A117484 Number of triangular numbers mod n.

1, 2, 2, 4, 3, 4, 4, 8, 4, 6, 6, 8, 7, 8, 6, 16, 9, 8, 10, 12, 8, 12, 12, 16, 11, 14, 11, 16, 15, 12, 16, 32, 12, 18, 12, 16, 19, 20, 14, 24, 21, 16, 22, 24, 12, 24, 24, 32, 22, 22, 18, 28, 27, 22, 18, 32, 20, 30, 30, 24, 31, 32, 16, 64, 21, 24, 34, 36, 24, 24, 36, 32, 37, 38, 22

Offset

1,2

Comments

Same as A000224 (number of squares mod n) for n odd, since there we can divide by 2 and then complete the square.

From David Morales Marciel, Jul 13 2015: (Start)

a(n) is also the total number of vertices of an n-gon that are a "final vertex" of a bouncing pattern representing the modulo-n series (an image of the bouncing pattern is included in the LINKS section). It is defined by the following algorithm:

(1) Define counter c=1.

(2) Start at any desired vertex.

(3) Mark the current vertex as a "final vertex".

(4) Advance clockwise c vertices.

(5) Set c=c+1.

(6) Repeat from (3).

The pattern of "final vertices" is cyclic: after some repetitions of steps (3)-(6), the marking of vertices is repeated and it is possible to count how many vertices of the n-gon contain a "final vertex" mark.

Examples: trivial case: a(n)=1 (one vertex is always a "final vertex"). From that point following the algorithm: a(2)=2 (segment, both vertices are a "final vertex"), a(3)=2 (triangle, only two vertices are "final vertex"), etc.

(End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

David Morales Marciel, Bouncing patterns of some modulo-n series (n-gons 3 to 14, 16, 17 and 27), according to the algorithm explained in the comments. Current sequence belongs to the bouncing pattern of the 3-gon sample (first at the top).

Formula

Multiplicative with a(2^e) = 2^e, a(p^e) = floor(p^(e+1)/(2p+2))+1 for p > 2.

Example

When n=3, there is no triangular number which is congruent to 2 (mod 3) but only == 0 or 1 (mod 3), so a(3) = 2. - Robert G. Wilson v, Sep 16 2015

Maple

a:= proc(n) local F, f;
F:= ifactors(n)[2];
mul(seq(`if`(f[1]=2, 2^f[2], floor(f[1]^(f[2]+1)/(2*f[1]+2))+1), f=F))
end proc:
map(a, [$1..100]); # Robert Israel, Jul 13 2015

Mathematica

f[n_] := Block[{fi = FactorInteger@ n, k = t = 1}, lng = 1 + Length@ fi; While[k < lng, t = t*If[ fi[[k, 1]] == 2, 2^fi[[k, 2]], Floor[1 + fi[[k, 1]]^(fi[[k, 2]] + 1)/(2 + 2fi[[k, 1]]) ]]; k++]; t]; Array[f, 75] (* Robert G. Wilson v, Sep 16 2015, after Robert Israel *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 2^f[i, 2], (f[i, 1]^(f[i, 2]+1)\(2*f[i, 1] + 2)) + 1)); } \\ Amiram Eldar, Sep 05 2023

Crossrefs

Cf. A000224.

Sequence in context: A087874 A363341 A166267 * A086835 A046701 A140472

Adjacent sequences: A117481 A117482 A117483 * A117485 A117486 A117487

Keyword

mult,easy,nonn

Author

Franklin T. Adams-Watters, Apr 25 2006

Status

approved