A273375 Squares ending in digit 4.

4, 64, 144, 324, 484, 784, 1024, 1444, 1764, 2304, 2704, 3364, 3844, 4624, 5184, 6084, 6724, 7744, 8464, 9604, 10404, 11664, 12544, 13924, 14884, 16384, 17424, 19044, 20164, 21904, 23104, 24964, 26244, 28224, 29584, 31684, 33124, 35344, 36864, 39204, 40804, 43264

Offset: 1

Author

Vincenzo Librandi, May 24 2016

Keywords

• nonn
• base
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000290, A047209.

Cf. A017317 (numbers ending in 4), A017319 (cubes ending in 4).

Cf. similar sequences listed in A273373.

Programs

Magma

/* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 4];

Magma

[(10*n+(-1)^n-5)^2/4: n in [1..50]];

Mathematica

Table[(10 n + (-1)^n - 5)^2/4, {n, 1, 50}] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {4, 64, 144, 324, 484}, 50]
Select[Range[200]^2,Mod[#,10]==4&] (* or *) LinearRecurrence[{1,1,-1},{2,8,12},40]^2(* Harvey P. Dale, Aug 06 2017 *)

Formula

G.f.: 4*x*(1 + 15*x + 18*x^2 + 15*x^3 + x^4) /((1+x)^2*(1-x)^3).

a(n) = 4*A047209(n)^2 = (10*n + (-1)^n - 5)^2/4.

Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5-sqrt(5))). - Amiram Eldar, Feb 16 2023

E.g.f.: (4 - 5*x + 25*x^2)*cosh(x) + (9 + 5*x + 25*x^2)*sinh(x) - 4. - Stefano Spezia, Feb 21 2025

Extensions

Edited by Bruno Berselli, May 24 2016