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%I #51 Dec 14 2023 05:05:48
%S 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,
%T 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,
%U 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
%N a(n) = n^2 mod 12, or alternately n^4 mod 12.
%C Period 6: repeat [0,1,4,9,4,1].
%C 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
%C 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
%C n^z mod 12, if z even number. Example: n^180 mod 12. etc... - _Zerinvary Lajos_, Nov 06 2009
%C Equivalently: n^(2*m + 2) mod 12. - _G. C. Greubel_, Apr 01 2016
%D 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
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 1).
%F 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
%F 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
%F a(n) = A260686(n)^2. - _Wesley Ivan Hurt_, Apr 01 2016
%p A070435:=n->n^2 mod 12: seq(A070435(n), n=0..100); # _Wesley Ivan Hurt_, Apr 01 2016
%t Table[Mod[n^2,12],{n,0,200}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 21 2011 *)
%t LinearRecurrence[{0, 0, 0, 0, 0, 1},{0, 1, 4, 9, 4, 1},101] (* _Ray Chandler_, Aug 26 2015 *)
%t PowerMod[Range[0, 100], 2, 12] (* _Wesley Ivan Hurt_, Apr 02 2016 *)
%o (Sage) [power_mod(n,4,12) for n in range(0, 101)] # _Zerinvary Lajos_, Oct 31 2009
%o (PARI) a(n)=n^2%12 \\ _Charles R Greathouse IV_, Sep 23 2013
%o (Magma) [Modexp(n, 2, 12): n in [0..100]]; // _Wesley Ivan Hurt_, Apr 01 2016
%Y Cf. A000290, A008959, A070431, A070438, A070442, A070452, A159852, A260686.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, May 12 2002
%E Incorrect g.f. removed by _Georg Fischer_, May 15 2019
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