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

Author

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

Keywords

• nonn
• easy

Comments

The averages of four consecutive even squares are in A027575.

Examples

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

Links

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

Crossrefs

Cf. A027575.

Programs

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}]

PARI

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

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

Extensions

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