A056107 Third spoke of a hexagonal spiral.

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

Offset: 0

Author

Henry Bottomley, Jun 09 2000

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002648 (prime terms), A201053.

Cf. A000578, A003136, A132111, A181123.

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

Other spirals: A054552.

Programs

GAP

List([0..40], n -> 3*n^2 + 1); # G. C. Greubel, Dec 02 2018

Magma

[3*n^2 + 1: n in [0..40]]; // G. C. Greubel, Dec 02 2018

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 *)

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

Sage

[3*n^2 + 1 for n in range(40)] # G. C. Greubel, Dec 02 2018

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)