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A061579
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Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.
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20
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0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66
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listen;
history;
text;
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OFFSET
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0,2
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COMMENTS
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A self-inverse permutation of the nonnegative numbers.
a(n) is the smallest nonnegative integer not yet in the sequence such that n + a(n) is one less than a square. [Franklin T. Adams-Watters, Apr 06 2009]
Array T(n,k) = (n+k)^2/2 + (n+3*k)/2 for n,k >=0 read by descending antidiagonals.
Array T(n,k) = (n+k)^2/2 + (3*n+k)/2 for n,k >=0 read by ascending antidiagonals. (End)
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LINKS
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FORMULA
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a(n) = floor(sqrt(2n+1)-1/2)*floor(sqrt(2n+1)+3/2) - n = A005563(A003056(n)) - n.
Row (or antidiagonal) n = 0, 1, 2, ... contains the integers from A000217(n) to A000217(n+1)-1 in reverse order (for diagonals, "reversed" with respect to the canonical "falling" order, cf. A001477/table). - M. F. Hasler, Nov 09 2021
T(n,k) = n*(n+3)/2 - k.
Sum_{k=0..n} k * T(n,k) = A002419(n).
Sum_{k=0..n} k^2 * T(n,k) = A119771(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A226725(n). (End)
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EXAMPLE
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Read as a triangle, the sequence is:
0
2 1
5 4 3
9 8 7 6
14 13 12 11 10
(...)
As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:
0 2 5 9 14 20 ...
1 4 8 13 19 22 ...
3 7 12 18 23 30 ...
6 11 17 24 31 39 ...
(...)
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MAPLE
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T:= (n, k)-> n*(n+3)/2-k:
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MATHEMATICA
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Module[{nn=20}, Reverse/@TakeList[Range[0, (nn(nn+1))/2], Range[nn]]]// Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jul 06 2018 *)
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PROG
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(PARI) A061579_row(n)=vector(n+=1, j, n*(n+1)\2-j)
A061579_upto(n)=concat([A061579_row(r)|r<-[0..sqrtint(2*n)]]) \\ yields approximately n terms: actual number differs by less than +- sqrt(n). - M. F. Hasler, Nov 09 2021
(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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