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

%I A140950 #39 Dec 24 2014 23:10:12
%S A140950 1,-3,-1,5,-6,3,-11,10,-12,-5,21,-22,20,-24,11,-43,42,-44,40,-48,-21,
%T A140950 85,-86,84,-88,80,-96,43,-171,170,-172,168,-176,160,-192,-85,341,-342,
%U A140950 340,-344,336,-352,320,-384,171,-683,682,-684,680,-688
%N A140950 a(n) = A140944(n+1) - 3*A140944(n).
%C A140950 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.
%C A140950 Consider the permutation of the nonnegative numbers
%C A140950 0, 2, 5,  9, 14, 20, 27,
%C A140950 1, 3, 6, 10, 15, 21, 28,
%C A140950    4, 7, 11, 16, 22, 29,
%C A140950       8, 12, 17, 23, 30,
%C A140950          13, 18, 24, 31,
%C A140950              19, 25, 32,
%C A140950                  26, 33,
%C A140950                      34, etc.
%C A140950 The corresponding distribution of a(n) is
%C A140950 1,  -1,   3,  -5,  11, -21,   43,
%C A140950 -3,  5, -11,  21, -43,  85, -171,
%C A140950     -6,  10, -22,  42, -86,  170,
%C A140950         -12,  20, -44,  84, -172,
%C A140950              -24,  40, -88,  168,
%C A140950                   -48,  80, -176,
%C A140950                        -96,  160,
%C A140950                             -192, etc.
%C A140950 Column sums: -2, -2, -10, -10, -42, -42, -170, ... duplicate of a bisection of -A078008(n+2).
%C A140950 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).
%t A140950 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 *)
%Y A140950 Cf. A000096, A000217, A005015, A001045, A007283, A020988, A077925, A078008, A140503, A146523, A151575, A175805, A084247.
%K A140950 sign,tabl
%O A140950 0,2
%A A140950 _Paul Curtz_, Jul 25 2008
%E A140950 More terms and a(19)=-48 instead of 42 corrected by _Jean-François Alcover_, Dec 22 2014