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

%I A017330 #42 Aug 04 2024 22:05:18
%S A017330 25,225,625,1225,2025,3025,4225,5625,7225,9025,11025,13225,15625,
%T A017330 18225,21025,24025,27225,30625,34225,38025,42025,46225,50625,55225,
%U A017330 60025,65025,70225,75625,81225,87025,93025,99225,105625,112225,119025,126025,133225,140625
%N A017330 a(n) = (10*n + 5)^2.
%C A017330 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
%D A017330 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 136.
%H A017330 Vincenzo Librandi, <a href="/A017330/b017330.txt">Table of n, a(n) for n = 0..10000</a>
%H A017330 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A017330 G.f.: -25*(x^2 + 6*x + 1)/(x - 1)^3. - _Colin Barker_, Nov 14 2012
%F A017330 a(n) = A017329(n)^2. - _Michel Marcus_, Dec 23 2013
%F A017330 a(0) = 25, a(n) = a(n - 1) + 200*n. - _Alonso del Arte_, Feb 25 2020
%F A017330 From _Amiram Eldar_, Apr 18 2023: (Start)
%F A017330 a(n) = 5^2 * A016754(n).
%F A017330 Sum_{n>=0} 1/a(n) = Pi^2/200.
%F A017330 Sum_{n>=0} (-1)^n/a(n) = G/25, where G is Catalan's constant (A006752). (End)
%F A017330 E.g.f.: 25*exp(x)*(1 + 8*x + 4*x^2). - _Stefano Spezia_, Aug 04 2024
%e A017330 5^2 = 25;
%e A017330 15^2 = (1 * 2) concatenate 25 = 225;
%e A017330 25^2 = (2 * 3) concatenate 25 = 625;
%e A017330 35^2 = (3 * 4) concatenate 25 = 1225;
%e A017330 45^2 = (4 * 5) concatenate 25 = 2025;
%e A017330 55^2 = (5 * 6) concatenate 25 = 3025;
%e A017330 65^2 = (6 * 7) concatenate 25 = 4225, etc.
%t A017330 25Range[1, 89, 2]^2 (* _Alonso del Arte_, Feb 25 2020 *)
%o A017330 (Magma) [(10*n+5)^2: n in [0..35]]; // _Vincenzo Librandi_, Aug 02 2011
%o A017330 (PARI) a(n)=(10*n+5)^2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A017330 (Scala) (5 to 445 by 10).map(n => n * n) // _Alonso del Arte_, Feb 25 2020
%Y A017330 Cf. A002378, A006752, A016754.
%K A017330 nonn,easy
%O A017330 0,1
%A A017330 _N. J. A. Sloane_
%E A017330 More terms from _Michel Marcus_, Dec 23 2013