OFFSET
0,2
COMMENTS
a(n) = sum of (n+1)-th row terms of triangle A134234. - Gary W. Adamson, Oct 14 2007
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
Equals binomial transform of [1, 4, 6, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008
From A.K. Devaraj, Sep 18 2009: (Start)
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod(f(x)); here n belongs to N.
There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z.
However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x.
The present sequence is the integer part when the polynomial is x^2 + x + 1 and x = sqrt(2),
f(x+n*f(x))/f(x) = a(n) + A005563(n)*sqrt(2).
Equals triangle A128229 as an infinite lower triangular matrix * A016777 as a vector, where A016777 = (3n+1).
(End)
Numbers of the form ((-h^2+h+1)^2+(h^2-h+1)^2+(h^2+h-1)^2)/(h^2+h+1) for h=n+1. - Bruno Berselli, Mar 13 2013
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 3*n^2 + n + 1.
a(n) = a(n-1) + 6*n - 2 = 2*a(n-1) - a(n-2) + 6
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A096777(3n+1) . - Reinhard Zumkeller, Dec 29 2007
a(n) = 6*n+a(n-1)-2 with n>0, a(0)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: (1+2*x+3*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
a(-n) = A056106(n). - Bruno Berselli, Mar 13 2013
E.g.f.: (3*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jul 19 2017
MATHEMATICA
Table[3 n^2 + n + 1, {n, 0, 50}] (* Bruno Berselli, Mar 13 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 5, 15}, 50] (* Harvey P. Dale, Dec 26 2023 *)
PROG
(Magma) [3*n^2+n+1: n in [0..50]]; // Bruno Berselli, Mar 13 2013
(PARI) a(n)=3*n^2+n+1 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Jun 09 2000
STATUS
approved