A098354 -id:A098354 - cp's OEIS frontend

A098355 Multiplication table of the powers of three read by antidiagonals.

1, 3, 3, 9, 9, 9, 27, 27, 27, 27, 81, 81, 81, 81, 81, 243, 243, 243, 243, 243, 243, 729, 729, 729, 729, 729, 729, 729, 2187, 2187, 2187, 2187, 2187, 2187, 2187, 2187, 6561, 6561, 6561, 6561, 6561, 6561, 6561, 6561, 6561, 19683, 19683, 19683, 19683, 19683, 19683

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

3^A003056: 3^n appears n+1 times.

			1; 3,3; 9,9,9; 27,27,27,27;

Jeremy Gardiner, Table of n, a(n) for n = 0..349

Cf. A000244, A003056, A003991, A098354, A137688.

Flatten[Table[3^x, {x, 0, 13}, {y, 0, x}]] (* Alonso del Arte, Nov 29 2011 *)

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

cp's OEIS Frontend

A098355 Multiplication table of the powers of three read by antidiagonals.

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