A260637 Sums of seven consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2 + (n+6)^2.

28, 35, 56, 91, 140, 203, 280, 371, 476, 595, 728, 875, 1036, 1211, 1400, 1603, 1820, 2051, 2296, 2555, 2828, 3115, 3416, 3731, 4060, 4403, 4760, 5131, 5516, 5915, 6328, 6755, 7196, 7651, 8120, 8603, 9100, 9611, 10136, 10675, 11228, 11795, 12376, 12971

Offset

-3,1

Comments

a(n) is defined for any n in Z and a(-n) = a(n-6).

There are no primes or squares in the sequence because a(n) is a multiple of 7 and 7 is with multiplicity 1: a(n) = 7*((n+3)^2 + 4), and the factor (n+3)^2 + 4 is not a multiple of 7 for any n. A001032 gives the integers k such that the sum of k consecutive squares is a square.

Links

Jean-Christophe Hervé, Table of n, a(n) for n = -3..1000

Patrick De Geest, World!Of Numbers.

Index entries for two-way infinite sequences.

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

Formula

a(n) = 7*n^2 + 42*n + 91 = 7*(n^2 + 6*n + 13) = 7*((n+3)^2 + 4).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 7*(2*n+7).

G.f.: 7*(4 - 7*x + 5*x^2) / (x^3*(1-x)^3). - Colin Barker, Nov 12 2015

From Amiram Eldar, Sep 15 2022: (Start)

Sum_{n>=-3} 1/a(n) = coth(2*Pi)*Pi/28 + 1/56.

Sum_{n>=-3} (-1)^(n+1)/a(n) = cosech(2*Pi)*Pi/28 + 1/56. (End)

Maple

A260637:=n->7*((n+3)^2 + 4): seq(A260637(n), n=-3..50); # Wesley Ivan Hurt, Nov 17 2015

Mathematica

Table[Plus@@(Range[n, n + 6]^2), {n, -3, 96}]
Total/@Partition[Range[-3, 50]^2, 7, 1] (* or *) LinearRecurrence[{3, -3, 1}, {28, 35, 56}, 50] (* Harvey P. Dale, Oct 05 2022 *)

Prog

(PARI) vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2+(n+5)^2+(n+6)^2).
(PARI) a(n) = 7*n^2 + 42*n + 91;
vector(50, n, a(n-4)) \\ Altug Alkan, Nov 11 2015
(PARI) Vec(-7*(5*x^2-7*x+4)/(x^3*(x-1)^3) + O(x^100)) \\ Colin Barker, Nov 12 2015
(Magma) [7*((n+3)^2 + 4) : n in [-3..50]]; // Wesley Ivan Hurt, Nov 17 2015
(SageMath) [7*((n+3)^2 +4) for n in (-3..50)] # G. C. Greubel, Aug 24 2022

Crossrefs

Cf. A000290, A001844, A027575, A027578, A027865, A120328, A276026.

Cf. A001032.

Sequence in context: A055576 A281916 A146077 * A143186 A101421 A039347

Adjacent sequences: A260634 A260635 A260636 * A260638 A260639 A260640

Keyword

nonn,easy

Author

Jean-Christophe Hervé, Nov 11 2015

Status

approved