A017330 a(n) = (10*n + 5)^2.

25, 225, 625, 1225, 2025, 3025, 4225, 5625, 7225, 9025, 11025, 13225, 15625, 18225, 21025, 24025, 27225, 30625, 34225, 38025, 42025, 46225, 50625, 55225, 60025, 65025, 70225, 75625, 81225, 87025, 93025, 99225, 105625, 112225, 119025, 126025, 133225, 140625

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This is an old school trick which says that a square of an integer that ends with 5 is easy to compute. Remove the 5, multiply the remaining number by (itself + 1), and concatenate 25 at the end. So, a(n)\100 = A002378(n). - Michel Marcus, Dec 23 2013

Examples

			5^2 = 25;
15^2 = (1 * 2) concatenate 25 = 225;
25^2 = (2 * 3) concatenate 25 = 625;
35^2 = (3 * 4) concatenate 25 = 1225;
45^2 = (4 * 5) concatenate 25 = 2025;
55^2 = (5 * 6) concatenate 25 = 3025;
65^2 = (6 * 7) concatenate 25 = 4225, etc.
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 136.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002378, A006752, A016754.

Programs

Magma

[(10*n+5)^2: n in [0..35]]; // Vincenzo Librandi, Aug 02 2011

Mathematica

25Range[1, 89, 2]^2 (* Alonso del Arte, Feb 25 2020 *)

PARI

a(n)=(10*n+5)^2 \\ Charles R Greathouse IV, Jun 17 2017

Scala

(5 to 445 by 10).map(n => n * n) // Alonso del Arte, Feb 25 2020

Formula

G.f.: -25*(x^2 + 6*x + 1)/(x - 1)^3. - Colin Barker, Nov 14 2012

a(n) = A017329(n)^2. - Michel Marcus, Dec 23 2013

a(0) = 25, a(n) = a(n - 1) + 200*n. - Alonso del Arte, Feb 25 2020

From Amiram Eldar, Apr 18 2023: (Start)

a(n) = 5^2 * A016754(n).

Sum_{n>=0} 1/a(n) = Pi^2/200.

Sum_{n>=0} (-1)^n/a(n) = G/25, where G is Catalan's constant (A006752). (End)

E.g.f.: 25*exp(x)*(1 + 8*x + 4*x^2). - Stefano Spezia, Aug 04 2024

Extensions

More terms from Michel Marcus, Dec 23 2013