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A025581
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Triangle T(n,k) = n-k, n >= 0, 0<=k<=n. Integers m to 0 followed by integers m+1 to 0 etc.
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83
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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_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
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FORMULA
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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
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EXAMPLE
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0; 1,0; 2,1,0; 3,2,1,0; 4,3,2,1,0; ...
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MAPLE
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A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))), 2) - (n+1);
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MATHEMATICA
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Flatten[NestList[Prepend[#, #[[1]]+1]&, {0}, 13]] (* Jean-François Alcover, May 17 2011 *)
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PROG
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(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 = a025581_tabl !! n !! k
a025581_row n = a025581_tabl !! n
a025581_tabl = iterate (\row@(h:_) -> (h + 1) : row) [0]
-- Reinhard Zumkeller, Jul 22 2012, Mar 07 2011
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CROSSREFS
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A004736(n+1)=1+A025581(n)
Cf. A025669, A025676, A025683, A002262, A004736, A001477.
Sequence in context: * A025669 A025676 A025683 A025660 A025677 A025651
Adjacent sequences: A025578 A025579 A025580 * A025582 A025583 A025584
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Typo in definition corrected by Arkadiusz Wesolowski, Nov 24 2011
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STATUS
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approved
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