A140950 a(n) = A140944(n+1) - 3*A140944(n).

1, -3, -1, 5, -6, 3, -11, 10, -12, -5, 21, -22, 20, -24, 11, -43, 42, -44, 40, -48, -21, 85, -86, 84, -88, 80, -96, 43, -171, 170, -172, 168, -176, 160, -192, -85, 341, -342, 340, -344, 336, -352, 320, -384, 171, -683, 682, -684, 680, -688

Offset: 0

Author

Paul Curtz, Jul 25 2008

Keywords

• sign
• tabl

Comments

Jacobsthal numbers appear twice: 1) A001045(n+2) signed, terms 0, 1, 3, 6, 10 (A000217); 2) A001045(n+1) signed, terms 0, 2, 5, 9 (n*(n+3)/2=A000096); between them are -3; 5, -6; -11, 10, -12; which appear (opposite sign) by rows in A140503 (1, -1, 2, 3, -2, 4) square.

Consider the permutation of the nonnegative numbers

0, 2, 5, 9, 14, 20, 27,

1, 3, 6, 10, 15, 21, 28,

4, 7, 11, 16, 22, 29,

8, 12, 17, 23, 30,

13, 18, 24, 31,

19, 25, 32,

26, 33,

34, etc.

The corresponding distribution of a(n) is

1, -1, 3, -5, 11, -21, 43,

-3, 5, -11, 21, -43, 85, -171,

-6, 10, -22, 42, -86, 170,

-12, 20, -44, 84, -172,

-24, 40, -88, 168,

-48, 80, -176,

-96, 160,

-192, etc.

Column sums: -2, -2, -10, -10, -42, -42, -170, ... duplicate of a bisection of -A078008(n+2).

b(n)= 1, -1, 3, -5, 11, 21, ... = (-1)^n*A001045(n+1) = A077925(n). Every row is b(n) or b(n+2) multiplied by 1, -1, -2, -4, -8, -16, ..., essentially -A011782(n).

Crossrefs

Cf. A000096, A000217, A005015, A001045, A007283, A020988, A077925, A078008, A140503, A146523, A151575, A175805, A084247.

Programs

Mathematica

T[0, 0] = 0; T[1, 0] = T[0, 1] = 1; T[0, n_] := T[0, n] = T[0, n - 1] + 2*T[0, n - 2]; T[d_, d_] = 0; T[d_, n_] := T[d, n] = T[d - 1, n + 1] - T[d - 1, n]; A140944 = Table[T[d, n], {d, 0, 10}, {n, 0, d}] // Flatten; a[n_] := A140944[[n + 2]] - 3*A140944[[n + 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 18 2014 *)

Extensions

More terms and a(19)=-48 instead of 42 corrected by Jean-François Alcover, Dec 22 2014