A186707 Partial sums of A007202 (crystal ball sequence for hexagonal close-packing).

1, 14, 71, 224, 547, 1134, 2101, 3584, 5741, 8750, 12811, 18144, 24991, 33614, 44297, 57344, 73081, 91854, 114031, 140000, 170171, 204974, 244861, 290304, 341797, 399854, 465011, 537824, 618871, 708750, 808081, 917504, 1037681, 1169294, 1313047, 1469664

Offset: 0

Author

Jonathan Vos Post, Feb 25 2011

Keywords

• nonn
• easy
• less

Comments

Subsequence of primes begins 71, 547, 5741, 114031, 244861, 465011, 808081, 1037681. Subsequence of powers includes 537824 = 2^5 * 7^5.

The sequence is a quasipolynomial, so under the Bunyakovsky conjecture there are infinitely many primes in this sequence. - Charles R Greathouse IV, Aug 21 2011

Let s(0) = 0 and s(n) = A186707(n-1) for n > 0. Then s(n) is the number of 4-tuples (w,x,y,z) having all terms in {1, ..., n} and |w - x| < w + |y - z|. - Clark Kimberling, May 24 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A007202.

Programs

Magma

[7*n*(n^3+4*n^2+6*n+4)/8+(15+(-1)^n)/16: n in [0..40] ]; // Vincenzo Librandi, Aug 22 2011

Mathematica

CoefficientList[Series[ (-1-10 x-20 x^2-10 x^3-x^4)/((x-1)^5 (1+x)),{x,0,40}],x]  (* Harvey P. Dale, Apr 04 2011 *)
Table[7*n*(n^3 + 4*n^2 + 6*n + 4)/8 + (15 + (-1)^n)/16, {n, 0, 40}] (* T. D. Noe, Apr 04 2011 *)

PARI

a(n)=7*n*(n^3+4*n^2+6*n+4)/8+(15+(-1)^n)/16 \\ Charles R Greathouse IV, Aug 21 2011

Formula

From R. J. Mathar, Mar 24 2011: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) = 7*n*(n^3/8 + n^2/2 + 3*n/4 + 1/2) + (15 + (-1)^n)/16.

G.f.: ( -1 - 10*x - 20*x^2 - 10*x^3 - x^4 ) / ( (1 + x)*(x - 1)^5 ). (End)

E.g.f.: (cosh(x) + 7*exp(x)*(1 + 15*x + 25*x^2 + 10*x^3 + x^4))/8. - Franck Maminirina Ramaharo, Nov 09 2018