A013574 Minimal scope of an (n,2) difference triangle.

3, 7, 10, 12, 15, 19, 22, 24, 27, 31, 34, 36, 39, 43, 46, 48, 51, 55, 58, 60, 63, 67, 70, 72, 75, 79, 82, 84, 87, 91, 94, 96, 99, 103, 106, 108, 111, 115, 118, 120, 123, 127, 130, 132, 135, 139, 142, 144, 147, 151, 154, 156, 159, 163, 166, 168, 171, 175, 178, 180, 183, 187, 190

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n)*Pi is also the total length of irregular spiral (center points: 1, 2, 5, 3, 4) after n-rotations. - Kival Ngaokrajang, Jan 08 2014

References

• CRC Handbook of Combinatorial Designs, 1996, p. 315.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Y. M. Chee, C. J. Colbourn, Constructions for difference triangle sets, arXiv:0712.2553 [cs.IT], 2007.
• Kival Ngaokrajang, Illustration of irregular spiral (center points: 1, 2, 5, 3, 4)
• J. B. Shearer, Difference Triangle Sets: Known optimal solutions.
• J. B. Shearer, Difference Triangle Sets: Discoverers
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

A row or column of array in A234943.

A319279 is an essentially identical sequence.

Programs

Maple

A013574 := proc(n)
    if modp(n,4) in {0,1} then
        3*n ;
    else
        3*n+1 ;
    end if;
end proc: # R. J. Mathar, Nov 28 2016

Mathematica

LinearRecurrence[{2, -2, 2, -1}, {3, 7, 10, 12}, 63] (* Jean-François Alcover, Nov 24 2017 *)

PARI

Vec(x*(3 + x + 2*x^2) / ((1 - x)^2*(1 + x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2017

Formula

a(n) = 3n if n = {0,1} (mod 4). a(n) = 3n+1 if n = {2,3} (mod 4). [Chee Theor. 2] - R. J. Mathar, Nov 28 2016

G.f.: x*(3+x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 28 2016

From Colin Barker, Nov 25 2017: (Start)

a(n) = (-1/4 - i/4) * ((-1+i) + (-i)^n - i*i^n - (6-6*i)*n).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

(End)