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Third spoke of a hexagonal spiral.
46

%I #121 Jan 09 2023 17:27:49

%S 1,4,13,28,49,76,109,148,193,244,301,364,433,508,589,676,769,868,973,

%T 1084,1201,1324,1453,1588,1729,1876,2029,2188,2353,2524,2701,2884,

%U 3073,3268,3469,3676,3889,4108,4333,4564,4801,5044,5293,5548,5809,6076,6349

%N Third spoke of a hexagonal spiral.

%C a(n+1) is the number of lines crossing n cells of an n X n X n cube. - _Lekraj Beedassy_, Jul 29 2005

%C Equals binomial transform of [1, 3, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, May 03 2008

%C Each term a(n), with n>1 represents the area of the right trapezoid with bases whose values are equal to hex number A003215(n) and A003215(n+1)and height equal to 1. The right trapezoid is formed by a rectangle with the sides equal to A003215(n) and 1 and a right triangle whose area is 3*n with the greater cathetus equal to the difference A003215(n+1)-A003215(n). - _Giacomo Fecondo_, Jun 11 2010

%C 2*a(n)^2 is of the form x^4+y^4+(x+y)^4. In fact, 2*a(n)^2 = (n-1)^4+(n+1)^4+(2n)^4. - _Bruno Berselli_, Jul 16 2013

%C Numbers m such that m+(m-1)+(m-2) is a square. - _César Aguilera_, May 26 2015

%C After 4, twice each term belongs to A181123: 2*a(n) = (n+1)^3 - (n-1)^3. - _Bruno Berselli_, Mar 09 2016

%C This is a subsequence of A003136: a(n) = (n-1)^2 + (n-1)*(n+1) + (n+1)^2. - _Bruno Berselli_, Feb 08 2017

%C For n > 3, also the number of (not necessarily maximal) cliques in the n X n torus grid graph. - _Eric W. Weisstein_, Nov 30 2017

%D Edward J. Barbeau, Murray S. Klamkin and William O. J. Moser, Five Hundred Mathematical Challenges, MAA, Washington DC, 1995, Problem 444, pp. 42 and 195.

%D Ben Hamilton, Brainteasers and Mindbenders, Fireside, 1992, p. 107.

%H Nathaniel Johnston, <a href="/A056107/b056107.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 A. J. C. Cunningham, <a href="/A056107/a056107.pdf">Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n]</a>, Messenger Math., 54 (1924), 17-21 [Incomplete annotated scanned copy]

%H Gabriele 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 A. L. Rubinoff and Leo Moser, <a href="http://www.jstor.org/stable/2305750">Solution to Problem E773</a>, The American Mathematical Monthly, Vol. 55, No. 2 (Feb., 1948), p. 99.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>.

%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 + 1.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.

%F G.f.: (1+x+4*x^2)/(1-x)^3.

%F a(n) = a(n-1) + 6*n - 3 for n>0.

%F a(n) = 2*a(n-1) - a(n-2) + 6 for n>1.

%F a(n) = A056105(n) + 2*n = A056106(n) + n.

%F a(n) = A056108(n) - n = A056109(n) - 2*n = A003215(n) - 3*n.

%F a(n) = (A000578(n+1) - A000578(n-1))/2. - _Lekraj Beedassy_, Jul 29 2005

%F a(n) = A132111(n+1,n-1) for n>1. - _Reinhard Zumkeller_, Aug 10 2007

%F E.g.f.: (1 + 3*x + 3*x^2)*exp(x). - _G. C. Greubel_, Dec 02 2018

%F From _Amiram Eldar_, Jul 15 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(3))*coth(Pi/sqrt(3)))/2.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(3))*csch(Pi/sqrt(3)))/2. (End)

%F From _Amiram Eldar_, Feb 05 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(3))*sinh(sqrt(2/3)*Pi).

%F Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(3))*csch(Pi/sqrt(3)). (End)

%p seq(3*n^2+1, n=0..46); # _Nathaniel Johnston_, Jun 26 2011

%t Table[3 n^2 + 1, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)

%t LinearRecurrence[{3, -3, 1}, {1, 4, 13}, 47] (* _Michael De Vlieger_, Feb 08 2017 *)

%t CoefficientList[Series[(1 + x + 4 x^2)/(1 - x)^3, {x, 0, 46}], x] (* _Michael De Vlieger_, Feb 08 2017 *)

%t 1 + 3 Range[0, 20]^2 (* _Eric W. Weisstein_, Nov 30 2017 *)

%o (PARI) for(n=0,1000,if(issquare(n+(n-1)+(n-2)),print1(n", "))); \\ _César Aguilera_, May 26 2015

%o (PARI) a(n) = 3*n^2 + 1; \\ _Altug Alkan_, Feb 08 2017

%o (Magma) [3*n^2 + 1: n in [0..40]]; // _G. C. Greubel_, Dec 02 2018

%o (Sage) [3*n^2 + 1 for n in range(40)] # _G. C. Greubel_, Dec 02 2018

%o (GAP) List([0..40], n -> 3*n^2 + 1); # _G. C. Greubel_, Dec 02 2018

%Y Cf. A002648 (prime terms), A201053.

%Y Cf. A000578, A003136, A132111, A181123.

%Y Other spokes: A003215, A056105, A056106, A056108, A056109.

%Y Other spirals: A054552.

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Jun 09 2000