A060588 If the final two digits of n written in base 3 are the same then this digit, otherwise mod 3-sum of these two digits.

0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 0

Offset

0,2

Comments

From William Walkington, Sep 14 2016: (Start)

With offset 1, the y-coordinates of position vectors from the origin (point 1) to the points numbered 1 to N^2 of the magic tori that display the Agrippa odd-order-N magic squares can be expressed as follows: a(n) = (-(n-1)-floor((n-1)/N)) mod N.

This generates the y-coordinates of the magic tori that display the Agrippa order-3 "Saturn," order-5 "Mars," order-7 "Venus," order-9 "Luna," and higher-odd-order-N magic squares.

Therefore, if the odd-order-N of the torus is 3, then the resulting sequence 0,2,1,2,1,0,1,0,2 represents the y-coordinates of position vectors from the origin (point number 1) to the point numbered 1 to 9 of the magic torus that displays the Agrippa order-3 "Saturn" magic square. (End)

References

H.C. Agrippa, "De occulta philosophia Libri tres," (1533) translated by "J.F." (John French?) and printed by Moule, London, in 1651, Book II, chapter XXII entitled "Of the tables of the Planets, their vertues,forms, and what Divine names, Intelligencies, and Spirits are set over them."

Links

Table of n, a(n) for n=0..104.

H. C. Agrippa, De Occulta Philosophia libri tres (Three books of occult philosophy), Book II, chapter XXII, Digital edition Peterson J.F.

W. Walkington, Magic torus coordinate and vector symmetries.

William Walkington, Agrippa odd-order magic squares

Wikipedia, Agrippa's magic squares.

Index entries for sequences related to final digits of numbers

Formula

a(n) = a(n-9) = (-[n/3]-n) mod 3 = A060587(n) mod 3.

a(n) = (-n - floor(n/3)) mod 3. - William Walkington, Sep 14 2016

Example

a(22)=1 since 22 is written in base 3 as 211 and the final two digits are 1; a(23)=0 since 23 is written in base 3 as 212 and the final two digits are 1 and 2 and 3-(1+2)=0.

Mathematica

b3d[n_]:=Module[{d3=Take[IntegerDigits[n, 3], -2]}, If[MatchQ[d3, {x_, x_}], d3[[1]], 3-Total[d3]]]; Join[{0, 2, 1}, Array[b3d, 110, 3]] (* Harvey P. Dale, Feb 29 2016 *)
Table[If[MatchQ @@ #, First@ #, Mod[3 - Total@ #, 3]] &@ Take[PadLeft[#, 2], -2] &@ IntegerDigits[n, 3], {n, 0, 120}] (* or *)
Table[Mod[-n - Floor[n/3], 3], {n, 0, 120}] (* Michael De Vlieger, Sep 14 2016 *)

Crossrefs

Cf. A060582, A270740.

Sequence in context: A360536 A226519 A066057 * A221167 A286134 A336922

Adjacent sequences: A060585 A060586 A060587 * A060589 A060590 A060591

Keyword

base,nonn

Author

Henry Bottomley, Apr 04 2001

Status

approved