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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

Mariotte, Regles pour les jets d'eau, pp. 508-518. In Divers ouvrages de mathematique et de physique par Messieurs de l'Academie 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

Table of n, a(n) for n=0..100.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

FORMULA

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]}, with n >= 0. - Paolo P. Lava, Nov 27 2008

G.f.: -x*(1+4*x+9*x^2+4*x^3+x^4) / ((x-1)*(1+x)*(1+x+x^2)*(x^2-x+1)). - R. J. Mathar, Jul 23 2009

From G. C. Greubel, Apr 01 2016: (Start)

a(6*m) = 0.

a(n) = (1/6)*(19 - 3*(-1)^n - 24*cos(n*Pi/3) + 8*cos(2*n*Pi/3)).

G.f.: x*(1 +4*x + 3*x^2 + 4*x^3 + x^4)/(1 - x^6). (End)

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 xrange(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

Cf. A000290, A008959, A070431, A070438, A070442, A070452, A159852, A260686.

Sequence in context: A285323 A178143 * A070516 A143298 A177839 A013669

Adjacent sequences:  A070432 A070433 A070434 * A070436 A070437 A070438

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 12 2002

STATUS

approved

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Last modified November 18 12:21 EST 2017. Contains 294891 sequences.