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A070435 a(n) = n^2 mod 12, or alternately n^4 mod 12. 15
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 (list; graph; refs; listen; history; text; internal format)
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
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
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 *)
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
Sequence in context: A365325 A321219 A178143 * A070516 A143298 A177839
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 12 2002
EXTENSIONS
Incorrect g.f. removed by Georg Fischer, May 15 2019
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)