OFFSET
0,3
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).
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
MAPLE
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 *)
PROG
(Sage) [power_mod(n, 4, 12) for n in range(0, 101)] # Zerinvary Lajos, Oct 31 2009
(PARI) a(n)=n^2%12 \\ Charles R Greathouse IV, Sep 23 2013
(Magma) [Modexp(n, 2, 12): n in [0..100]]; // Wesley Ivan Hurt, Apr 01 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 12 2002
EXTENSIONS
Incorrect g.f. removed by Georg Fischer, May 15 2019
STATUS
approved