A064999 Partial sums of sequence (essentially A002378): 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...

1, 3, 9, 21, 41, 71, 113, 169, 241, 331, 441, 573, 729, 911, 1121, 1361, 1633, 1939, 2281, 2661, 3081, 3543, 4049, 4601, 5201, 5851, 6553, 7309, 8121, 8991, 9921, 10913, 11969, 13091, 14281, 15541, 16873, 18279, 19761, 21321, 22961, 24683

Offset: 0

Author

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Oct 31 2001

Keywords

• easy
• nonn

Comments

Equals triangle A144328 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 18 2008

a(n) is the number of parking functions of size n+1 avoiding the patterns 123 and 312. - Lara Pudwell, Apr 10 2023

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• 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.
• Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002378, A007290.

Cf. A144328. - Gary W. Adamson, Sep 18 2008

Programs

Magma

[(n^3+3*n^2+2*n+3)/3: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^2-n od: seq(a[n], n=0..42); # Zerinvary Lajos, Jun 05 2008

Mathematica

Table[(x^3 - x + 3)/3, {x, 1, 100}] (* Artur Jasinski, Feb 14 2007 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 3, 9, 21}, 50] (* Vincenzo Librandi, Feb 28 2016 *)

PARI

{ for (n=0, 1000, if (n, a+=n*(n + 1), a=1); write("b064999.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009

PARI

a(n) = (n^3+3*n^2+2*n+3)/3; \\ Altug Alkan, May 16 2018

Formula

a(n) = A007290(n+2) + 1 = (n^3 + 3*n^2 + 2*n + 3)/3.

a(0) = 1, a(n) = n*(n+1) + a(n-1) for n > 1. - Gerald McGarvey, Sep 26 2004

O.g.f.: (1 - x + 3x^2 - x^3)/(1 - x)^4.

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Nov 12 2001