A027575 a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2.

14, 30, 54, 86, 126, 174, 230, 294, 366, 446, 534, 630, 734, 846, 966, 1094, 1230, 1374, 1526, 1686, 1854, 2030, 2214, 2406, 2606, 2814, 3030, 3254, 3486, 3726, 3974, 4230, 4494, 4766, 5046, 5334, 5630, 5934, 6246, 6566, 6894, 7230, 7574, 7926, 8286, 8654, 9030

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

Summation of n^2 taken 4 at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 20 2009

Terms are congruent to (2,0,0) mod 6. - Ezhilarasu Velayutham, Apr 04 2019

Links

• G. C. Greubel, 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, A027578, A027865, A120328, A260637, A276026.

Cf. A016754, A027577.

Programs

Magma

[2*(2*n^2 +6*n +7): n in [0..50]]; // G. C. Greubel, Aug 25 2022

Mathematica

Table[n^2 + (n + 1)^2 + (n + 2)^2 + (n + 3)^2, {n, 0, 42}] (* Alonso del Arte, Feb 17 2012 *)
Table[Total[Range[n,n+3]^2],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{14,30,54},50] (* Harvey P. Dale, Jan 23 2017 *)
Total/@Partition[Range[0,50]^2,4,1] (* Harvey P. Dale, Feb 08 2020 *)

PARI

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

Sage

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

Formula

a(n) = 4*n^2 + 12*n + 14. - Al Hakanson (hawkuu(AT)gmail.com), May 20 2009

a(n) = a(n-1) + 8*(n+1) for n>0, a(0)=14. - Vincenzo Librandi, Nov 19 2010

G.f.: 2*(7-6*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 17 2012

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

a(n) = (2*n+3)^2 + 5 = A016754(n+1) + 5, hence a(n) is never square.

The last formula defines a(n) for n < 0; then we have a(-n) = a(n-3) for all n. (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Apr 16 2021

E.g.f.: 2*(7 + 8*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 25 2022

Sum_{n>=0} 1/a(n) = tanh(sqrt(5)*Pi/2)*Pi/(4*sqrt(5)) - 1/6. - Amiram Eldar, Sep 15 2022