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%I #19 Mar 12 2024 10:54:10
%S 1,2,3,4,5,6,8,10,11,12,16,20,21,22,24,32,40,42,43,44,48,64,80,84,85,
%T 86,88,96,128,160,168,170,171,172,176,192,256,320,336,340,341,342,344,
%U 352,384,512,640,672,680,682,683,684,688,704,768,1024,1280,1344,1360
%N Triangle of sorted absolute values of Jacobsthal successive differences.
%C The triangle is generated from the set of Jacobsthal numbers A001045 and all the iterated differences (see A078008, A084247), taking the absolute values and sorting into natural order.
%C The first differences generated individually along any row of this triangle here are all in A000079.
%F Row sums: A113861(n+2).
%e The triangle starts
%e 1;
%e 2, 3;
%e 4, 5, 6;
%e 8, 10, 11, 12;
%e 16, 20, 21, 22, 24;
%e The Jacobsthal sequence and its differences in successive rows start:
%e 0, 1, 1, 3, 5, 11, 21, 43, 85, ...
%e 1, 0, 2, 2, 6, 10, 22, 42, 86, ...
%e -1, 2, 0, 4, 4, 12, 20, 44, 84, ...
%e 3, -2, 4, 0, 8, 8, 24, 40, 88, ...
%e -5, 6, -4, 8, 0, 16, 16, 48, 80, ...
%e 11, -10, 12, -8, 16, 0, 32, 32, 96, ...
%e -21, 22, -20, 24, -16, 32, 0, 64, 64, ...
%e 43, -42, 44, -40, 48, -32, 64, 0, 128, ...
%e The values +-7, +-9, +-13, for example, are missing there, so 7, 9 and 13 are not in the triangle.
%t maxTerm = 384; FixedPoint[(nMax++; Print["nMax = ", nMax]; jj = Table[(2^n - (-1)^n)/3, {n, 0, nMax}]; Table[Differences[jj, n], {n, 0, nMax}] // Flatten // Abs // Union // Select[#, 0 < # <= maxTerm &] &) &, nMax = 5 ] (* _Jean-François Alcover_, Dec 16 2014 *)
%o (PARI) a(n) = n++; my(A = (sqrtint(8*n) + 1)\2, B = n - A*(A - 1)/2); B = if(B <= (A+1)\2, 2*B - 1, 2*(A - B + 1)); my(v1); v1 = vector(A, i, 0); for(i = 1, B\2, v1[2*i - 1] = 1); v1[B] = 1; fromdigits(v1, 2) \\ _Mikhail Kurkov_, Mar 12 2024
%Y Cf. A000079, A003945, A078008, A084247, A113861.
%K nonn,tabl
%O 0,2
%A _Paul Curtz_, Jul 08 2008
%E Edited by _R. J. Mathar_, Dec 05 2008
%E a(45)-a(58) from _Stefano Spezia_, Mar 12 2024
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