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Triangle of sorted absolute values of Jacobsthal successive differences.
3

%I #25 Apr 24 2024 09:18:09

%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 *)

%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