|
| |
| |
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
