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A025581 Triangle T(n, k) = n-k, 0 <= k <= n. 100
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

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

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

a(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. 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

REFERENCES

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 203.

LINKS

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

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

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

FORMULA

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

T(n, k) = sqrt((4*A105125(n, k) - A051162(n, k)^3)/(3*A051162(n, k))). See a comment above. - Wolfdieter Lang, May 15 2015

EXAMPLE

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

... formatted, - Wolfdieter Lang, May 12 2015

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 *)

With[{nn=20}, Flatten[Table[Join[{0}, Reverse[Range[i]]], {i, nn}]]] (* Harvey P. Dale, Dec 31 2014 *)

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+a(n).

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

Cf. A141418 (partial sums per row).

Cf. A079904, A257238, A051162, A105125.

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

Edited: part of name moved to first comment; added definition of trinv in formula. - Wolfdieter Lang, May 12 2015

STATUS

approved

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Last modified July 4 07:29 EDT 2015. Contains 259191 sequences.