A081134 -id:A081134 - cp's OEIS frontend

A081249 Partial sums of A081134.

0, 1, 1, 2, 4, 7, 9, 10, 10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 63, 70, 76, 81, 85, 88, 90, 91, 91, 92, 94, 97, 101, 106, 112, 119, 127, 136, 146, 157, 169, 182, 196, 211, 227, 244, 262, 281, 301, 322, 344, 367, 391, 416, 442, 469, 495, 520, 544, 567, 589, 610, 630

Offset: 1

Klaus Brockhaus, Mar 17 2003

			First seven terms of A081134 are 0,1,0,1,2,3,2, so a(7) = 9.

Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251

Accumulate[Flatten[Table[Join[Range[0,3^n],Range[3^n-1,1,-1]],{n,0,4}]]] (* Harvey P. Dale, Sep 21 2016 *)

{s=0; for(n=1,62,p=3^floor(0.1^25+log(n)/log(3)); print1(s=s+min(n-p,3*p-n),","))}

a(n) = sum{j=1..n, A081134(j)}.

A053646 Distance to nearest power of 2.

Original entry on oeis.org

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

A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

Original entry on oeis.org

1, 3, 11, 33, 101, 303, 911, 2733, 8201, 24603, 73811, 221433, 664301, 1992903, 5978711, 17936133, 53808401, 161425203, 484275611, 1452826833, 4358480501, 13075441503, 39226324511, 117678973533, 353036920601, 1059110761803

Offset: 0

Klaus Brockhaus, Mar 17 2003

The limit of the local minima, lim_{n->infinity} A081249(n)/n^2 = 1/10. For local maxima cf. A081251.

			11 is a term since A081249(10)/10^2 = 11/100 = 0.110, A081249(11)/11^2 = 13/121 = 0.107, A081249(12)/12^2 = 16/144 = 0.111.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A005052, A062107, A081134, A081249, A081251.

List([0..30], n-> (5*3^n +(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

[Floor(3^n*5/4): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+2 od: seq(a[n], n=0..30); # Zerinvary Lajos, Apr 28 2008

Floor[5*3^Range[0, 30]/4] (* Wesley Ivan Hurt, Mar 30 2017 *)

vector(30, n, n--; (5*3^n +(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

[(5*3^n +(-1)^n -2)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = floor(3^n*5/4).
G.f.: x*(1+x^2)/((1-x)*(1+x)*(1-3*x)).
a(n) = 3*a(n-1) + 1*a(n-2) - 3*a(n-3).
a(n) = (5*3^n + (-1)^n - 2)/4. - Paul Barry, May 19 2003
a(n) = a(n-2) + 10*3^(n-2) for n > 1.
a(n+2) - a(n) = A005052(n).
a(2*n) = Sum_{j=1..n+1} A062107(2*j).
a(2*n+1) = Sum_{j=1..n+1} A062107(2*j+1).
With a leading 0, this is a(n) = (5*3^n - 6 + 4*0^n - 3*(-1)^n)/12, the binomial transform of A084183. - Paul Barry, May 19 2003
Convolution of 3^n and {1, 0, 2, 0, 2, 0, ...}. a(n) = Sum_{k=0..n} ((1 + (-1)^k) - 0^k)*3^(n-k) = Sum_{k=0..n} ((1 + (-1)^(n-k)) - 0^(n-k))3^k. - Paul Barry, Jul 19 2004
a(n) = 2*a(n-1) + 3*a(n-2) + 2, a(0)=1, a(1)=3. - Zerinvary Lajos, Apr 28 2008

Offset changed from 1 to 0 by Vincenzo Librandi, Jun 10 2011

A081251 Numbers n such that A081249(m)/m^2 has a local maximum for m = n.

Original entry on oeis.org

2, 6, 20, 60, 182, 546, 1640, 4920, 14762, 44286, 132860, 398580, 1195742, 3587226, 10761680, 32285040, 96855122, 290565366, 871696100, 2615088300, 7845264902, 23535794706, 70607384120, 211822152360, 635466457082, 1906399371246

Offset: 1

Klaus Brockhaus, Mar 17 2003

nonn

The limit of the local maxima, lim A081249(n)/n^2 = 1/6. For local minima cf. A081250.
Also the number of different 4- and 3-colorings for the vertices of all triangulated planar polygons on a base with n+2 vertices, if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010
From Toby Gottfried, Apr 18 2010: (Start)
a(n) = the number of ternary sequences of length n+1 where the numbers of (0's, 1's) are both odd.
A015518 covers the (odd, even) and (even, odd) cases, and A122983 covers (even, even). (End)

			6 is a term since A081249(5)/5^2 = 4/25 = 0.160, A081249(6)/6^2 = 7/36 = 0.194, A081249(7)/7^2 = 9/49 = 0.184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A008776, A033113, A054880, A081134, A081249, A081250.

List([1..30], n-> (9*3^(n-1) -(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

[Floor(3^(n+1)/4) : n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(3^(n+1)/4), n=1..30). # Mircea Merca, Dec 27 2010

a[n_]:= Floor[3^(n+1)/4]; Array[a, 30]
Table[(9*3^(n-1) -(-1)^n -2)/4, {n, 1, 30}] (* G. C. Greubel, Jul 14 2019 *)

vector(30, n, (9*3^(n-1) -(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

[(9*3^(n-1) -(-1)^n -2)/4 for n in (1..30)] # G. C. Greubel, Jul 14 2019

G.f.: 2/((1-x)*(1+x)*(1-3*x)).
a(n) = a(n-2) + 2*3^(n) for n > 1.
a(n+2) - a(n) = A008776(n).
a(n) = 2*A033113(n+1).
a(2*n+1) = A054880(n+1).
a(n) = floor(3^(n+1)/4). - Mircea Merca, Dec 26 2010
From G. C. Greubel, Jul 14 2019: (Start)
a(n) = (9*3^(n-1) -(-1)^n -2)/4.
E.g.f.: (3*exp(3*x) - 2*exp(x) - exp(-x))/4. (End)

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

A006166 a(0)=0, a(1)=a(2)=1; for n >= 1, a(3n+2) = 2a(n+1) + a(n), a(3n+1) = a(n+1) + 2a(n), a(3n) = 3a(n).

Original entry on oeis.org

0, 1, 1, 3, 3, 3, 3, 5, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69

Offset: 0

N. J. A. Sloane

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
vN. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. [Preprint.]
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi 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, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47.

a(n) + n = A003605(n). Cf. A006165, A080678, A081134.

From Peter Bala, Oct 08 2022: (Start)
a(n+2) - a(n) = 0 or 2.
a(3^k + j) = 3^k for k >= 0 and for 0 <= j <= 3^k.
a(2*3^k + j) = 3^k + 2*j for k >= 0 and for 0 <= j <= 3^k.
A081134(n) = n - a(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

A342872 Distance to nearest product of 3 consecutive numbers (three-dimensional promic number, A007531).

Original entry on oeis.org

0, 1, 2, 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, 11, 12, 13, 14, 15, 16, 17, 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

Offset: 0

Lamine Ngom, Mar 28 2021

			a(13) = 7 since 6 is the closest three-dimensional promic to 13 and 13 - 6 = 7.

Lamine Ngom, Table of n, a(n) for n = 0..10000

Cf. A007531, A342873.

Other distance to nearest: A081134, A053646, A201053.

def a(n): return min(abs(n-k*(k+1)*(k+2)) for k in range(int(n**1/3)+1))
print([a(n) for n in range(78)]) # Michael S. Branicky, Mar 28 2021

A342873 Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).

Original entry on oeis.org

0, 7, 16, 62, 92, 213, 276, 508, 616, 995, 1160, 1722, 1956, 2737, 3052, 4088, 4496, 5823, 6336, 7990, 8620, 10637, 11396, 13812, 14712, 17563, 18616, 21938, 23156, 26985, 28380, 32752, 34336, 39287, 41072, 46638, 48636, 54853, 57076, 63980, 66440, 74067

Offset: 1

Lamine Ngom, Mar 28 2021

That is, numbers k such that A074989(k) = A342872(k).
They form 2 partitions:
7, 62, 213, ... = 8*k^3 - k = k*A157914(k).
0, 16, 92,  ... = 8*k^3 + 6*k^2 + 2*k = 2*k*A033951(k).

Cf. A074989, A342872.

Cf. A157914, A033951, A074378, A201053, A000578, A007531, A081134.

def aupto(limit):
  cubes = [k**3 for k in range(int((limit+1)**1/3)+2)]
  proms = [k*(k+1)*(k+2) for k in range(int((limit+1)**1/3)+1)]
  A074989 = [min(abs(n-c) for c in cubes) for n in range(limit+1)]
  A342872 = [min(abs(n-p) for p in proms) for n in range(limit+1)]
  return [m for m in range(limit+1) if A074989[m] == A342872[m]]
print(aupto(10**4)) # Michael S. Branicky, Mar 28 2021

A343766 Lexicographically earliest sequence of distinct integers such that a(0) = 0 and the balanced ternary expansions of two consecutive terms differ by a single digit, as far to the right as possible.

Original entry on oeis.org

0, -1, 1, -2, -4, -3, 3, 2, 4, -5, -7, -6, -12, -13, -11, -8, -10, -9, 9, 8, 10, 7, 5, 6, 12, 11, 13, -14, -16, -15, -21, -22, -20, -17, -19, -18, -36, -37, -35, -38, -40, -39, -33, -34, -32, -23, -25, -24, -30, -31, -29, -26, -28, -27, 27, 26, 28, 25, 23, 24

Offset: 0

Rémy Sigrist, Apr 28 2021

This sequence has similarities with A003188 and with A341055.
A007949 gives the positions of the digit that is altered from one term to the other.
To compute a(n):
- consider the ternary representation of A128173(n),
- replace 1's by -1's and 2's by 1's,
- convert back to decimal.

			The first terms, alongside their balanced ternary expansion (with T's denoting -1's), are:
  n   a(n)  bter(a(n))
  --  ----  ----------
   0     0           0
   1    -1           T
   2     1           1
   3    -2          T1
   4    -4          TT
   5    -3          T0
   6     3          10
   7     2          1T
   8     4          11
   9    -5         T11
  10    -7         T1T
  11    -6         T10
  12   -12         TT0
  13   -13         TTT
  14   -11         TT1

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Scatterplot of the first 3^10 terms
Rémy Sigrist, PARi program for A343766
Index entries for sequences that are permutations of the integers

Cf. A003188, A007949, A024023, A081134, A128173, A341055.

PARI
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See Links section.
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a(n) = -A117966(A128173(n)).
Sum_{k=0..n-1} sign(a(k)) = -A081134(n).
Sum_{k=0..n} a(k) = 0 iff n belongs to A024023.

cp's OEIS Frontend

A081249 Partial sums of A081134.

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A053646 Distance to nearest power of 2.

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A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

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A081251 Numbers n such that A081249(m)/m^2 has a local maximum for m = n.

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

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A006166 a(0)=0, a(1)=a(2)=1; for n >= 1, a(3n+2) = 2a(n+1) + a(n), a(3n+1) = a(n+1) + 2a(n), a(3n) = 3a(n).

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