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A025581 Triangle T(n,k) = n-k, n >= 0, 0<=k<=n. Integers m to 0 followed by integers m+1 to 0 etc. 96
0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals upwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002

Riordan array (x/(1-x)^2, x). - Philippe Deléham, Feb 18 2012

A025581(n,k) = (A214604(n,k) - A214661(n,k)) / 2. - Reinhard Zumkeller, Jul 25 2012

Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A025581 is the reverse reluctant sequence of sequence 0,1,2,3,..., the nonnegative integers A001477. - Boris Putievskiy, Dec 13 2012

LINKS

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]

M. Somos, Sequences used for indexing triangular or square arrays

FORMULA

a(n) = (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) # Cf. A002262

G.f.: y / [(1-x)^2 * (1-xy) ]. - Ralf Stephan, Jan 25 2005

a(n)=A001477(m), where m=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Dec 13 2012

EXAMPLE

0; 1,0; 2,1,0; 3,2,1,0; 4,3,2,1,0; ...

MAPLE

A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))), 2) - (n+1);

MATHEMATICA

Flatten[NestList[Prepend[#, #[[1]]+1]&, {0}, 13]] (* Jean-François Alcover, May 17 2011 *)

PROG

(PARI) a(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1) /* produces a(n) */

(PARI) t1(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1) /* A025581 */

(PARI) t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) /* A002262 */

(Haskell)

a025581 n k = n - k

a025581_row n = [n, n-1 .. 0]

a025581_tabl = iterate (\xs@(x:_) -> (x + 1) : xs) [0]

-- Reinhard Zumkeller, Aug 04 2014, Jul 22 2012, Mar 07 2011

CROSSREFS

A004736(n+1)=1+A025581(n)

Cf. A025669, A025676, A025683, A002262, A004736, A001477.

Cf. A141418 (partial sums per row).

Sequence in context: * A025669 A025676 A025683 A025660 A025677 A025651

Adjacent sequences:  A025578 A025579 A025580 * A025582 A025583 A025584

KEYWORD

nonn,tabl,easy,nice

AUTHOR

David W. Wilson

EXTENSIONS

Typo in definition corrected by Arkadiusz Wesolowski, Nov 24 2011

STATUS

approved

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Last modified November 25 15:28 EST 2014. Contains 249997 sequences.