A153057 -id:A153057 - 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

A179904 a(n) = A056520(n)+1 for n>0, a(0)=1.

Original entry on oeis.org

1, 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

Gary W. Adamson, Jul 31 2010

Original name: (1,3,5,7,9,..) = A005408 convolved with (1,0,2,3,4,..) = 1 followed by A087156.

			a(3) = 16 = 1 + A056520(3) = (1 + 15).
a(4) = 32 = (9, 7, 5, 3, 1) dot (1, 0, 2, 3, 4) = (9 + 0 + 10 + 9 + 4).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A056520, A153056, A153057, A153058.

LinearRecurrence[{4,-6,4,-1},{1,3,7,16,32},50] (* Harvey P. Dale, Apr 25 2020 *)

From Bruno Berselli, Aug 26 2011: (Start)
G.f.: (1 + x)*(1 - 2*x + 3*x^2 - x^3)/(1 - x)^4.
a(n) = (1/6)*(2*n^3 + 3*n^2 + n + 12) for n>0, a(0)=1. (End)
a(n) = A153056(n) for n > 0. - Georg Fischer, Oct 24 2018

More terms and a(20) added by Bruno Berselli, Aug 26 2011

A155753 a(n) = (n^3 - n + 9)/3.

Original entry on oeis.org

3, 5, 11, 23, 43, 73, 115, 171, 243, 333, 443, 575, 731, 913, 1123, 1363, 1635, 1941, 2283, 2663, 3083, 3545, 4051, 4603, 5203, 5853, 6555, 7311, 8123, 8993, 9923, 10915, 11971, 13093, 14283, 15543, 16875, 18281, 19763, 21323, 22963

Offset: 1

Vincenzo Librandi, Jan 26 2009

Bruno Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A007290, A153057.

f[n_]:=(n^3 -n +9)/3; f[Range[1,100]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011*)
LinearRecurrence[{4,-6,4,-1}, {3,5,11,23}, 50] (* Harvey P. Dale, Oct 20 2011 *)

a(n)=(n^3-n)/3+3 \\ Charles R Greathouse IV, Jan 11 2012

[(n^3 -n +9)/3 for n in (1..50)] # G. C. Greubel, Jun 05 2021

a(n) = a(n-1) + n*(n-1), with a(1)=3 .
From Bruno Berselli, Jun 21 2010: (Start)
G.f.: x*(3 -9*x +11*x^2 -3*x^3)/(1-x)^4.
a(n) + a(n-1) = 2*A153057(n-1) (n>1).
a(n) + A000217(n) = A153057(n) (n>0).
a(n) - 4*a(n-1) + 6*a(n-2) - 4*a(n-3) + a(n-4) = 0 with n>4.
a(n) = 3 + A007290(n+1) = (n^3 - n + 9)/3. (End)
E.g.f.: (1/3)*(-9 + (9 + 3*x^2 + x^3)*exp(x)). - G. C. Greubel, Jun 05 2021

Entries confirmed by John W. Layman, Jun 17 2010
Edited by Bruno Berselli, Aug 12 2010
New name from Charles R Greathouse IV, Jan 11 2012

cp's OEIS Frontend

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

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

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

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A179904 a(n) = A056520(n)+1 for n>0, a(0)=1.

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A155753 a(n) = (n^3 - n + 9)/3.

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