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A070435
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a(n) = n^2 mod 12, or alternately n^4 mod 12.
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15
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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
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LINKS
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Table of n, a(n) for n=0..100.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).
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FORMULA
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a(n) = (1/45)*(17*(n mod 6) + 32*((n+1) mod 6) + 47*((n+2) mod 6) - 28*((n+3) mod 6) - 13*((n+4) mod 6) + 2*((n+5) mod 6)). - Paolo P. Lava, Nov 27 2008
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
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MAPLE
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A070435:=n->n^2 mod 12: seq(A070435(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000290, A008959, A070431, A070438, A070442, A070452, A159852, A260686.
Sequence in context: A285323 A321219 A178143 * A070516 A143298 A177839
Adjacent sequences: A070432 A070433 A070434 * A070436 A070437 A070438
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, May 12 2002
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EXTENSIONS
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Incorrect g.f. removed by Georg Fischer, May 15 2019
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STATUS
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approved
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