A232604 -id:A232604 - cp's OEIS frontend

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

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

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

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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

Offset: 0

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

Growth of happy bug population in GCSE math course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - Wolfdieter Lang, Jun 25 2010
With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013
From Paul Curtz, Nov 02 2021 (Start)
a(n-2) difference table (from 0, 0, a(n)):
    0    0    1    0    3    2    9    12    31    54  ...
    0    1   -1    3   -1    7    3    19    23    63  ...
    1   -2    4   -4    8   -4   16     4    40    44  ...
   -3    6   -8   12  -12   20  -12    36     4    84  ...
    9  -14   20  -24   32  -32   48   -32    80     0  ...
  -23   34  -44   56  -64   80  -80   112   -80   176  ...
   57  -78  100 -120  144 -160  192  -192   256  -192  ...
   ... .
The signature is valid for every row.
a(n-2) + a(n-1) = A001045(n).
a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
a(n-2) + a(n+3) = see A144472(n+1).
Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
First subdiagonal: -A036895(n) = -2*A001787(n).
Main diagonal: A001787(n) = -first and -third upper diagonals.
Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

G.f.: 1 / (1-3*x^2-2*x^3).
With offset 1: a(1)=1; a(n) = 2*a(n-1) - (-1)^n*n; a(n) = (1/9)*(2^(n+1) - (-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002
a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004
a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012
E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

More terms from James Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000

A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

Original entry on oeis.org

0, -2, 6, -18, 46, -114, 270, -626, 1422, -3186, 7054, -15474, 33678, -72818, 156558, -334962, 713614, -1514610, 3203982, -6757490, 14214030, -29826162, 62448526, -130489458, 272163726, -566697074, 1178133390, -2445745266, 5070447502, -10498808946, 21713445774

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.

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 (-3,0,4).

Cf. A045883, A140960 (absolute values), 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=q=2), A036827 (p=3, q=2), A077925 (p=0, 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. A045883.

[2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232600:= n-> 2*((-2)^n*(3*n+1) -1)/9; seq(A232600(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* Bruno Berselli, Nov 28 2013 *)

PARI
```
a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;
```

[2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.
abs(a(n)) = 2*A045883(n) = A140960(n).
From Bruno Berselli, Nov 28 2013: (Start)
G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by Georg Fischer, May 11 2019]
a(n) = -3*a(n-1) +4*a(n-3). (End)
From G. C. Greubel, Mar 31 2021: (Start)
E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).
a(n) = 2*(-1)^n*A045883(n). (End)

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

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

Original entry on oeis.org

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

A213575 Antidiagonal sums of the convolution array A213573.

Original entry on oeis.org

1, 10, 47, 158, 441, 1098, 2539, 5590, 11909, 24818, 50967, 103662, 209521, 421786, 846947, 1697990, 3400893, 6807618, 13622095, 27252190, 54513641, 109037930, 218088027, 436189878, 872395381, 1744808338, 3489636359

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..500
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 (6,-14,16,-9,2).

Cf. A213564, A213500, A232603, A232604.

List([1..30], n-> 13*2^(n+1)-(n^3+6*n^2+18*n+26)); # G. C. Greubel, Jul 25 2019

[13*2^(n+1)-(n^3+6*n^2+18*n+26): n in [1..30]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n^2;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213573 *)
d = Table[t[n, n], {n, 1, 40}] (* A213574 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
(* Additional programs *)
Table[Sum[k^3*2^(n-k),{k,0,n}],{n,1,30}] (* Vaclav Kotesovec, Nov 28 2013 *)

vector(30, n, 13*2^(n+1)-(n^3+6*n^2+18*n+26)) \\ G. C. Greubel, Jul 25 2019

[13*2^(n+1)-(n^3+6*n^2+18*n+26) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
G.f.: x*(1 + 4 x + x^2)/((1 - 2*x)*(1 - x)^4).
From Stanislav Sykora, Nov 27 2013: (Start)
a(n) = 2^n*Sum_{k=0..n} k^p*q^k, for p=3, q=1/2.
a(n) = 2^(n+1)*13 - (n^3 + 6*n^2 + 18*n + 26). (End)
a(n) = 2*a(n-1) + n^3. - Sochima Everton, Biereagu, Jul 14 2019
E.g.f.: 26*exp(2*x) - (26 +25*x +9*x^2 +x^3)*exp(x). - G. C. Greubel, Jul 25 2019

cp's OEIS Frontend

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

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

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A089594 Alternating sum of squares to n.

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A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

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

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A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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