A016850 a(n) = (5*n)^2.

0, 25, 100, 225, 400, 625, 900, 1225, 1600, 2025, 2500, 3025, 3600, 4225, 4900, 5625, 6400, 7225, 8100, 9025, 10000, 11025, 12100, 13225, 14400, 15625, 16900, 18225, 19600, 21025, 22500, 24025, 25600, 27225, 28900, 30625, 32400, 34225, 36100, 38025, 40000, 42025

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If we define C(n) = (5*n)^2 (n > 0), the sequence is the first "square-sequence" such that for every n there exists p such that C(n) = C(p) + C(p+n). We observe in fact that p = 3*n because 25 = 3^2 + 4^2. The sequence without 0 is linked with the first nontrivial solution (trivial: n^2 = 0^2 + n^2) of the equation X^2 = 2*Y^2 + 2*n^2 where X = 2*k and Y = 2*p + n which is equivalent to k^2 = p^2 + (p+n)^2 for n given. The second such "square-sequence" is (29*n)^2 (n > 0) because 29^2 = 20^2 + 21^2 and with this relation we obtain (29*n)^2 = (20*n)^2 + (20*n+n)^2. - Richard Choulet, Dec 23 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A033429, A053742 (first differences), A008587, A008607.

Similar sequences listed in A244630.

Programs

Magma

[(5*n)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

Mathematica

(5Range[0, 31])^2 (* Alonso del Arte, Oct 08 2017 *)

PARI

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

Formula

a(n) = 25*n^2 = 25*A000290(n) = 5*A033429(n). - Omar E. Pol, Jul 03 2014

From Amiram Eldar, Jan 25 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/150.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/300.

Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/5)/(Pi/5).

Product_{n>=1} (1 - 1/a(n)) = sin(Pi/5)/(Pi/5) = 5*sqrt((5-sqrt(5))/2)/(2*Pi). (End)

a(n) = Sum_{i=0..n-1} A053742(i). - John Elias, Jun 30 2021

G.f.: 25*x*(1 + x)/(1 - x)^3. - Stefano Spezia, Jul 08 2023

From Elmo R. Oliveira, Nov 30 2024: (Start)

E.g.f.: 25*x*(1 + x)*exp(x).

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

a(n) = n*A008607(n) = A000290(A008587(n)) = A008587(n)^2. (End)