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A056108
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Fourth spoke of a hexagonal spiral.
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36
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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
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0,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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Table[3 n^2 + n + 1, {n, 0, 50}] (* Bruno Berselli, Mar 13 2013 *)
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PROG
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(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
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CROSSREFS
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Cf. A134234, A000217.
Cf. A005563, A016777, A128229.
Other spokes: A003215, A056105, A056106, A056107, A056109.
Other spirals: A054552.
Sequence in context: A048021 A225325 A133268 * A055831 A037984 A298032
Adjacent sequences: A056105 A056106 A056107 * A056109 A056110 A056111
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, Jun 09 2000
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STATUS
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approved
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