A179904 -id:A179904 - cp's OEIS frontend

A056520 a(n) = (n + 2)(2n^2 - n + 3)/6.

1, 2, 6, 15, 31, 56, 92, 141, 205, 286, 386, 507, 651, 820, 1016, 1241, 1497, 1786, 2110, 2471, 2871, 3312, 3796, 4325, 4901, 5526, 6202, 6931, 7715, 8556, 9456, 10417, 11441, 12530, 13686, 14911, 16207, 17576, 19020, 20541, 22141, 23822

Offset: 0

N. J. A. Sloane, Laura Kasavan (maui12129(AT)cswebmail.com), Aug 26 2000

Hankel transform of A030238. - Paul Barry, Oct 16 2007
Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 5, 7, 9, ...). - Gary W. Adamson, Jul 31 2010
a(n) equals n!^2 times the determinant of the n X n matrix whose (i,j)-entry is 1 + KroneckerDelta[i, j] (-1 + (1 + i^2)/i^2). - John M. Campbell, May 20 2011
Positions of ones in A253903 (with offset 1). - Harvey P. Dale, Mar 05 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A153056, A153057, A153058, A179904.

Cf. A030238, A253903.

[(n+2)*(2*n^2-n+3)/6: n in [0..40]]; // Vincenzo Librandi, May 24 2011

a[n_] := (n+2)*(2*n^2-n+3)/6; Table[a[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
s = 1; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 1, 41, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
Table[n!^2*Det[Array[KroneckerDelta[#1,#2](((#1^2+1)/(#1^2))-1)+1&,{n,n}]],{n,1,20}] (* John M. Campbell, May 20 2011 *)
FoldList[#1 + #2^2 &, 1, Range@ 40] (* Robert G. Wilson v, Oct 28 2011 *)

a(n)=(n+2)*(2*n^2-n+3)/6 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = a(n-1) + n^2.
a(n) = A000330(n) + 1.
G.f.: (1 - 2*x + 4*x^2 - x^3)/(1 - x)^4. - Paul Barry, Apr 14 2010
Let b(0) = b(1) = 1, b(n) = max(b(n-1) + (n - 1)^2, b(n-2) + (n - 2)^2) for n >= 2; then a(n) = b(n+1). - Yalcin Aktar, Jul 28 2011

A153058 a(0)=4; a(n)=n^2+a(n-1) for n>0.

Original entry on oeis.org

4, 5, 9, 18, 34, 59, 95, 144, 208, 289, 389, 510, 654, 823, 1019, 1244, 1500, 1789, 2113, 2474, 2874, 3315, 3799, 4328, 4904, 5529, 6205, 6934, 7718, 8559, 9459, 10420, 11444, 12533, 13689, 14914, 16210, 17579, 19023, 20544, 22144, 23825, 25589

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Indranil Ghosh, Table of n, a(n) for n = 0..64950
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A000330, A056520, A153056, A153057, A179904.

a=4;lst={};Do[a=n^2+a;AppendTo[lst,a],{n,0,5!}];lst
RecurrenceTable[{a[0]==4,a[n]==n^2+a[n-1]},a,{n,50}] (* Harvey P. Dale, Apr 27 2012 *)

G.f.: (4-11x+13x^2-4x^3)/(1-x)^4. a(n)=4+A000330(n). - R. J. Mathar, Jan 17 2009

Added indices to definition and corrected offset. - R. J. Mathar, Jan 17 2009

A153056 a(0)=2, a(n) = n^2+a(n-1).

Original entry on oeis.org

2, 3, 7, 16, 32, 57, 93, 142, 206, 287, 387, 508, 652, 821, 1017, 1242, 1498, 1787, 2111, 2472, 2872, 3313, 3797, 4326, 4902, 5527, 6203, 6932, 7716, 8557, 9457, 10418, 11442, 12531, 13687, 14912, 16208, 17577, 19021, 20542, 22142, 23823, 25587

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153057, A153058, A179904.

a=2;lst={};Do[a=n^2+a;AppendTo[lst,a],{n,0,5!}];lst
nxt[{n_,a_}]:={n+1,(n+1)^2+a}; NestList[nxt,{0,2},50][[;;,2]] (* or *) LinearRecurrence[{4,-6,4,-1},{2,3,7,16},50] (* Harvey P. Dale, Sep 05 2023 *)

a(n) = n*(n+1)*(2*n+1)/6 + 2; \\ Altug Alkan, Apr 30 2018

G.f.: (2-5x+7x^2-2x^3)/(1-x)^4. a(n)=2+n(1+2n^2+3n)/6 = 2+A000330(n). - R. J. Mathar, Jan 08 2009

A153057 a(0)=3; a(n) = n^2 + a(n-1) for n>0.

Original entry on oeis.org

3, 4, 8, 17, 33, 58, 94, 143, 207, 288, 388, 509, 653, 822, 1018, 1243, 1499, 1788, 2112, 2473, 2873, 3314, 3798, 4327, 4903, 5528, 6204, 6933, 7717, 8558, 9458, 10419, 11443, 12532, 13688, 14913, 16209, 17578, 19022, 20543, 22143, 23824, 25588

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153056, A153058, A179904.

I:=[3,4,8,17]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, May 09 2017

a=3;lst={};Do[a=n^2+a;AppendTo[lst,a],{n,0,5!}];lst
CoefficientList[Series[(3 - 8 x + 10 x^2 - 3 x^3) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 09 2017 *)

From R. J. Mathar, Jan 17 2009: (Start)
G.f.: (3-8*x + 10*x^2 - 3*x^3)/(1 - x)^4.
a(n) = 3+A000330(n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 09 2017

Added indices to definition. Corrected offset R. J. Mathar, Jan 17 2009

cp's OEIS Frontend

A056520 a(n) = (n + 2)*(2*n^2 - n + 3)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A153058 a(0)=4; a(n)=n^2+a(n-1) for n>0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A153056 a(0)=2, a(n) = n^2+a(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A153057 a(0)=3; a(n) = n^2 + a(n-1) for n>0.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A056520 a(n) = (n + 2)(2n^2 - n + 3)/6.