Joan Llobera Querol - cp's OEIS frontend

User: Joan Llobera Querol

Joan Llobera Querol has authored 2 sequences.

A365905 "2-peloton numbers": Numbers that appear at least twice in A365904.

15, 36, 43, 49, 64, 66, 78, 85, 99, 100, 118, 120, 134, 141, 151, 159, 168, 169, 190, 204, 210, 211, 219, 225, 241, 246, 253, 256, 270, 274, 279, 283, 288, 295, 309, 321, 323, 325, 345, 351, 355, 358, 364, 372, 376, 379, 386, 393, 394, 400, 405, 406, 423, 429, 435, 438, 440, 456, 463, 474, 484, 498

Offset: 1

Joan Llobera Querol, Sep 22 2023

nonn

Called "peloton" numbers after the original sequence idea in first link: the difference of a rhombus (a square number) and a triangular number, placed as points on a triangular grid, form the shape of a peloton in bicycle racing.

Contains all elements of A001110 other than 0 and 1.

			15 can be obtained as T(4,1) or T(5,4) following notation in A365904.
36 can be obtained as T(6,0) or T(8,7).

Eric Snyder, Table of n, a(n) for n = 1..10000
Zach Wissner-Gross, Can You Shape the Peloton?, Fiddler on the Proof, Sep 22, 2023.

Cf. A175035 (numbers appear at least once), A365904.

isok(n) = sum(m=sqrtint(n), (sqrtint(8*n+1)-1)\2, ispolygonal(m^2-n,3)) > 1 \\ Andrew Howroyd, Sep 24 2023
(Python/SageMath)
nmax, m, Out = 300, 2, []
Z = [ n^2 - (k^2 + k)/2 for n in [2..nmax] for k in [0..n-1] ]
for i in Z:
    if Z.count(i) >= m: Out.append(i)
Out=sorted(list(set(Out)))
for j in [1..10000]: print(j+1, Out[j])
\\ Eric Snyder, Sep 29 2023

A365904 Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k

Original entry on oeis.org

1, 4, 3, 9, 8, 6, 16, 15, 13, 10, 25, 24, 22, 19, 15, 36, 35, 33, 30, 26, 21, 49, 48, 46, 43, 39, 34, 28, 64, 63, 61, 58, 54, 49, 43, 36, 81, 80, 78, 75, 71, 66, 60, 53, 45, 100, 99, 97, 94, 90, 85, 79, 72, 64, 55, 121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66

Offset: 1

Joan Llobera Querol, Sep 22 2023

T(n,k) is the number of points in a rhombus that has 1 point in the first row, then 2 in the second, and following until the n-th row with n points, and then n-1 in the following row, n-2 in the following to end with a row with k+1 points.
T(n,0) are the perfect squares (A000290).
T(n,n-1) are the triangular numbers (A000217).

			Triangle begins:
    1;
    4,  3;
    9,  8,  6;
   16, 15, 13, 10;
   25, 24, 22, 19, 15;
   36, 35, 33, 30, 26, 21;
   49, 48, 46, 43, 39, 34, 28;
   64, 63, 61, 58, 54, 49, 43, 36;
   81, 80, 78, 75, 71, 66, 60, 53, 45;
  100, 99, 97, 94, 90, 85, 79, 72, 64, 55;
  ...

Joan Llobera Querol, Table of n, a(n) for n = 1..10000
Zach Wissner-Gross, Can You Shape the Peloton?, Fiddler on the Proof, Sep 22, 2023.

Row sums give A004068.

Cf. A214859.

G.f.: x*(1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Oct 05 2023

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User: Joan Llobera Querol

A365905 "2-peloton numbers": Numbers that appear at least twice in A365904.

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