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A025581
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Triangle read by rows: T(n, k) = n-k, for 0 <= k <= n.
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156
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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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Decreasing integers m to 0 followed by decreasing integers m+1 to 0, etc.
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
Sequence B is called a reverse reluctant sequence of sequence A if B is a triangular array read by rows such that row number k lists the first k terms of the sequence A in reverse order. This sequence is the reverse reluctant sequence of sequence 0,1,2,3,..., the nonnegative integers A001477. - Boris Putievskiy, Dec 13 2012
A problem posed by François Viète (Vieta) in his book Zeteticorum liber quinque (1593), liber 2, problem 19 (quoted in the Alten et al. reference, on p. 292) is to find for a rectangle (a >= b >= 1) with given a^3 - b^3, name it C, and a*b, name it F, the difference a-b, name it x. This is a simple exercise which Viète found remarkable. It reduces to a standard cubic equation for x, namely x^3 + 3*F*x = C. Proof: Use the square of the diagonal d^2 = a^2 + b^2. Then (i) C = a^3 - b^3 = (a - b)*(a^2 + b^2 + a*b) = x*(d^2 + F). (ii) use the trivial relation d^2 = (a-b)^2 + 2*a*b = x^2 + 2*F, to eliminate d^2 in (i). End of the Proof. Here for positive integers a = n and b = k: (T(n, k)^2 + 3*A079904(n, k))*T(n, k) = A257238(n, k) (also true for n = k = 0). - Wolfdieter Lang, May 12 2015
See a comment on A051162 on the cubic equation for S = a+b in terms of Cplus = a^3 + b^3 and D = a - b. This equation leads to a - b = sqrt((4*Cplus -S^3)/(3*S)). - Wolfdieter Lang, May 15 2015
The entries correspond to the first of the 2 coordinates of the Cantor Pairs, specifically x=w-(CPKey-(w^2+w)/2), where w=floor((sqrt(8*CPKey+1)-1)/2) and CPKey=Cantor Pair key (A001477). The second of the coordinate pairs is A002262. - Bill McEachen, Sep 12 2015
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REFERENCES
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H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 203.
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LINKS
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FORMULA
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T(n, k) = n-k, for 0 <= k <= n.
As a sequence: a(n) = (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), with trinv(n) = floor((1+sqrt(1+8*n))/2). Cf. A002262.
G.f. for T(n,k): y / ((1-x)^2 * (1-x*y)). - Ralf Stephan, Jan 25 2005
For the cubic equation satisfied by T(n, k) see the comment on a problem by Viète above. - Wolfdieter Lang, May 12 2015
G.f. for a(n): -(1-x)^(-2) + (1-x)^(-1) * Sum_{n>=0} (n+1)*x^(n*(n+1)/2)). The sum is related to Jacobi theta functions. - Robert Israel, May 12 2015
a(n) = (1/2)*(t^2 + t - 2*n - 2), where t = floor(sqrt(2*n+1) + 1/2) = round(sqrt(2*n+1)). - Ridouane Oudra, Dec 01 2019
a(n) = ((1/2) * ceiling((-1 + sqrt(9 + 8 * n))/2) * ceiling((1 + sqrt(9 + 8 * n))/2)) - n - 1. - Ryan Jean, Apr 22 2022
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EXAMPLE
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The triangle T(n, k) begins (note that one could use l <= k <= n, for any integer l, especially 1):
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 0
1: 1 0
2: 2 1 0
3: 3 2 1 0
4: 4 3 2 1 0
5: 5 4 3 2 1 0
6: 6 5 4 3 2 1 0
7: 7 6 5 4 3 2 1 0
8: 8 7 6 5 4 3 2 1 0
9: 9 8 7 6 5 4 3 2 1 0
10: 10 9 8 7 6 5 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): seq(A025581(n), n=0..100);
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MATHEMATICA
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With[{nn=20}, Flatten[Table[Join[{0}, Reverse[Range[i]]], {i, nn}]]] (* Harvey P. Dale, Dec 31 2014 *)
Table[Range[n, 0, -1], {n, 0, 15}]//Flatten (* Harvey P. Dale, Aug 01 2020 *)
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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 */
(PARI) apply( {A025581(n)=binomial(sqrtint(8*n+1)\/2+1, 2)-n-1}, [0..90]) \\ M. F. Hasler, Dec 06 2019
(Haskell)
a025581 n k = n - k
a025581_row n = [n, n-1 .. 0]
a025581_tabl = iterate (\xs@(x:_) -> (x + 1) : xs) [0]
(Magma) /* As triangle */ [[(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Sep 13 2015
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CROSSREFS
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Cf. A141418 (partial sums per row).
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited (part of name moved to first comment; definition of trinv added in formula) by Wolfdieter Lang, May 12 2015
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STATUS
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approved
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