A186707 - OEIS

%I #38 Sep 08 2022 08:45:55

%S 1,14,71,224,547,1134,2101,3584,5741,8750,12811,18144,24991,33614,

%T 44297,57344,73081,91854,114031,140000,170171,204974,244861,290304,

%U 341797,399854,465011,537824,618871,708750,808081,917504,1037681,1169294,1313047,1469664

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

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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A186707/b186707.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).

%F From _R. J. Mathar_, Mar 24 2011: (Start)

%F 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.

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

%F 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

%t 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 *)

%t 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 *)

%o (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

%o (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

%Y Cf. A007202.

%K nonn,easy,less

%O 0,2

%A _Jonathan Vos Post_, Feb 25 2011