A056108 Fourth spoke of a hexagonal spiral.

1, 5, 15, 31, 53, 81, 115, 155, 201, 253, 311, 375, 445, 521, 603, 691, 785, 885, 991, 1103, 1221, 1345, 1475, 1611, 1753, 1901, 2055, 2215, 2381, 2553, 2731, 2915, 3105, 3301, 3503, 3711, 3925, 4145, 4371, 4603, 4841, 5085, 5335, 5591, 5853, 6121, 6395

Offset: 0

Author

Henry Bottomley, Jun 09 2000

Keywords

• easy
• nonn

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(n) = (3*n+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).

Crossrefs

Cf. A134234, A000217.

Cf. A005563, A016777, A128229.

Other spokes: A003215, A056105, A056106, A056107, A056109.

Other spirals: A054552.

Programs

Magma

[3*n^2+n+1: n in [0..50]]; // Bruno Berselli, Mar 13 2013

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 *)

PARI

a(n)=3*n^2+n+1 \\ Charles R Greathouse IV, Jun 17 2017

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) = A056105(n) + 3*n = A056106(n) + 2*n = A056107(n) + n = A056109(n) - n = A003215(n) - 2*n.

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