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%I #63 Dec 26 2023 17:36:52
%S 1,5,15,31,53,81,115,155,201,253,311,375,445,521,603,691,785,885,991,
%T 1103,1221,1345,1475,1611,1753,1901,2055,2215,2381,2553,2731,2915,
%U 3105,3301,3503,3711,3925,4145,4371,4603,4841,5085,5335,5591,5853,6121,6395
%N Fourth spoke of a hexagonal spiral.
%C a(n) = sum of (n+1)-th row terms of triangle A134234. - _Gary W. Adamson_, Oct 14 2007
%C 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
%C Equals binomial transform of [1, 4, 6, 0, 0, 0, ...] - _Gary W. Adamson_, Apr 30 2008
%C From _A.K. Devaraj_, Sep 18 2009: (Start)
%C 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.
%C There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z.
%C However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x.
%C The present sequence is the integer part when the polynomial is x^2 + x + 1 and x = sqrt(2),
%C f(x+n*f(x))/f(x) = a(n) + A005563(n)*sqrt(2).
%C Equals triangle A128229 as an infinite lower triangular matrix * A016777 as a vector, where A016777 = (3n+1).
%C (End)
%C 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
%H G. C. Greubel, <a href="/A056108/b056108.txt">Table of n, a(n) for n = 0..5000</a>
%H Henry Bottomley, <a href="/A003215/a003215.gif">Illustration of initial terms</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*n^2 + n + 1.
%F a(n) = a(n-1) + 6*n - 2 = 2*a(n-1) - a(n-2) + 6
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = A056105(n) + 3*n = A056106(n) + 2*n = A056107(n) + n = A056109(n) - n = A003215(n) - 2*n.
%F a(n) = A096777(3n+1) . - _Reinhard Zumkeller_, Dec 29 2007
%F a(n) = 6*n+a(n-1)-2 with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 07 2010
%F G.f.: (1+2*x+3*x^2)/(1-3*x+3*x^2-x^3). - _Colin Barker_, Jan 04 2012
%F a(-n) = A056106(n). - _Bruno Berselli_, Mar 13 2013
%F E.g.f.: (3*x^2 + 4*x + 1)*exp(x). - _G. C. Greubel_, Jul 19 2017
%t Table[3 n^2 + n + 1, {n, 0, 50}] (* _Bruno Berselli_, Mar 13 2013 *)
%t LinearRecurrence[{3,-3,1},{1,5,15},50] (* _Harvey P. Dale_, Dec 26 2023 *)
%o (Magma) [3*n^2+n+1: n in [0..50]]; // _Bruno Berselli_, Mar 13 2013
%o (PARI) a(n)=3*n^2+n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A134234, A000217.
%Y Cf. A005563, A016777, A128229.
%Y Other spokes: A003215, A056105, A056106, A056107, A056109.
%Y Other spirals: A054552.
%K easy,nonn
%O 0,2
%A _Henry Bottomley_, Jun 09 2000
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