cp's OEIS Frontend

A056107 Third spoke of a hexagonal spiral.

%I A056107 #122 Feb 16 2025 08:32:43
%S A056107 1,4,13,28,49,76,109,148,193,244,301,364,433,508,589,676,769,868,973,
%T A056107 1084,1201,1324,1453,1588,1729,1876,2029,2188,2353,2524,2701,2884,
%U A056107 3073,3268,3469,3676,3889,4108,4333,4564,4801,5044,5293,5548,5809,6076,6349
%N A056107 Third spoke of a hexagonal spiral.
%C A056107 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 A056107 Equals binomial transform of [1, 3, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, May 03 2008
%C A056107 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 A056107 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 A056107 Numbers m such that m+(m-1)+(m-2) is a square. - _César Aguilera_, May 26 2015
%C A056107 After 4, twice each term belongs to A181123: 2*a(n) = (n+1)^3 - (n-1)^3. - _Bruno Berselli_, Mar 09 2016
%C A056107 This is a subsequence of A003136: a(n) = (n-1)^2 + (n-1)*(n+1) + (n+1)^2. - _Bruno Berselli_, Feb 08 2017
%C A056107 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 A056107 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 A056107 Ben Hamilton, Brainteasers and Mindbenders, Fireside, 1992, p. 107.
%H A056107 Nathaniel Johnston, <a href="/A056107/b056107.txt">Table of n, a(n) for n = 0..5000</a>
%H A056107 Henry Bottomley, <a href="/A003215/a003215.gif">Illustration of initial terms</a>
%H A056107 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 A056107 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 A056107 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 A056107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Clique.html">Clique</a>.
%H A056107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>.
%H A056107 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A056107 a(n) = 3*n^2 + 1.
%F A056107 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
%F A056107 G.f.: (1+x+4*x^2)/(1-x)^3.
%F A056107 a(n) = a(n-1) + 6*n - 3 for n>0.
%F A056107 a(n) = 2*a(n-1) - a(n-2) + 6 for n>1.
%F A056107 a(n) = A056105(n) + 2*n = A056106(n) + n.
%F A056107 a(n) = A056108(n) - n = A056109(n) - 2*n = A003215(n) - 3*n.
%F A056107 a(n) = (A000578(n+1) - A000578(n-1))/2. - _Lekraj Beedassy_, Jul 29 2005
%F A056107 a(n) = A132111(n+1,n-1) for n>1. - _Reinhard Zumkeller_, Aug 10 2007
%F A056107 E.g.f.: (1 + 3*x + 3*x^2)*exp(x). - _G. C. Greubel_, Dec 02 2018
%F A056107 From _Amiram Eldar_, Jul 15 2020: (Start)
%F A056107 Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(3))*coth(Pi/sqrt(3)))/2.
%F A056107 Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(3))*csch(Pi/sqrt(3)))/2. (End)
%F A056107 From _Amiram Eldar_, Feb 05 2021: (Start)
%F A056107 Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(3))*sinh(sqrt(2/3)*Pi).
%F A056107 Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(3))*csch(Pi/sqrt(3)). (End)
%p A056107 seq(3*n^2+1, n=0..46); # _Nathaniel Johnston_, Jun 26 2011
%t A056107 Table[3 n^2 + 1, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%t A056107 LinearRecurrence[{3, -3, 1}, {1, 4, 13}, 47] (* _Michael De Vlieger_, Feb 08 2017 *)
%t A056107 CoefficientList[Series[(1 + x + 4 x^2)/(1 - x)^3, {x, 0, 46}], x] (* _Michael De Vlieger_, Feb 08 2017 *)
%t A056107 1 + 3 Range[0, 20]^2 (* _Eric W. Weisstein_, Nov 30 2017 *)
%o A056107 (PARI) for(n=0,1000,if(issquare(n+(n-1)+(n-2)),print1(n", "))); \\ _César Aguilera_, May 26 2015
%o A056107 (PARI) a(n) = 3*n^2 + 1; \\ _Altug Alkan_, Feb 08 2017
%o A056107 (Magma) [3*n^2 + 1: n in [0..40]]; // _G. C. Greubel_, Dec 02 2018
%o A056107 (Sage) [3*n^2 + 1 for n in range(40)] # _G. C. Greubel_, Dec 02 2018
%o A056107 (GAP) List([0..40], n -> 3*n^2 + 1); # _G. C. Greubel_, Dec 02 2018
%Y A056107 Cf. A002648 (prime terms), A201053.
%Y A056107 Cf. A000578, A003136, A132111, A181123.
%Y A056107 Other spokes: A003215, A056105, A056106, A056108, A056109.
%Y A056107 Other spirals: A054552.
%K A056107 nonn,easy
%O A056107 0,2
%A A056107 _Henry Bottomley_, Jun 09 2000