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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
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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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FORMULA
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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
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MAPLE
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MATHEMATICA
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LinearRecurrence[{0, 0, 0, 0, 0, 1}, {0, 1, 4, 9, 4, 1}, 101] (* Ray Chandler, Aug 26 2015 *)
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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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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