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

0,2

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) = 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

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

Cf. A134234, A000217.

Cf. A005563, A016777, A128229.

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

Other spirals: A054552.

Sequence in context: A048021 A225325 A133268 * A055831 A346823 A037984

Adjacent sequences: A056105 A056106 A056107 * A056109 A056110 A056111

Keyword

easy,nonn

Author

Henry Bottomley, Jun 09 2000

Status

approved