A053188 -id:A053188 - cp's OEIS frontend

A053646 Distance to nearest power of 2.

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 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, 10, 11, 12, 13, 14, 15, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Offset: 1

Henry Bottomley, Mar 22 2000

Sum_{j=1..2^(k+1)} a(j) = A002450(k) = (4^k - 1)/3. - Klaus Brockhaus, Mar 17 2003

			a(10)=2 since 8 is closest power of 2 to 10 and |8-10| = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Index entries for sequences related to distance to nearest element of some set

Cf. A006165, A053188, A060973, A081134, A002450, A081252, A081253, A081254.

a:= n-> (h-> min(n-h, 2*h-n))(2^ilog2(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

np2[n_]:=Module[{min=Floor[Log[2,n]],max},max=min+1;If[2^max-nHarvey P. Dale, Feb 21 2012 *)

a(n)=vecmin(vector(n,i,abs(n-2^(i-1))))

for(n=1,89,p=2^floor(0.1^25+log(n)/log(2)); print1(min(n-p,2*p-n),","))

a(n) = my (p=#binary(n)); return (min(n-2^(p-1), 2^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A053646(n): return min(n-(m := 2**(len(bin(n))-3)),2*m-n) # Chai Wah Wu, Mar 08 2022

a(2^k+i) = i for 1 <= i <= 2^(k-1); a(3*2^k+i) = 2^k-i for 1 <= i <= 2^k; (Sum_{k=1..n} a(k))/n^2 is bounded. - Benoit Cloitre, Aug 17 2002
a(n) = min(n-2^floor(log(n)/log(2)), 2*2^floor(log(n)/log(2))-n). - Klaus Brockhaus, Mar 08 2003
From Peter Bala, Aug 04 2022: (Start)
a(n) = a( 1 + floor((n-1)/2) ) + a( ceiling((n-1)/2) ).
a(2*n) = 2*a(n); a(2*n+1) = a(n) + a(n+1) for n >= 2. Cf. A006165. (End)
a(n) = 2*A006165(n) - n for n >= 2. - Peter Bala, Sep 25 2022

A053187 Square nearest to n.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Henry Bottomley, Mar 01 2000

Apart from 0, k^2 appears 2k times from a(k^2-k+1) through to a(k^2+k) inclusive.

			a(7) = 9 since 7 is closer to 9 than to 4.
G.f. = x + x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 4*x^6 + 9*x^7 + 9*x^8 + 9*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A048760, A053188, A002483, A245552.

Cf. A061023, A201053 (nearest cube), A000290, A000194.

a053187 n = a053187_list !! n
a053187_list = 0 : concatMap (\x -> replicate (2*x) (x ^ 2)) [1..]
-- Reinhard Zumkeller, Nov 28 2011

seq(ceil((-1+sqrt(4*n+1))/2)^2, n=0..20); # Robert Israel, Jan 05 2015

nearestSq[n_] := Block[{a = Floor@ Sqrt@ n}, If[a^2 + a + 1/2 > n, a^2, a^2 + 2 a + 1]]; Array[ nearestSq, 75, 0] (* Robert G. Wilson v, Aug 01 2014 *)

from math import isqrt
def A053187(n): return ((m:=isqrt(n))+int(n>m*(m+1)))**2 # Chai Wah Wu, Jun 06 2025

a(n) = ceiling((-1 + sqrt(4*n+1))/2)^2. - Robert Israel, Aug 01 2014
G.f.: (1/(1-x))*Sum_{n>=0} (2*n+1)*x^(n^2+n+1). - Robert Israel, Aug 01 2014.  This is related to the Jacobi theta-function theta'_1(q), see A002483 and A245552.
G.f.: x / (1-x) * Sum_{k>0} (2*k - 1) * x^(k^2 - k). - Michael Somos, Jan 05 2015
a(n) = floor(sqrt(n)+1/2)^2. - Mikael Aaltonen, Jan 17 2015
Sum_{n>=1} 1/a(n)^2 = 2*zeta(3). - Amiram Eldar, Aug 15 2022
a(n) = A000194(n)^2. - Chai Wah Wu, Jun 06 2025

Title improved by Jon E. Schoenfield, Jun 09 2019

A053615 Pyramidal sequence: distance to nearest product of two consecutive integers (promic or heteromecic numbers).

Original entry on oeis.org

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

Henry Bottomley, Mar 20 2000

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

			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)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences related to distance to nearest element of some set

Cf. A002262, A002378, A004738, A049581, A053188, A196199.

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

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 *)

PARI
```
a(n)=sqrtint(n)-a(n-sqrtint(n))
```

apply( {A053615(n)=(t=sqrt(n)\/1)-abs(t^2-n)}, [0..99]) \\ M. F. Hasler, Feb 01 2025

A053615 = lambda n: (t := round(n**.5)) - abs(t**2 - n) # M. F. Hasler, Feb 01 2025

from math import isqrt
def A053615(n): return abs((t:=isqrt(n))*(t+1)-n) # Chai Wah Wu, Mar 01 2025

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

A053616 Pyramidal sequence: distance to nearest triangular number.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1

Offset: 0

Henry Bottomley, Mar 20 2000

From Wolfdieter Lang, Oct 24 2020: (Start)
If this sequence is written with offset 1 as a number triangle T(n, k), with n the length of row n, for n >= 1, then row n gives the primitive period of the periodic sequence {k (mod* n)}_{k>=0}, where k (mod* n) = k (mod n) if k <= floor(n/2) and otherwise it is -k (mod n). Such a modified modular relation mod* n has been used by Brändli and Beyne, but for integers relative prime to n.
These periodic sequences are given in A000007, A000035, A011655, A007877, |A117444|, A260686, A279316, for n = 1, 2, ..., 7. For n = 10 A271751, n = 12 A271832, n = 14 A279313. (End)

			a(12) = |12 - 10| = 2 since 10 is the nearest triangular number to 12.
From _M. F. Hasler_, Dec 06 2019: (Start)
Ignoring a(0) = 0, the sequence can be written as triangle indexed by m >= k >= 1, in which case the terms are (m - |k - |m-k||)/2, as follows:
   0,      (Row 0: ignore)
   0,      (Row m=1, k=1: For k=m, m - |k - |m-k|| = m - |m - 0| = 0.)
   1, 0,        (Row m=2: for k=1, |m-k| = 1, k-|m-k| = 0, m-0 = 2, (...)/2 = 1.)
   1, 1, 0,
   1, 2, 1, 0,    (Row m=4: for k=2, we have twice the value of (m=2, k=1) => 2.)
   1, 2, 2, 1, 0,
   (...)
This is related to the non-associative operation A049581(x,y) = |x - y| =: x @ y. Specifically, @ is commutative and any x is its own inverse, so non-associativity of @ can be measured through the commutator ((x @ y) @ y) @ x which equals twice the element indexed {m,k} = {x,y} in the above triangle.
(End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2015-2016.
Index entries for sequences related to distance to nearest element of some set

Cf. A000217, A002262, A053188, A172471.

a(n) = abs(A305258(n)).

a[n_] := (k =.; k = Reduce[k > 0 && k*(k+1)/2 == n, Reals][[2]] // Floor; Min[(k+1)*(k+2)/2 - n, n - k*(k+1)/2]); Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jan 08 2013 *)
Module[{trms=120,t},t=Accumulate[Range[Ceiling[(Sqrt[8*trms+1]-1)/2]]]; Join[{0},Flatten[Table[Abs[Nearest[t,n][[1]]-n],{n,trms}]]]] (* Harvey P. Dale, Nov 08 2013 *)

print1(x=0, ", ");for(stride=1,13,x+=stride;y=x+stride+1;for(k=x,y-1,print1(min(k-x,y-k), ", "))) \\ Hugo Pfoertner, Jun 02 2018

apply( {a(n)=if(n,-abs(n*2-(n=sqrtint(8*n-7)\/2)^2)+n)\2}, [0..40]) \\ same as (i - |j - |i-j||)/2 with i=sqrtint(8*n-7)\/2, j=n-i(i-1)/2. - M. F. Hasler, Dec 06 2019

from math import isqrt
def A053616(n): return abs((m:=isqrt(k:=n<<1))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022

a(n) = (x - |y - |x-y||)/2, when (x,y) is the n-th element in the triangle x >= y >= 1. - M. F. Hasler, Dec 06 2019
a(n) = (1/2)*abs(t^2 + t - 2*n), where t = floor(sqrt(2*n)) = A172471. - Ridouane Oudra, Dec 15 2021
From Ctibor O. Zizka, Nov 12 2024: (Start)
For s >= 1, t from [0, s] :
a(2*s^2 + t) = s - t.
a(2*s^2 - t) = s - t.
a(2*s^2 + 2*s - t) = s - t.
a(2*s^2 + 2*s + 1 + t) = s - t. (End)

A074989 Distance from n to nearest cube.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 0

Zak Seidov, Oct 02 2002

nonn

a(n)=0 when n is a cube; between zeros local maxima are of form 3/2 k(k-1).

			a(3) = 2 because the nearest cube to 3 is 1 and distance from 3 to 1 is 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to distance to nearest element of some set

Cf. A053188 (distance from n to nearest square).

Cf. A201053, A000578.

a074989 0 = 0
a074989 n = min (n - last xs) (head ys - n) where
   (xs,ys) = span (< n) a000578_list
-- Reinhard Zumkeller, Nov 28 2011

A074989 := proc(n) local iscbr ; iroot(n,3,'iscbr') ; if iscbr then 0; else iscbr := floor(n^(1/3)) ; min((iscbr+1)^3-n, n-iscbr^3) ; end if; end proc; # R. J. Mathar, Nov 01 2009

dnc[n_]:=Module[{cr=Surd[n,3]},Min[n-Floor[cr]^3,Ceiling[cr]^3-n]]; Array[ dnc,90,0] (* Harvey P. Dale, Jan 24 2015 *)

from sympy import integer_nthroot
def A074989(n):
    a = integer_nthroot(n,3)[0]
    return min(n-a**3,(a+1)**3-n) # Chai Wah Wu, Mar 31 2021

a(0) added and offset changed by Reinhard Zumkeller, Nov 28 2011

A080883 Distance of n to next square.

Original entry on oeis.org

1, 3, 2, 1, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13

Offset: 0

Ralf Stephan, Mar 29 2003

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446, A080883. - Jeremy Gardiner, Dec 30 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A075555.

Cf. A066635, A053188. - R. J. Mathar, Aug 08 2009

List([0..90], n-> Int(1+RootInt(n))^2 -n); # G. C. Greubel, Nov 07 2019

[Floor(1+Sqrt(n))^2 -n: n in [0..90]]; // G. C. Greubel, Nov 07 2019

A080883 := proc(n) (floor(sqrt(n)+1))^2 -n ; end: seq( A080883(n),n=0..40) ; # R. J. Mathar, Aug 08 2009

Table[Floor[1+Sqrt[n]]^2 -n, {n,0,90}] (* G. C. Greubel, Nov 07 2019 *)

a(n) = (sqrtint(n)+1)^2-n; \\ Michel Marcus, May 22 2024

[floor(1+sqrt(n))^2 -n for n in (0..90)] # G. C. Greubel, Nov 07 2019

a(n) = floor( sqrt(n)+1 )^2 - n.

A351830 Distance from the n-th square pyramidal number (sum of the first n positive squares) to the nearest square.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 9, 4, 8, 4, 15, 22, 25, 22, 9, 15, 25, 21, 7, 30, 46, 53, 49, 32, 0, 49, 40, 41, 30, 91, 46, 12, 9, 15, 4, 26, 77, 114, 25, 91, 61, 105, 15, 122, 129, 66, 22, 1, 1, 24, 76, 157, 170, 37, 131, 141, 91, 139, 165, 15, 174, 247, 150, 80, 39, 29

Offset: 0

Paolo Xausa, Feb 21 2022

As noted by Conway and Sloane (1999), the only zero terms appear at n = 0, n = 1 and n = 24, and the n = 24 case allows for the Lorentzian construction of the Leech lattice through the A351831 vector.

The zero terms are equivalently the subject of the "pile of cannonballs" problem posed by Lucas and solved by Watson. - Peter Munn, Aug 03 2023

			a(4) = 5 because the sum of the first 4 positive squares is 1 + 4 + 9 + 16 = 30, the nearest square is 25 and 30 - 25 = 5. - _Paolo Xausa_, Jul 05 2022

W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidsskr. 34 (1952), pp 65-72.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, entry 24, p 101.

Paolo Xausa, Table of n, a(n) for n = 0..9999
Michael A. Bennett, Lucas' Square Pyramid Problem Revisited.
Richard E. Borcherds, How to construct the Leech lattice, YouTube video, 2022.
J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bulletin (New Series) of the American Mathematical Society, Volume 6, Number 2, March 1982.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer, New York, NY, 1999, pp. 524-528.
Anji Dong, Katerina Saettone, Kendra Song, and Alexandru Zaharescu, An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II, arXiv:2505.04166 [math.NT], 2025.
E. Lucas, Problem 1180, Nouvelles Ann. Math. (2) 14 (1875), p 336.
G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.
Wikipedia, Leech lattice.

Cf. A000290, A000330, A053188, A351831, A353295, A354330, A363284.

nterms=66;Array[Abs[(s=#(#+1)(2#+1)/6)-Round[Sqrt[s]]^2]&,nterms,0]

from math import isqrt
def a(n):
    t = n*(n+1)*(2*n+1)//6
    r = isqrt(t)
    return min(t - r**2, (r+1)**2 - t)
print([a(n) for n in range(66)]) # Michael S. Branicky, Feb 21 2022

From Paolo Xausa, Jul 05 2022: (Start)
a(n) = A053188(A000330(n)).
a(n) = abs(A000330(n) - A353295(n)). (End)

Name edited by Peter Munn, Aug 04 2023

A038760 a(n) = n - floor(sqrt(n)) * ceiling(sqrt(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, -1, 0, 1, 2, 0, -2, -1, 0, 1, 2, 3, 0, -3, -2, -1, 0, 1, 2, 3, 4, 0, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 0, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 0, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 0, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, -8, -7, -6, -5, -4

Offset: 0

Henry Bottomley, May 03 2000

			Sqrt(31) is between 5 and 6, and 31 - 6*5 = 1, so a(31)=1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A053188.

a:= n-> n -(x-> floor(x)*ceil(x))(sqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 03 2015

f[n_]:=n-Floor[Sqrt[n]]*Ceiling[Sqrt[n]];Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)

a(n)=if(issquare(n),0,my(s=sqrtint(n));n-s^2-s) \\ Charles R Greathouse IV, Feb 07 2013

from math import isqrt
def A038760(n): return m-k if (m:=n-(k:=isqrt(n))**2) else 0 # Chai Wah Wu, Jul 28 2022

a(n) = n - A000196(n)*A003059(n) = n - A038759(n).

A296239 a(n) = distance from n to nearest Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Rémy Sigrist, Dec 09 2017

The Fibonacci numbers correspond to sequence A000045.
This sequence is analogous to:
- A051699 (distance to nearest prime),
- A053188 (distance to nearest square),
- A053646 (distance to nearest power of 2),
- A053615 (distance to nearest oblong number),
- A053616 (distance to nearest triangular number),
- A061670 (distance to nearest power),
- A074989 (distance to nearest cube),
- A081134 (distance to nearest power of 3),
The local maxima of the sequence correspond to positive terms of A004695.
a(n) = 0 iff n = A000045(k) for some k >= 0.
a(n) = 1 iff n = A061489(k) for some k > 4.
For any n >= 0, abs(a(n+1) - a(n)) <= 1.
For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

			For n = 42:
- A000045(9) = 34 <= 42 <= 55 = A000045(10),
- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* Robert G. Wilson v, Dec 11 2017 *)
With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

cp's OEIS Frontend

A074378 Even triangular numbers halved.

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A053646 Distance to nearest power of 2.

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A053187 Square nearest to n.

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A053615 Pyramidal sequence: distance to nearest product of two consecutive integers (promic or heteromecic numbers).

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A053616 Pyramidal sequence: distance to nearest triangular number.

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A074989 Distance from n to nearest cube.

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