A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.

30, 55, 90, 135, 190, 255, 330, 415, 510, 615, 730, 855, 990, 1135, 1290, 1455, 1630, 1815, 2010, 2215, 2430, 2655, 2890, 3135, 3390, 3655, 3930, 4215, 4510, 4815, 5130, 5455, 5790, 6135, 6490, 6855, 7230, 7615, 8010, 8415, 8830, 9255, 9690, 10135, 10590, 11055

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

a(n) is defined for n < 0 and a(-n) = a(n-4) for any n; a(-3) = a(-1) = 15, a(-2) = 10. - Jean-Christophe Hervé, Nov 11 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Patrick De Geest, Palindromic Sums of Squares of Consecutive Integers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
• Index entries for two-way infinite sequences.

Crossrefs

Cf. A000290, A001844, A027575, A027865, A120328, A260637, A276026.

Cf. A027580.

Subsequence of A174069, A174070, A174071.

Programs

Magma

[n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2: n in [0..50] ]; // Vincenzo Librandi, Jun 17 2011

Maple

A027578:=n->5*(n+2)^2+10: seq(A027578(n), n=0..50); # Wesley Ivan Hurt, Nov 12 2015

Mathematica

Table[5 (n + 2)^2 + 10, {n, 0, 50}] (* Bruno Berselli, Jul 29 2015 *)
Total/@Partition[Range[0,50]^2,5,1] (* or *) LinearRecurrence[{3,-3,1},{30,55,90},50] (* Harvey P. Dale, Mar 06 2018 *)

PARI

vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2) \\ Altug Alkan, Nov 11 2015

Sage

[i^2+(i+1)^2+(i+2)^2+(i+3)^2+(i+4)^2 for i in range(0,50)] # Zerinvary Lajos, Jul 03 2008

Formula

a(n) = 5*A059100(n+2).

From Colin Barker, Mar 29 2012: (Start)

G.f.: 5*(6-7*x+3*x^2)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)

a(n) = 5*(n + 2)^2 + 10. a(n) is never square. - Bruno Berselli, Jul 29 2015

E.g.f.: 5*(6 + 5*x + x^2)*exp(x). - G. C. Greubel, Aug 24 2022

From Amiram Eldar, Sep 15 2022: (Start)

Sum_{n>=0} 1/a(n) = coth(sqrt(2)*Pi)*Pi/(10*sqrt(2)) - 7/60.

Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2)*Pi)*Pi/(10*sqrt(2)) + 1/60. (End)