A008852 Numbers n such that n^2 and n have same last 2 digits.

0, 1, 25, 76, 100, 101, 125, 176, 200, 201, 225, 276, 300, 301, 325, 376, 400, 401, 425, 476, 500, 501, 525, 576, 600, 601, 625, 676, 700, 701, 725, 776, 800, 801, 825, 876, 900, 901, 925, 976, 1000, 1001, 1025, 1076, 1100, 1101, 1125, 1176, 1200, 1201

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base

References

• L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 459.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Programs

GAP

a:=[0,1,25,76,100];; for n in [6..60] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # G. C. Greubel, Sep 13 2019

Magma

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1 +24*x +51*x^2 +24*x^3)/((1-x)*(1-x^4)) )); // G. C. Greubel, Sep 13 2019

Maple

for n to 2000 do if n^2 - n mod 100 = 0 then print(n); fi; od;

Mathematica

Select[Range[1200], Mod[ #, 100] == Mod[ #^2, 100] &] (* Stefan Steinerberger, Apr 15 2006 *)
LinearRecurrence[{1,0,0,1,-1}, {0,1,25,76,100}, 60] (* G. C. Greubel, Sep 13 2019 *)

PARI

my(x='x+O('x^60)); concat([0], Vec(x*(1 +24*x +51*x^2 +24*x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Sep 13 2019

Sage

[n for n in (0..1250) if mod(n,100)==mod(n^2,100)] # G. C. Greubel, Sep 13 2019

Formula

a(4n)=100*n, a(4n+1)=100*n+1, a(4n+2)=100*n+25, a(4n+3)=100*n+76. - Franklin T. Adams-Watters, Mar 13 2006

From G. C. Greubel, Sep 13 2019: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5).

G.f.: x*(1 +24*x +51*x^2 +24*x^3)/((1-x)*(1-x^4)). (End)