A038122 Start with {1,2,...,n}, replace any two numbers a,b with |a^2-b^2|, repeat until single number k remains; a(n) = minimal value of k.

1, 3, 0, 16, 15, 63, 8, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0

Offset

1,2

Comments

Due mostly to the efforts of Dean Hickerson, supported by David W. Wilson and Michael Kleber, it is now known that this has period 12 beginning at n=8.

Links

Table of n, a(n) for n=1..104.

Dean Hickerson and Michael Kleber, Reducing a Set by Subtracting Squares, J. Integer Sequences, Vol. 2, 1999, #4.

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

Formula

For n<4 and n>7, a(n) = n*(n+1)/2 mod 6 = A010875(A000217(n)). - Henry Bottomley, Feb 24 2003

a(n) = a(n-3)-a(n-6)+a(n-9) for n>16. - Colin Barker, Oct 01 2014

G.f.: x*(4*x^15 +60*x^14 +12*x^13 +8*x^12 -60*x^11 -12*x^10 -8*x^9 +60*x^8 +12*x^7 +7*x^6 -63*x^5 -12*x^4 -15*x^3 -3*x -1) / ((x -1)*(x^2 +1)*(x^2 +x +1)*(x^4 -x^2 +1)). - Colin Barker, Oct 01 2014

Example

a(2) = 3 from (1,2); a(3) = 0 from ((1,2),3); a(4) = 16 from (((1,2),3),4); a(5) = 15 from ((((2,3),5),1),4)
a(6) = 63 from (((1,4),(3,5)),(2,6)) [ Michael Kleber ]
a(7) = 8 from (((((4,5),6),(2,7)),1),3) [ Kleber ]
a(8) = 0 from ((((4,5),7)(2,6))((1,3),8)) [ Guy ]
a(9) = 3 from (2,(1,(((6,7),((3,4),8)),(5,9)))) [ Kleber ]
a(10)= 1 from ((((((((4,5),9),6),(8,10)),2),3),7),1) [ This and the following are due to Dean Hickerson ]
a(11)= 0 from ((((((3,7),(9,11)),6),(8,10)),(1,2)),(4,5))
a(12)= 0 from ((((((1,3),7),(8,10)),(((5,6),9),(11,12))),2),4)
a(13)= 1 from (((((((((3,7),(9,11)),6),(8,10)),5),(12,13)),2),4),1) ...

Mathematica

LinearRecurrence[{0, 0, 1, 0, 0, -1, 0, 0, 1}, {1, 3, 0, 16, 15, 63, 8, 0, 3, 1, 0, 0, 1, 3, 0, 4}, 120] (* Harvey P. Dale, Jul 29 2015 *)

Prog

(PARI) a(n)=if(n<4||n>7, n*(n+1)/2%6, [16, 15, 63, 8][n-3]) \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A013494 A365972 A290580 * A359862 A143779 A240244

Adjacent sequences: A038119 A038120 A038121 * A038123 A038124 A038125

Keyword

nonn,nice,easy

Author

R. K. Guy

Status

approved