OFFSET
1,3
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).
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
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 *)
PROG
(PARI) a(n)=(n^3-7*n)/6+2 \\ Charles R Greathouse IV, Mar 26 2012
(Maxima) A105163(n):=(n^3 - 7*n + 12)/6$ makelist(A105163(n), n, 1, 20); /* Martin Ettl, Dec 18 2012 */
(Python) for n in range(1, 45): print((n**3 - 7*n + 12)/6, end=', ') # Stefano Spezia, Jan 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Apr 10 2005
STATUS
approved