A140642 Triangle of sorted absolute values of Jacobsthal successive differences.

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, 86, 88, 96, 128, 160, 168, 170, 171, 172, 176, 192, 256, 320, 336, 340, 341, 342, 344, 352, 384, 512, 640, 672, 680, 682, 683, 684, 688, 704, 768, 1024, 1280, 1344, 1360

Offset

0,2

Comments

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.

The first differences generated individually along any row of this triangle here are all in A000079.

Links

Table of n, a(n) for n=0..58.

Formula

Row sums: A113861(n+2).

Example

The triangle starts
   1;
   2,  3;
   4,  5,  6;
   8, 10, 11, 12;
  16, 20, 21, 22, 24;
The Jacobsthal sequence and its differences in successive rows start:
    0,   1,   1,   3,   5,  11,  21,  43,  85, ...
    1,   0,   2,   2,   6,  10,  22,  42,  86, ...
   -1,   2,   0,   4,   4,  12,  20,  44,  84, ...
    3,  -2,   4,   0,   8,   8,  24,  40,  88, ...
   -5,   6,  -4,   8,   0,  16,  16,  48,  80, ...
   11, -10,  12,  -8,  16,   0,  32,  32,  96, ...
  -21,  22, -20,  24, -16,  32,   0,  64,  64, ...
   43, -42,  44, -40,  48, -32,  64,   0, 128, ...
The values +-7, +-9, +-13, for example, are missing there, so 7, 9 and 13 are not in the triangle.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000079, A003945, A078008, A084247, A113861.

Sequence in context: A067936 A087721 A226244 * A188936 A260317 A072666

Adjacent sequences: A140639 A140640 A140641 * A140643 A140644 A140645

Keyword

nonn,tabl

Author

Paul Curtz, Jul 08 2008

Extensions

Edited by R. J. Mathar, Dec 05 2008

a(45)-a(58) from Stefano Spezia, Mar 12 2024

Status

approved