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 A056107 Third spoke of a hexagonal spiral. 45
 1, 4, 13, 28, 49, 76, 109, 148, 193, 244, 301, 364, 433, 508, 589, 676, 769, 868, 973, 1084, 1201, 1324, 1453, 1588, 1729, 1876, 2029, 2188, 2353, 2524, 2701, 2884, 3073, 3268, 3469, 3676, 3889, 4108, 4333, 4564, 4801, 5044, 5293, 5548, 5809, 6076, 6349 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n+1) is the number of lines crossing n cells of an n X n X n cube. - Lekraj Beedassy, Jul 29 2005 Equals binomial transform of [1, 3, 6, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008 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 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 Numbers m such that m+(m-1)+(m-2) is a square. - César Aguilera, May 26 2015 After 4, twice each term belongs to A181123: 2*a(n) = (n+1)^3 - (n-1)^3. - Bruno Berselli, Mar 09 2016 This is a subsequence of A003136: a(n) = (n-1)^2 + (n-1)*(n+1) + (n+1)^2. - Bruno Berselli, Feb 08 2017 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 REFERENCES 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. Ben Hamilton, Brainteasers and Mindbenders, Fireside, 1992, p. 107. LINKS Nathaniel Johnston, Table of n, a(n) for n = 0..5000 Henry Bottomley, Illustration of initial terms A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21 [Incomplete annotated scanned copy] Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2. A. L. Rubinoff and Leo Moser, Solution to Problem E773, The American Mathematical Monthly, Vol. 55, No. 2 (Feb., 1948), p. 99. Eric Weisstein's World of Mathematics, Clique. Eric Weisstein's World of Mathematics, Torus Grid Graph. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 3*n^2 + 1. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. G.f.: (1+x+4*x^2)/(1-x)^3. a(n) = a(n-1) + 6*n - 3 for n>0. a(n) = 2*a(n-1) - a(n-2) + 6 for n>1. a(n) = A056105(n) + 2*n = A056106(n) + n. a(n) = A056108(n) - n = A056109(n) - 2*n = A003215(n) - 3*n. a(n) = (A000578(n+1) - A000578(n-1))/2. - Lekraj Beedassy, Jul 29 2005 a(n) = A132111(n+1,n-1) for n>1. - Reinhard Zumkeller, Aug 10 2007 E.g.f.: (1 + 3*x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018 From Amiram Eldar, Jul 15 2020: (Start) Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(3))*coth(Pi/sqrt(3)))/2. Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(3))*csch(Pi/sqrt(3)))/2. (End) From Amiram Eldar, Feb 05 2021: (Start) Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(3))*sinh(sqrt(2/3)*Pi). Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(3))*csch(Pi/sqrt(3)). (End) MAPLE seq(3*n^2+1, n=0..46); # Nathaniel Johnston, Jun 26 2011 MATHEMATICA Table[3 n^2 + 1, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *) LinearRecurrence[{3, -3, 1}, {1, 4, 13}, 47] (* Michael De Vlieger, Feb 08 2017 *) CoefficientList[Series[(1 + x + 4 x^2)/(1 - x)^3, {x, 0, 46}], x] (* Michael De Vlieger, Feb 08 2017 *) 1 + 3 Range[0, 20]^2 (* Eric W. Weisstein, Nov 30 2017 *) PROG (PARI) for(n=0, 1000, if(issquare(n+(n-1)+(n-2)), print1(n", "))); \\ César Aguilera, May 26 2015 (PARI) a(n) = 3*n^2 + 1; \\ Altug Alkan, Feb 08 2017 (Magma) [3*n^2 + 1: n in [0..40]]; // G. C. Greubel, Dec 02 2018 (Sage) [3*n^2 + 1 for n in range(40)] # G. C. Greubel, Dec 02 2018 (GAP) List([0..40], n -> 3*n^2 + 1); # G. C. Greubel, Dec 02 2018 CROSSREFS Cf. A002648 (prime terms), A201053. Cf. A000578, A003136, A132111, A181123. Other spokes: A003215, A056105, A056106, A056108, A056109. Other spirals: A054552. Sequence in context: A298017 A356984 A307272 * A155433 A272746 A273557 Adjacent sequences: A056104 A056105 A056106 * A056108 A056109 A056110 KEYWORD nonn,easy AUTHOR Henry Bottomley, Jun 09 2000 STATUS approved

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Last modified December 6 16:13 EST 2023. Contains 367612 sequences. (Running on oeis4.)