A174452 -id:A174452 - cp's OEIS frontend

A008959 Final digit of squares: a(n) = n^2 mod 10.

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

Offset: 0

N. J. A. Sloane, Mar 15 1996

a(m*n) = a(m)*a(n) mod 10; a(5*n+k) = a(5*n-k) for k <= 5*n. - Reinhard Zumkeller, Apr 24 2009
a(n) = n^6 mod 10. - Zerinvary Lajos, Nov 06 2009
a(n) = A002015(n) mod 10 = A174452(n) mod 10. - Reinhard Zumkeller, Mar 21 2010
Decimal expansion of 166285490/1111111111. - Alexander R. Povolotsky, Mar 09 2013

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A000290, A070431, A070435, A070438, A070442, A070452, A159852, A010879, A008960, A070514.

[0] cat [Intseq(n^2)[1]: n in [1..80]]; // Bruno Berselli, Feb 14 2013

[n^2 - 10*Floor(n^2/10): n in [0..80]]; // Vincenzo Librandi, Jun 16 2015

A008959:=n->(n^2 mod 10); seq(A008959(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[n^2,10],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
PowerMod[Range[0,80],2,10] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,0,1},{0,1,4,9,6,5,6,9,4,1},120] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=n^2%10 \\ Charles R Greathouse IV, Sep 24 2015

[power_mod(n,2,10) for n in range(0, 81)] # Zerinvary Lajos, Nov 06 2009

Periodic with period 10. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = 4.5 - (1 + 5^(1/2))*cos(Pi*n/5) + (-1 - 3/5*5^(1/2))*cos(2*Pi*n/5) + (5^(1/2) - 1)*cos(3*Pi*n/5) + (-1 + 3/5*5^(1/2))*cos(4*Pi*n/5) - 0.5*(-1)^n. - Richard Choulet, Dec 12 2008
a(n) = A010879(A000290(n)). - Reinhard Zumkeller, Jan 04 2009
G.f.: (x^9+4*x^8+9*x^7+6*x^6+5*x^5+6*x^4+9*x^3+4*x^2+x)/(-x^10+1). - Colin Barker, Aug 14 2012
a(n) = n^2 - 10*floor(n^2/10). - Wesley Ivan Hurt, Jun 12 2013
a(n) = (n - 5*A002266(n + 2))^2 + 5*(5*A002266(n + 2) mod 2). - Wesley Ivan Hurt, Jun 06 2014
a(n) = A033569(n+3) mod 10. - Wesley Ivan Hurt, Dec 06 2014
a(n) = n^k mod 10; for k > 0 where k mod 4 = 2. - Doug Bell, Jun 15 2015

A122986 Squares mod 1000.

Original entry on oeis.org

0, 1, 4, 9, 16, 24, 25, 36, 41, 44, 49, 56, 64, 76, 81, 84, 89, 96, 100, 104, 116, 121, 124, 129, 136, 144, 156, 161, 164, 169, 176, 184, 196, 201, 204, 209, 216, 224, 225, 236, 241, 244, 249, 256, 264, 276, 281, 284, 289, 296, 304, 316, 321, 324, 329, 336, 344

Offset: 1

Sergio Pimentel, Sep 22 2006

Possible last three digits of n^2 (leading zeros omitted).

Range of A174452; A010461 is a subset; and also all squares less than 1000 belong to this sequence; the sequence is finite with A000993(3)=159 terms: a(159)=996 is the last term.

			The last three digits of n^2 can be 000, 001, 236, 241, 996, etc. but not 002, 003, 237, 238, etc.

R. Zumkeller, Table of n, a(n) for n = 1..159 (full sequence)
Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to final digits of numbers

Cf. A036688, A010382, A010411, A010462, A010421, A174452.
Cf. A006716, A053975, A027676, A027678, A122987, A122988.
Row 1000 of A096008.

[n: n in [0..999] | IsSquare(R! n) where R:= ResidueClassRing(1000)]; // Vincenzo Librandi, Dec 29 2019

s:={}: for n from 0 to 999 do s:=s union {n^2 mod 1000}: od: op(s); # Nathaniel Johnston, Jun 22 2011

Union[PowerMod[Range[1000], 2, 1000]] (* Vincenzo Librandi, Dec 29 2019 *)

More terms and additional comments from Reinhard Zumkeller, Mar 21 2010

Edited by N. J. A. Sloane, Apr 10 2010

A002015 a(n) = n^2 reduced mod 100.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, 25, 76, 29, 84, 41, 0, 61, 24, 89, 56, 25, 96, 69, 44, 21, 0, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81

Offset: 0

N. J. A. Sloane

Periodic with period 50: (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, 25, 76, 29, 84, 41, 0, 61, 24, 89, 56, 25, 96, 69, 44, 21, 0, 81, 64, 49, 36, 25, 16, 9, 4, 1) and next term is 0. The period is symmetrical about the "midpoint" 25. - Zak Seidov, Oct 26 2009

A010461 gives the range of this sequence. - Reinhard Zumkeller, Mar 21 2010

Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,1).

Cf. A053879, A070430, A070431, A070432, A070433, A070434, A070435, A070438, A070442, A070452, A159852.

PowerMod[Range[0,60],2,100] (* Harvey P. Dale, Nov 28 2012 *)

a(n)=n^2%100 \\ Charles R Greathouse IV, Oct 07 2015

From Reinhard Zumkeller, Mar 21 2010: (Start)
a(n) = (n mod 10) * ((n mod 10) + 20 * ((n\10) mod 10)) mod 100.
a(n) = A174452(n) mod 100; A008959(n) = a(n) mod 10;
a(m*n) = a(m)*a(n) mod 100;
a(n) = (n mod 100)^2 mod 100;
a(n) = n for n = 0, 1, and 25. (End)

Definition rephrased at the suggestion of Zak Seidov, Oct 26 2009

A210251 Residues modulo 100 of odd squares.

Original entry on oeis.org

1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89

Offset: 1

M. F. Hasler, Mar 19 2012

Range of A016754. Odd terms from A010461. See also A002015, A008959, A174452.

Also the range of A030156 and A192775 without initial term.

Mod[#,100]&/@(Range[1,55,2]^2)//Union (* Harvey P. Dale, Jul 27 2017 *)

PARI
```
vecsort(vector(12,n,(2*n-1)^2)%100,,8)
```

{1,9} + {0,1,2,3,4}*20 union {25}.

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A008959 Final digit of squares: a(n) = n^2 mod 10.

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A122986 Squares mod 1000.

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A002015 a(n) = n^2 reduced mod 100.

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A210251 Residues modulo 100 of odd squares.

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