A049879 - OEIS

%I #14 Nov 20 2019 02:30:47

%S 1,2,3,2,2,3,4,6,4,7,9,7,10,12,12,14,12,15,17,17,19,17,20,22,5,26,25,

%T 19,27,16,29,13,31,33,33,31,8,6,37,3,39,19,40,42,42,44,4,46,6,48,48,8,

%U 50,11,53,53,55,53,16,58,58,17,60,19,63,23,65,25,67,11,70,70,68,33,58,35,74

%N a(n) = |j - k|, where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UlamSequence.html">Ulam Sequence</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_number">Ulam number</a>.

%e From _Petros Hadjicostas_, Nov 20 2019: (Start)

%e A002859(3) = 4 = 1 + 3 = A002859(1) + A002859(2), so a(3) = |1-2| = 1.

%e A002859(4) = 5 = 1 + 4 = A002859(1) + A002859(3), so a(4) = |1-3| = 2.

%e A002859(5) = 6 = 1 + 5 = A002859(1) + A002859(4), so a(5) = |1-4| = 3.

%e A002859(6) = 8 = 3 + 5 = A002859(2) + A002859(4), so a(6) = |2-4| = 2.

%e A002859(7) = 10 = 4 + 6 = A002859(3) + A002859(5), so a(7) = |3-5| = 2.

%e (End)

%p # First we modify _Peter Luschny_'s program from A002858 (with len >= 3):

%p UlamList := proc(len) local isUlam, nextUlam, behead; behead := u -> u[2 .. numelems(u)]; isUlam := proc(n, h, u, r) local hu, tu, hr, tr; hu := u[1]; hr := r[1]; if h = 2 then return false; end if; if hr <= hu then return evalb(h = 1); end if; if hr + hu = n then tu := behead(u); tr := behead(r); return isUlam(n, h + 1, tu, tr); end if; if hr + hu < n then tu := behead(u); return isUlam(n, h, tu, r); end if; tr := behead(r); isUlam(n, h, u, tr); end proc; nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len - 1 then return [op(u), n]; end if; nextUlam(n + 1, [op(u), n], [n, op(r)]); else nextUlam(n + 1, u, r); end if; end proc; nextUlam(3, [1, 3], [3, 1]); end proc:

%p # Next we create a function to calculate a(n) for given n >= 3:

%p a := proc(n) local u, a, i, j: u := 0: if 3 <= n then a := UlamList(n): for i to n - 2 do for j from i + 1 to n - 1 do if a[n] = a[i] + a[j] then u := abs(i-j): end if: end do: end do: end if: u: end proc:

%p # Finally, we create a list of values for a(n):

%p seq(a(n), n=3..100); # _Petros Hadjicostas_, Nov 20 2019

%Y Cf. A002858, A002859, A049821, A049877, A049878.

%K nonn

%O 3,2

%A _Clark Kimberling_

%E Name edited by and more terms from _Petros Hadjicostas_, Nov 20 2019