

A056108


Fourth spoke of a hexagonal spiral.


34



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
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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 4subset of an nset X then, for n>=4, a(n4) is the number of 4subsets 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^2h+1)^2+(h^2+h1)^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
H. 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 kcommuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 20142015.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 3*n^2 + n + 1.
a(n) = a(n1) + 6*n  2 = 2*a(n1)  a(n2) + 6
a(n) = 3*a(n1)  3*a(n2) + a(n3).
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(n1)2 with n>0, a(0)=1.  Vincenzo Librandi, Aug 07 2010
G.f.: (1+2*x+3*x^2)/(13*x+3*x^2x^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 *)


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. A054552 for example of square (or octagonal) spiral spoke.
Cf. A134234, A000217.
Sequence in context: A048021 A225325 A133268 * A055831 A037984 A298032
Adjacent sequences: A056105 A056106 A056107 * A056109 A056110 A056111


KEYWORD

easy,nonn


AUTHOR

Henry Bottomley, Jun 09 2000


STATUS

approved



