

A120328


Sum of three consecutive squares: a(n) = n^2 + (n + 1)^2 + (n + 2)^2.


13



2, 5, 14, 29, 50, 77, 110, 149, 194, 245, 302, 365, 434, 509, 590, 677, 770, 869, 974, 1085, 1202, 1325, 1454, 1589, 1730, 1877, 2030, 2189, 2354, 2525, 2702, 2885, 3074, 3269, 3470, 3677, 3890, 4109, 4334, 4565, 4802, 5045, 5294, 5549, 5810, 6077, 6350
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OFFSET

1,1


COMMENTS

A rectangular prism with sides n, n + 1, and n + 2 will have four diagonals of different lengths. The sum of the squares of all four is three times the numbers in this sequence beginning with 14 (third term in the sequence for n = 1).  J. M. Bergot, Sep 15 2011
From JeanChristophe Hervé, Nov 11 2015: (Start)
This sequence differs from A005918 only in the first term.
a(n) is also defined for any negative number and a(n) = a(n2).
If a 2set Y and a 3set Z are disjoint subsets of an nset (n >= 5) X then a(n5) is the number of 4subsets of X intersecting both Y and Z (from comment in A005918 by Milan Janjic, Sep 08 2007).
(End)


LINKS

JeanChristophe Hervé, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (3,3,1).
Index entries for twoway infinite sequences


FORMULA

From R. J. Mathar, Aug 07 2008: (Start)
a(n) = A005918(n + 1), n >= 0.
O.g.f.: (2  x + 5*x^2)/(x*(1  x)^3). (End)
a(n) = 3*(2*n + 1) + a(n  1) (with a(1) = 2).  Vincenzo Librandi, Nov 13 2010
a(n) = 3*n^2 + 6*n + 5.  T. D. Noe by way of Alonso del Arte, Oct 29 2012
From JeanChristophe Hervé, Nov 11 2015: (Start)
a(n) = 3*(n + 1)^2 + 2 == 2 (mod 3), hence a(n) is never square.
a(n) = 3*a(n1)  3*a(n2) + a(n3) for all n in Z. (End)


MAPLE

[seq(n^2+(n+1)^2+(n+2)^2, n=1..45)];


MATHEMATICA

Table[Total[Range[n, n + 2]^2], {n, 1, 45}] (* Harvey P. Dale, Jan 23 2011 *)


PROG

(Sage) [i^2+(i+1)^2+(i+2)^2 for i in xrange(1, 46)] # Zerinvary Lajos, Jul 03 2008
(PARI) a(n) = n^2 + (n + 1)^2 + (n + 2)^2; \\ Altug Alkan, Nov 11 2015
(MAGMA) [3*n^2 + 6*n + 5 : n in [1..50]]; // Wesley Ivan Hurt, Nov 12 2015


CROSSREFS

Cf. A001844, A005918, A027575, A027578, A027865.
Cf. A027574, A027602.
Sequence in context: A022630 A047133 A031874 * A026011 A212393 A056358
Adjacent sequences: A120325 A120326 A120327 * A120329 A120330 A120331


KEYWORD

easy,nonn


AUTHOR

Zerinvary Lajos, Jun 21 2006


STATUS

approved



