A056520 a(n) = (n + 2)*(2*n^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

Author

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

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

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

Cf. A030238, A253903.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Formula

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