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

0,2

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).

Links

Table of n, a(n) for n=0..49.

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 *)

Crossrefs

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

Sequence in context: A208904 A344479 A209754 * A256504 A205713 A394451

Adjacent sequences: A140947 A140948 A140949 * A140951 A140952 A140953

Keyword

sign,tabl

Author

Paul Curtz, Jul 25 2008

Extensions

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

Status

approved