A173960 Averages of four consecutive odd squares.

21, 41, 69, 105, 149, 201, 261, 329, 405, 489, 581, 681, 789, 905, 1029, 1161, 1301, 1449, 1605, 1769, 1941, 2121, 2309, 2505, 2709, 2921, 3141, 3369, 3605, 3849, 4101, 4361, 4629, 4905, 5189, 5481, 5781, 6089, 6405, 6729, 7061, 7401, 7749, 8105, 8469

Offset

1,1

Comments

The averages of four consecutive even squares are in A027575.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = ((2*n-1)^2 + (2*n+1)^2 + (2*n+3)^2 + (2*n+5)^2)/4 = 4*n^2 + 8*n + 9.

From Colin Barker, Apr 15 2012: (Start)

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

G.f.: x*(21-22*x+9*x^2)/(1-x)^3. (End)

E.g.f.: exp(x)*(4*x^2 + 12*x + 9) - 9. - Elmo R. Oliveira, Nov 01 2024

Example

(1^2 + 3^2 + 5^2 + 7^2)/4 = 21.

Maple

A173960 := proc(n) 4*n^2+8*n+9 ; end proc: seq(A173960(n), n=1..100) ; # R. J. Mathar, Mar 31 2010

Mathematica

f[n_]:=(n^2+(n+2)^2+(n+4)^2+(n+6)^2)/4; Table[f[n], {n, 1, 6!, 2}]
Mean/@Partition[Range[1, 101, 2]^2, 4, 1] (* or *) LinearRecurrence[{3, -3, 1}, {21, 41, 69}, 50] (* Harvey P. Dale, Sep 16 2025 *)

Prog

(PARI) a(n)=4*n^2+8*n+9 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A027575.

Sequence in context: A261522 A215145 A154576 * A147273 A195034 A067344

Adjacent sequences: A173957 A173958 A173959 * A173961 A173962 A173963

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

Extensions

Formula corrected by R. J. Mathar, Mar 31 2010

Status

approved