A016756 a(n) = (2*n+1)^4.

1, 81, 625, 2401, 6561, 14641, 28561, 50625, 83521, 130321, 194481, 279841, 390625, 531441, 707281, 923521, 1185921, 1500625, 1874161, 2313441, 2825761, 3418801, 4100625, 4879681, 5764801, 6765201, 7890481, 9150625, 10556001, 12117361, 13845841, 15752961, 17850625

Offset

0,2

Comments

a(n) is the number of ordered pairs of lattice points (vectors in R^2 with integer coordinates) that are in or on a square centered at the origin with side length 2*n. - Geoffrey Critzer, Apr 20 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Formula

From Wolfdieter Lang, Mar 12 2017: (Start)

G.f.: (1+76*x+230*x^2+76*x^3+x^4)/(1-x)^5; see row n=5 of A060187.

E.g.f.: (1 + 80*x + 232*x^2 + 128*x^3 + 16*x^4)*exp(x); see row n=4 of A154537. (End)

Sum_{n>=0} 1/a(n) = Pi^4/96 (A300707). - Amiram Eldar, Oct 10 2020

From Amiram Eldar, Jan 28 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = (cos(Pi/sqrt(2)) + cosh(Pi/sqrt(2)))/2.

Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(Pi/2)/8. (End)

Example

a(1) = 81 because there are 9 lattice points in or on the 2 x 2 square centered at the origin, so there are 9*9 =81 ordered pairs. - Geoffrey Critzer, Apr 20 2013

Mathematica

Table[(2n+1)^4, {n, 0, 25}]  (* Geoffrey Critzer, Apr 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 81, 625, 2401, 6561}, 30] (* Harvey P. Dale, Mar 24 2020 *)

Prog

(Magma) [(2*n+1)^4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011
(PARI) vector(40, n, n--; (2*n+1)^4) \\ G. C. Greubel, Sep 15 2018

Crossrefs

Cf. A016755, A060187, A154537, A300707.

Sequence in context: A235432 A206064 A359607 * A182647 A256590 A322240

Adjacent sequences: A016753 A016754 A016755 * A016757 A016758 A016759

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved