A027865 Sums of six consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2.

55, 91, 139, 199, 271, 355, 451, 559, 679, 811, 955, 1111, 1279, 1459, 1651, 1855, 2071, 2299, 2539, 2791, 3055, 3331, 3619, 3919, 4231, 4555, 4891, 5239, 5599, 5971, 6355, 6751, 7159, 7579, 8011, 8455, 8911, 9379, 9859, 10351, 10855, 11371, 11899, 12439, 12991

Offset

0,1

Comments

From Jean-Christophe Hervé, Nov 11 2015: (Start)

a(n) is defined for n < 0 and a(-n) = a(n-5) for any n; a(-4) = a(-1) = 31, a(-3) = a(-2) = 19.

a(n) == 3 (mod 4) for all n in Z, hence a(n) is never square.

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Patrick De Geest, World!Of Numbers.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Index entries for two-way infinite sequences.

Formula

a(n) = 6*n^2 + 30*n + 55.

G.f.: (55 - 74*x + 31*x^2) / (1-x)^3. - R. J. Mathar, Jun 11 2013

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 12*(n+2). - Jean-Christophe Hervé, Nov 11 2015

E.g.f.: (55 + 36*x + 6*x^2)*exp(x). - G. C. Greubel, Aug 25 2022

Sum_{n>=0} 1/a(n) = tanh(sqrt(35/3)*Pi/2)*Pi/(2*sqrt(105)) - 50/589. - Amiram Eldar, Sep 15 2022

Maple

A027865:=n->6*n^2 + 30*n + 55: seq(A027865(n), n=0..60); # Wesley Ivan Hurt, Nov 12 2015

Mathematica

Table[Total@ Map[#^2 &, n + Range[0, 5]], {n, 0, 34}] (* Michael De Vlieger, Nov 12 2015 *)
Total/@Partition[Range[0, 40]^2, 6, 1] (* or *) Table[6x^2+30x+55, {x, 0, 40}] (* Harvey P. Dale, Mar 23 2018 *)

Prog

(PARI) a(n)=6*n^2+30*n+55 \\ Charles R Greathouse IV, Jul 28 2015
(PARI) Vec((-31*x^2+74*x-55)/(x-1)^3 + O(x^50)) \\ Altug Alkan, Nov 12 2015

Crossrefs

Cf. A000290, A001844, A027575, A027578, A120328, A260637, A276026.

Sequence in context: A039533 A157484 A230125 * A247681 A195118 A143205

Adjacent sequences: A027862 A027863 A027864 * A027866 A027867 A027868

Keyword

nonn,easy

Author

Patrick De Geest

Extensions

Corrected by Ralf Stephan, Jan 02 2005

Edited by Charles R Greathouse IV, Jul 25 2010

Renamed by Jean-Christophe Hervé, Nov 12 2015

Status

approved