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A053615
Pyramidal sequence: distance to nearest product of two consecutive integers (promic or heteromecic numbers).
12
0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7
OFFSET
0,5
COMMENTS
a(A002378(n)) = 0; a(n^2) = n.
Table A049581 T(n,k) = |n-k| read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 29 2013
FORMULA
a(n) = A004738(n+1) - 1.
Let u(1)=1, u(n) = n - u(n-sqrtint(n)) (cf. A037458); then a(0)=0 and for n > 0 a(n) = 2*u(n) - n. - Benoit Cloitre, Dec 22 2002
a(0)=0 then a(n) = floor(sqrt(n)) - a(n - floor(sqrt(n))). - Benoit Cloitre, May 03 2004
a(n) = |A196199(n)|. a(n) = |n - t^2 - t|, where t = floor(sqrt(n)). - Boris Putievskiy, Jan 29 2013 [corrected by Ridouane Oudra, May 11 2019]
a(n) = A000194(n) - A053188(n) = t - |t^2 - n|, where t = floor(sqrt(n)+1/2). - Ridouane Oudra, May 11 2019
EXAMPLE
a(10) = |10 - 3*4| = 2.
From Boris Putievskiy, Jan 29 2013: (Start)
The start of the sequence as table:
0, 1, 2, 3, 4, 5, 6, 7, ...
1, 0, 1, 2, 3, 4, 5, 6, ...
2, 1, 0, 1, 2, 3, 4, 5, ...
3, 2, 1, 0, 1, 2, 3, 4, ...
4, 3, 2, 1, 0, 1, 2, 3, ...
5, 4, 3, 2, 1, 0, 1, 2, ...
6, 5, 4, 3, 2, 1, 0, 1, ...
...
The start of the sequence as triangle array read by rows:
0;
1, 0, 1;
2, 1, 0, 1, 2;
3, 2, 1, 0, 1, 2, 3;
4, 3, 2, 1, 0, 1, 2, 3, 4;
5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5;
6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6;
7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7;
...
Row number r contains 2*r-1 numbers: r-1, r-2, ..., 0, 1, 2, ..., r-1. (End)
MAPLE
A053615 := proc(n)
A004738(n+1)-1 ; # reuses code of A004738
end proc:
seq(A053615(n), n=0..30) ; # R. J. Mathar, Feb 14 2019
MATHEMATICA
a[0] = 0; a[n_] := Floor[Sqrt[n]] - a[n - Floor[Sqrt[n]]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Dec 16 2011, after Benoit Cloitre *)
Join[{0}, Module[{nn=150, ptci}, ptci=Times@@@Partition[Range[nn/2+1], 2, 1]; Table[Abs[n-Nearest[ptci, n]], {n, nn}][[All, 1]]]] (* Harvey P. Dale, Aug 29 2020 *)
PROG
(PARI) a(n)=sqrtint(n)-a(n-sqrtint(n))
(PARI) apply( {A053615(n)=(t=sqrt(n)\/1)-abs(t^2-n)}, [0..99]) \\ M. F. Hasler, Feb 01 2025
(Python) A053615 = lambda n: (t := round(n**.5)) - abs(t**2 - n) \\ M. F. Hasler, Feb 01 2025
CROSSREFS
KEYWORD
easy,nice,nonn,changed
AUTHOR
Henry Bottomley, Mar 20 2000
STATUS
approved