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

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

Author

Paul Curtz, Jun 30 2008

Keywords

• sign
• tabl
• easy

Comments

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.

Examples

			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).
		

Crossrefs

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

Programs

PARI

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

Formula

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

Extensions

Edited by R. J. Mathar, Jul 14 2008