A140944 -id:A140944 - cp's OEIS frontend

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

Paul Curtz, Jul 25 2008

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).

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

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 *)

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

A140503 Triangle T(d,n) read by rows, the n-th term of the d-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

1, -1, 2, 3, -2, 4, -5, 6, -4, 8, 11, -10, 12, -8, 16, -21, 22, -20, 24, -16, 32, 43, -42, 44, -40, 48, -32, 64, -85, 86, -84, 88, -80, 96, -64, 128, 171, -170, 172, -168, 176, -160, 192, -128, 256, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 683, -682, 684, -680

Offset: 1

Paul Curtz, Jun 30 2008

If interpreted as a flat sequence a(j), we obtain a(j+1)-2a(j)= -3, 4, -1, -8, 8, -13, 16, -16, 16, -5, -32, 32, -32, 32, -53, 64, ... which is essentially the negative values of A096773 padded by groups of one, then two, then three etc. signed elements of A098354.

			A001045 and its d times iterated differences are
.0,.1,.1,.3,.5,11,21,43,...
.1,.0,.2,.2,.6,10,22,... < d=1
-1,.2,.0,.4,.4,12,... < d=2
.3,-2,.4,.0,.8,.. < d=3
-5,.6,-4,.8,.0,...
The sequence contains the first d elements of the d-th row, those up to the diagonal (which contains zeros).

Cf. A001045, A140944 (with an extra diagonal of 0's).

T(d,n) = (2^n - 2^d*(-1)^(d+n))/3 \\ Jianing Song, Aug 11 2022

T(d,n)=T(d-1,n+1)-T(d-1,n). T(0,n)=A001045(n).
Row sums: sum_{n=0..d-1} T(d,n) = A002450([(d+1)/2]).
Row sums of absolute values: sum_{n=0..d-1} |T(d,n)| = A045883(d).
T(d,n) = (2^n - 2^d*(-1)^(d+n))/3, for d > n >= 0. - Jianing Song, Aug 11 2022

Edited by R. J. Mathar, Jul 14 2008

A140946 Triangle T(n,k) = (-2)^n*(-1)^k if kA001045(n+1).

Original entry on oeis.org

1, -2, -1, 4, -4, 3, -8, 8, -8, -5, 16, -16, 16, -16, 11, -32, 32, -32, 32, -32, -21, 64, -64, 64, -64, 64, -64, 43, -128, 128, -128, 128, -128, 128, -128, -85, 256, -256, 256, -256, 256, -256, 256, -256, 171, -512, 512, -512, 512, -512, 512, -512, 512, -512, -341, 1024, -1024, 1024, -1024, 1024

Offset: 0

Paul Curtz, Jul 24 2008

The sequence appears if the values b(n+1)-2*b(n) are computed from the (flattened) sequence b(.)=A140944.

Reading the triangle by rows, taking absolute values and removing duplicates we obtain A112387.

			1;
-2,-1;
4,-4,3;
-8,8,-8,-5;
16,-16,16,-16,11;
-32,32,-32,32,-32,-21;
64,-64,64,-64,64,-64,43;
-128,128,-128,128,-128,128,-128,-85;

Cf. A140513, A140589.

(* A = A140944 *) A[0, 0] = 0; A[1, 0] = A[0, 1] = 1; A[0, k_] := A[0, k] = A[0, k-1] + 2*A[0, k-2]; A[n_, n_] = 0; A[n_, k_] := A[n, k] = A[n-1, k+1] - A[n-1, k];  T[n_, n_] := T[n, n] = A[n+1, 0] - 2*A[n, n]; T[n_, k_] := T[n, k] = A[n, k+1] - 2*A[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 17 2014 *)

T(n,k) = A140944(n,k+1)-2*A140944(n,k), k

T(n,n) = A140944(n+1,0) -2*A140944(n,n).

Edited by R. J. Mathar, Jul 06 2011

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cp's OEIS Frontend

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

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A140503 Triangle T(d,n) read by rows, the n-th term of the d-th differences of the Jacobsthal sequence A001045.

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A140946 Triangle T(n,k) = (-2)^n*(-1)^k if kA001045(n+1).

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