A070435 a(n) = n^2 mod 12, or alternately n^4 mod 12.

0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4

Offset: 0

Author

N. J. A. Sloane, May 12 2002

Keywords

• nonn
• easy

Comments

Period 6: repeat [0,1,4,9,4,1].

Occurs in Mariotte reference, pp. 511-512. Consider waterjets of heights 0,5,10, ... = A008587 up to 100 pieds (feet). a(n) is the difference in pouces (inches) between tank's heights (in feet and inches) and part in feet (0,5,10,15,21,..). Row with 0's is implicit. - Paul Curtz, Nov 18 2008

a(m*n) = a(m)*a(n) mod 12; a(6*n+k) = a(6*n-k) for k <= 6*n. - Reinhard Zumkeller, Apr 24 2009

n^z mod 12, if z even number. Example: n^180 mod 12. etc... - Zerinvary Lajos, Nov 06 2009

Equivalently: n^(2*m + 2) mod 12. - G. C. Greubel, Apr 01 2016

References

• Edme Mariotte, Règles pour les jets d'eau, pp. 508-518. In Divers ouvrages de mathématique et de physique par Messieurs de l'Académie Royale des Sciences, 6, 518, 1 p., Paris, 1693. Edme Mariotte (1620-1684) is known for the perfect gas law (1676, Essai sur l'air), but later than Robert Boyle (1662). - Paul Curtz, Nov 18 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Cf. A000290, A008959, A070431, A070438, A070442, A070452, A159852, A260686.

Programs

Magma

[Modexp(n, 2, 12): n in [0..100]]; // Wesley Ivan Hurt, Apr 01 2016

Maple

A070435:=n->n^2 mod 12: seq(A070435(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016

Mathematica

Table[Mod[n^2,12],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
LinearRecurrence[{0, 0, 0, 0, 0, 1},{0, 1, 4, 9, 4, 1},101] (* Ray Chandler, Aug 26 2015 *)
PowerMod[Range[0, 100], 2, 12] (* Wesley Ivan Hurt, Apr 02 2016 *)

PARI

a(n)=n^2%12 \\ Charles R Greathouse IV, Sep 23 2013

Sage

[power_mod(n,4,12) for n in range(0, 101)] # Zerinvary Lajos, Oct 31 2009

Formula

G.f.: x*(1 + 4*x + 9*x^2 + 4*x^3 + x^4)/((1 - x)*(1 + x)*(1 + x + x^2)*(1 - x + x^2)). - R. J. Mathar, Jul 23 2009

a(n) = (1/6)*(19 - 3*(-1)^n - 24*cos(n*Pi/3) + 8*cos(2*n*Pi/3)). - G. C. Greubel, Apr 01 2016

a(n) = A260686(n)^2. - Wesley Ivan Hurt, Apr 01 2016

Extensions

Incorrect g.f. removed by Georg Fischer, May 15 2019