990, -26233818, 58700838, -130567130, 288863270, -635980762, 1394062374, -3043511258, 6620165158

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^2*2^0 - 1^2*2^1 + 2^2*2^2 - 3^2*2^3 = -58.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Stanislav 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 (-5,-6,4,8).

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), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232601:= n-> 2*(1 - (-2)^n*(1-6*n-9*n^2))/27; seq(A232601(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-5,-6,4,8},{0,-2,14,-58},30] (* Harvey P. Dale, Aug 20 2015 *)

S2M2(n)=((-1)^n*2^(n+1)*(9*n^2+6*n-1)+2)/27;
v = vector(10001); for(k=1, #v, v[k]=S2M2(k-1))

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*((-2)^n * (9*n^2 + 6*n - 1) + 1)/27.
G.f.: 2*x*(-1 + 2*x) / ((1-x)*(1+2*x)^3). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -36*x^2)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 5*a(n-1) - 6*a(n-2) + 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232602 a(n) = Sum_{k=0..n} k^p*q^k, where p=3, q=-2.

Original entry on oeis.org

0, -2, 30, -186, 838, -3162, 10662, -33242, 97830, -275418, 748582, -1977306, 5100582, -12897242, 32060454, -78531546, 189903910, -454052826, 1074770982, -2521320410, 5867287590, -13554437082

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^3*2^0 - 1^3*2^1 + 2^3*2^2 - 3^3*2^3 = -186.

Stanislav Sykora, 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 (-7,-16,-8,16,16).

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), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232602:= n-> 2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27; seq(A232602(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-7,-16,-8,16,16}, {0,-2,30,-186,838}, 40] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*2^(n+1)*(27*n^3+27*n^2-9*n-3)+6)/81;

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = 2*(1 - (-2)^n*(1 +3*n -9*n^2 -9*n^3))/27.
G.f.: -2*x*(1-8*x+4*x^2) / ( (1-x)*(1+2*x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -144*x^2 +72*x^3)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 7*a(n-1) - 16*a(n-2) - 8*a(n-3) + 16*a(n-4) + 16*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

Original entry on oeis.org

0, -1, 2, -5, 6, -13, 10, -29, 6, -69, -38, -197, -250, -669, -1142, -2509, -4762, -9813, -19302, -38965, -77530, -155501, -310518, -621565, -1242554, -2485733, -4970790, -9942309, -19883834, -39768509, -79536118

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to keep the values integer.

See also A232600 and references therein for integer values of q.

			a(3) = 2^3 * [0^2/2^0 - 1^2/2^1 + 2^2/2^2 - 3^2/2^3] = -5.

Stanislav Sykora, 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 (-1,3,5,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232604 (p=3,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232603:= n-> ((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27; seq(A232603(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-1,3,5,2}, {0,-1,2,-5}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*(9*n^2+12*n+2)-2^(n+1))/27;

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(9*n^2+12*n+2) - 2^(n+1))/27.
G.f.: x*(-1+x)/( (1-2*x)*(1+x)^3 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(-2*exp(2*x) + (2 -21*x +9*x^2)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - a(n-1) + 3*a(n-2) + 5*a(n-3) + 2*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

Original entry on oeis.org

0, -1, 6, -15, 34, -57, 102, -139, 234, -261, 478, -375, 978, -241, 2262, 1149, 6394, 7875, 21582, 36305, 80610, 151959, 314566, 616965, 1247754, 2479883, 4977342, 9935001, 19891954, 39759519, 79546038, 159062285

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to make the values integers.
See also A232600 and references therein for integer values of q.
The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - Richard R. Forberg, Feb 17 2014.

			a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.

Stanislav Sykora, 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 (-2,2,8,7,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232604:= n-> (2^(n+1) +(-1)^n*(9*n^3 +18*n^2 +6*n -2))/27; seq(A232604(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-2,2,8,7,2}, {0,-1,6,-15,34}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = (2^(n+1) + (-1)^n*(9*n^3+18*n^2+6*n-2))/27.
G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(2*exp(2*x) - (2 +33*x -45*x^2 +9*x^3)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - 2*a(n-1) + 2*a(n-2) + 8*a(n-3) + 7*a(n-4) + 2*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

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