A117444 -id:A117444 - cp's OEIS frontend

A053616 Pyramidal sequence: distance to nearest triangular number.

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)

A271751 Period 10 zigzag sequence; repeat: [0, 1, 2, 3, 4, 5, 4, 3, 2, 1].

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5

Offset: 0

Wesley Ivan Hurt, Apr 13 2016

Decimal expansion of 11111/900009. - Elmo R. Oliveira, Mar 03 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,-1,1).

Cf. A076839, A117444.

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), this sequence (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

&cat[[0, 1, 2, 3, 4, 5, 4, 3, 2, 1]: n in [0..10]];

a:=n->[0, 1, 2, 3, 4, 5, 4, 3, 2, 1][(n mod 10)+1]: seq(a(n), n=0..100);

CoefficientList[Series[x*(1 + x + x^2 + x^3 + x^4)/(1 - x + x^5 - x^6), {x, 0, 30}], x]

a(n) = abs(n-10*round(n/10)); \\ Altug Alkan, Apr 13 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4)/(1 - x + x^5 - x^6).
a(n) = a(n-1) - a(n-5) + a(n-6) for n>5.
a(n) = abs(n - 10*round(n/10)).
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/5).
a(2n) = 2*abs(A117444(n)).
a(2n+7) = 2*A076839(n)-1 for n>0.
a(n) = a(n-10) for n >= 10. - Wesley Ivan Hurt, Sep 07 2022

A090223 Nonnegative integers with doubled multiples of 4.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 56, 56, 57, 58

Offset: 0

Wolfdieter Lang, Dec 01 2003

Degrees of row-polynomials of array A090222.

a(n) is the number of full orbits completed by body A for n full orbits completed by body B in a celestial system with two orbiting bodies A and B with orbital resonance A:B equal to 4:5. This resonance is exhibited by the planets Kepler-90b and Kepler-90c in the planetary system of the star Kepler-90. - Felix Fröhlich, May 03 2021

Felix Fröhlich, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A057353 and other floors of ratios references there.

Cf. A090222.

[4*n div 5: n in [0..80]]; // Bruno Berselli, Dec 07 2016

A090223:=n->floor(4*n/5); seq(A090223(n), n=0..100); # Wesley Ivan Hurt, Nov 15 2013

Table[Floor[4n/5], {n,0,100}] (* Wesley Ivan Hurt, Nov 15 2013 *)

a(n)=4*n\5 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = floor(4*n/5).
G.f.:  x^2 *(1+x^2)*(1+x)/((1-x^5)*(1-x)) = (x^2)*(1-x^4)/((1-x^5)*(1-x)^2).
a(n) = n - 1 - A002266(n - 1). - Wesley Ivan Hurt, Nov 15 2013
a(n) = A057354(2*n). - R. J. Mathar, Jul 21 2020
5*a(n) = 4*n-2+A117444(n+2) . - R. J. Mathar, Jul 21 2020
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/8. - Amiram Eldar, Sep 30 2022

A257145 a(n) = 5 * floor( (n+2) / 5) - n with a(0) = 1.

Original entry on oeis.org

1, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0

Offset: 0

Michael Somos, Apr 16 2015

Cycle period is 5, {0, -1, -2, 2, 1} after the first five terms. - Robert G. Wilson v, Aug 02 2018

			G.f. = 1 - x - 2*x^2 + 2*x^3 + x^4 - x^6 - 2*x^7 + 2*x^8 + x^9 - x^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Cf. A257143, A253262, A117444.

a257145 0 = 1
a257145 n = div (n + 2) 5 * 5 - n  -- Reinhard Zumkeller, Apr 17 2015

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1-x^2)^2/(1-x^5))); // G. C. Greubel, Aug 02 2018

a[ n_] := If[ n==0, 1, -Mod[ n, 5, -2]];
a[ n_] := If[ n==0, 1, Sign[n] SeriesCoefficient[ (1 - x) * (1 - x^2)^2 / (1 - x^5), {x, 0, Abs@n}]];
CoefficientList[Series[(1-x)*(1-x^2)^2/(1-x^5), {x,0,60}], x] (* G. C. Greubel, Aug 02 2018 *)
a[n_] := 5 Floor[(n + 2)/5] - n; Array[a, 77, 0] (* or *)
CoefficientList[ Series[(x - 1)^2 (x + 1)^2/(x^4 + x^3 + x^2 + x + 1), {x, 0, 76}], x] (* or *)
LinearRecurrence[{-1, -1, -1, -1}, {1, -1, -2, 2, 1, 0}, 76] (* Robert G. Wilson v, Aug 02 2018*)

{a(n) = if( n==0, 1, (n+2) \ 5 * 5 - n)};

{a(n) = if( n==0, 1, [0, -1, -2, 2, 1][n%5 + 1])};

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 - x) * (1 - x^2)^2 / (1 - x^5) + x * O(x^abs(n)), abs(n)))};

x='x+O('x^60); Vec((1-x)*(1-x^2)^2/(1-x^5)) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 5 sequence [-1, -2, 0, 0, 1].
a(5*n) = 0 for all n in Z except n=0.
a(n) = -a(-n) for all n in Z except n=0.
a(n) = a(n+5) for all n in Z except n=-5 or n=0.
Convolution inverse is A257143.
G.f.: (1 - x) * (1 - x^2)^2 / (1 - x^5).
G.f.: (1 - 2*x^2 + x^4) / (1 + x + x^2 + x^3 + x^4).
a(n) = -A117444(n), n>0. - R. J. Mathar, Oct 05 2017

A165190 G.f.: 1/((1-x^4)*(1-x^5)).

Original entry on oeis.org

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

Offset: 0

Alford Arnold, Sep 24 2009

A121262 convolved with A079998. The two sequences have very simple generating functions and can be mapped to the numeric partitions 4=4 and 5=5 respectively.

Number of partitions of n into parts 4 and 5. - Joerg Arndt, Aug 28 2015

Michael De Vlieger, Table of n, a(n) for n = 0..10000
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
Index to Molien series
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,0,0,0,-1).

Cf. A057077, A079998, A117444, A121262.

[Floor((n+4)/4) - Floor((n+4)/5) : n in [0..100]]; // Wesley Ivan Hurt, Aug 27 2015

A165190:=n->floor((n+4)/4) - floor((n+4)/5): seq(A165190(n), n=0..100); # Wesley Ivan Hurt, Aug 27 2015

CoefficientList[Series[1/((1-x^4)(1-x^5)),{x,0,110}],x] (* or *) LinearRecurrence[{0,0,0,1,1,0,0,0,-1},{1,0,0,0,1,1,0,0,1},110] (* Harvey P. Dale, Aug 16 2012 *)
Table[Floor[(n + 4)/4] - Floor[(n + 4)/5], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 27 2015 *)

1 followed by the Euler transform of the finite sequence [0,0,0,1,1].
G.f.: 1/((1-x)^2*(1+x)*(1+x^2)*(1+x+x^2+x^3+x^4)). [R. J. Mathar, Oct 07 2009]
a(n) = A117444(n+2)/5 + n/20 + 9/40 + (-1)^n/8 + A057077(n)/4. [R. J. Mathar, Oct 07 2009]
a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=1, a(6)=0, a(7)=0, a(8)=1, a(n) = a(n-4)+a(n-5)-a(n-9), n>8. - Harvey P. Dale, Aug 16 2012
a(n) = floor((n+4)/4) - floor((n+4)/5). - Wesley Ivan Hurt, Aug 27 2015
a(n)+a(n-2) = A008616(n). - R. J. Mathar, Jun 23 2021

Removed duplicate of comment in A165188; Euler transform formula corrected - R. J. Mathar, Oct 07 2009

A188125 Number of strictly increasing arrangements of 6 nonzero numbers in -(n+4)..(n+4) with sum zero.

Original entry on oeis.org

4, 16, 52, 137, 308, 624, 1154, 1999, 3278, 5144, 7772, 11387, 16230, 22602, 30830, 41303, 54440, 70734, 90706, 114963, 144146, 178984, 220244, 268797, 325548, 391514, 467756, 555449, 655816, 770208, 900020, 1046787, 1212094, 1397668

Offset: 0

R. H. Hardin, Mar 21 2011

nonn

Row 6 of A188122.

			4 + 16*x + 52*x^2 + 137*x^3 + 308*x^4 + 624*x^5 + 1154*x^6 + 1999*x^7 + 3278*x^8 + ...
Some solutions for n=6
-10...-8...-7...-8...-8...-9...-9...-9...-9...-7..-10...-9...-7..-10...-9...-9
.-8...-6...-5...-5...-6...-3...-7...-3...-2...-5...-6...-5...-5...-6...-4...-5
.-1....1...-1...-1...-1...-2...-2....1...-1...-2...-2...-1...-1...-2...-2...-4
..4....3....1....1....2....3....3....2....1....1....2....1....3....3....1....3
..7....4....2....3....5....4....5....4....2....4....6....6....4....5....5....6
..8....6...10...10....8....7...10....5....9....9...10....8....6...10....9....9

R. H. Hardin, Table of n, a(n) for n = 0..200 (a(0) inserted by _Georg Fischer_, Jan 24 2019)

Cf. A188122

{a(n) = local(v, c, m); m = n+4; forvec( v = vector( 6, i, [-m, m]), if( 0==prod( k=1, 6, v[k]), next); if( 0==sum( k=1, 6, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+2*a(n-8)-a(n-11)-a(n-13)+2*a(n-15)-a(n-16)
= 168587/43200 +187*n/32 +3593*n^3/2160 +619*n^2/144 +457*n^4/1440 +11*n^5/450 -(-1)^n/64-3*n*(-1)^n/32 +4*(-1)^n*A119910(n+1)/27 -2*A117444(n+2)/25 +A057077(n)/8.
Empirical: G.f. -x*(-16 -20*x -33*x^2 -50*x^3 -60*x^4 -59*x^5 -51*x^6 -41*x^7 -18*x^8 -3*x^9 -x^10 +x^11 +4*x^12 -2*x^13 -7*x^14 +4*x^15) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x)^2 *(x-1)^6 ). - R. J. Mathar, Mar 21 2011

A117445 Periodic {0,-1,1,4,-1,4,-4,-4,1,1,-4,-4,4,-1,4,1,-1} (period 17).

Original entry on oeis.org

0, -1, 1, 4, -1, 4, -4, -4, 1, 1, -4, -4, 4, -1, 4, 1, -1, 0, -1, 1, 4, -1, 4, -4, -4, 1, 1, -4, -4, 4, -1, 4, 1, -1, 0, -1, 1, 4, -1, 4, -4, -4, 1, 1, -4, -4, 4, -1, 4, 1, -1, 0, -1, 1, 4, -1, 4, -4, -4, 1, 1, -4, -4, 4, -1, 4, 1, -1, 0, -1, 1, 4, -1, 4, -4, -4, 1, 1, -4, -4

Offset: 0

Paul Barry, Mar 16 2006

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1).

Cf. A117444.

PadRight[{},60,{0,-1,1,4,-1,4,-4,-4,1,1,-4,-4,4,-1,4,1,-1}] (* Harvey P. Dale, Sep 11 2012 *)
LinearRecurrence[{-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1},{0,-1,1,4,-1,4,-4,-4,1,1,-4,-4,4,-1,4,1},80]
(* Ray Chandler, Jul 15 2015 *)

G.f.: (-1)*x*(1+x)*(1-x)^2*(1 -3*x^2 -3*x^3 -10*x^4 -6*x^5 -9*x^6 -6*x^7 -10*x^8 -3*x^9 -3*x^10 +x^12)/(1-x^17).
a(n) = (1/2)*Sum_{k=0..17} L(k*(k^2-n)/17), where L(j/p) is the Legendre symbol of j and p.
G.f.: (-x)*(1-x)*(1+x)*(1 -3*x^2 -3*x^3 -10*x^4 -6*x^5 -9*x^6 -6*x^7 -10*x^8 -3*x^9 -3*x^10 +x^12)/(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10 +x^11 +x^12 +x^13 +x^14 +x^15 +x^16). - R. J. Mathar, Feb 23 2015

A134301 Periodic sequence (0, 2, 6, 2, 0).

Original entry on oeis.org

0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, 2, 6, 2, 0

Offset: 0

Franz Vrabec, Jan 30 2008

Also: twice the partial sums of A117444. - R. J. Mathar, Feb 01 2008

Rozsa Peter, Leon Harkleroad, Mathematics is Beautiful, Math. Intellig., 12 (No. 1, 1990), 58-64.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

PadRight[{},70,{0,2,6,2,0}] (* Harvey P. Dale, Mar 22 2012 *)

a(n)=[0, 2, 6, 2, 0][n%5+1] \\ Charles R Greathouse IV, Jul 13 2016

a(n) = n(n+1) mod 10

O.g.f.: -2/(x-1)+(2*x^3+2*x^2-2*x-2)/(1+x+x^2+x^3+x^4). a(n)=a(n-5). - R. J. Mathar, Feb 01 2008

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A053616 Pyramidal sequence: distance to nearest triangular number.

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A271751 Period 10 zigzag sequence; repeat: [0, 1, 2, 3, 4, 5, 4, 3, 2, 1].

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A090223 Nonnegative integers with doubled multiples of 4.

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A257145 a(n) = 5 * floor( (n+2) / 5) - n with a(0) = 1.

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A165190 G.f.: 1/((1-x^4)*(1-x^5)).

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A188125 Number of strictly increasing arrangements of 6 nonzero numbers in -(n+4)..(n+4) with sum zero.

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