A105163 a(n) = (n^3 - 7*n + 12)/6.

1, 1, 3, 8, 17, 31, 51, 78, 113, 157, 211, 276, 353, 443, 547, 666, 801, 953, 1123, 1312, 1521, 1751, 2003, 2278, 2577, 2901, 3251, 3628, 4033, 4467, 4931, 5426, 5953, 6513, 7107, 7736, 8401, 9103, 9843, 10622, 11441, 12301, 13203, 14148, 15137, 16171

Offset: 1

Author

Creighton Dement, Apr 10 2005

Keywords

• nonn
• easy

Comments

A floretion-generated sequence relating to the sequence "A class of Boolean functions of n variables and rank 2" (among several others- see link "Sequences in Context").

a(n) is the number of P-position in 2-modular Nim with n-1 piles. - Tanya Khovanova and Karan Sarkar, Jan 10 2016

a(n) is the number of parking functions of size n-1 avoiding the patterns 123 and 231. - Lara Pudwell, Apr 10 2023

a(n) is the number of length (n-2) strings on the alphabet {0,1,2} with digit sum at most 3. - Daniel T. Martin, May 23 2023

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article #10.7.8.
• Nurul Hilda Syani Putri, Mashadi, and Sri Gemawati, Sequences from heptagonal pyramid corners of integer, International Mathematical Forum, Vol. 13, 2018, no. 4, 193-200.
• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A005581, A102169, A181971.

Programs

Maple

seq(binomial(n+1, n-2)-n+2, n=1..44); # Zerinvary Lajos, Mar 21 2008

Mathematica

Rest@ CoefficientList[Series[x (1 - 3 x + 5 x^2 - 2 x^3)/(1 - x)^4, {x, 0, 46}], x] (* or *)
Array[(#^3 - 7 # + 12)/6 &, 46] (* Michael De Vlieger, Nov 18 2019 *)

Maxima

A105163(n):=(n^3 - 7*n + 12)/6$ makelist(A105163(n),n,1,20); /* Martin Ettl, Dec 18 2012 */

PARI

a(n)=(n^3-7*n)/6+2 \\ Charles R Greathouse IV, Mar 26 2012

Python

for n in range(1,45): print((n**3 - 7*n + 12)/6, end=', ') # Stefano Spezia, Jan 05 2019

Formula

a(n) = A005581(n) + 1.

a(n) = C(n+1,n-2) - n + 2. - Zerinvary Lajos, Mar 21 2008

Sequence starting (1, 3, 8, 17, ...) = binomial transform of [1, 2, 3, 1, 0, 0, 0, ...]. - Gary W. Adamson, Apr 24 2008

G.f.: x*(1 - 3*x + 5*x^2 - 2*x^3)/(1 - x)^4. - Colin Barker, Mar 26 2012

a(n) = A181971(n,3) for n > 2. - Reinhard Zumkeller, Jul 09 2012

a(n) = 2*a(n-1) - a(n-2) + n - 1, for all n in Z. - Gionata Neri, Jul 28 2016

a(n) = A000292(n-2) + A000124(n-2). - Torlach Rush, Aug 06 2018