cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A359626 a(n) is equal to the number of filled unit triangles in a regular triangle whose coloring scheme is given in the comments.

Original entry on oeis.org

1, 4, 9, 15, 21, 27, 34, 43, 54, 66, 78, 90, 103, 118, 135, 153, 171, 189, 208, 229, 252, 276, 300, 324, 349, 376, 405, 435, 465, 495, 526, 559, 594, 630, 666, 702, 739, 778, 819, 861, 903, 945, 988, 1033, 1080, 1128, 1176, 1224, 1273, 1324, 1377, 1431, 1485, 1539, 1594, 1651, 1710, 1770, 1830, 1890, 1951, 2014, 2079
Offset: 1

Views

Author

Nicolay Avilov, Apr 20 2023

Keywords

Comments

A regular triangle with side n is divided by segments parallel to the sides of the triangle into n^2 unit triangles. In it, you can select triangular frames nested inside each other. Coloring them through one, starting from the outer one, we obtain a coloring of unit triangles corresponding to the given sequence. See link.

Examples

			a(7) = 7^2 - 4^2 + 1^2 = 34;
a(8) = 8^2 - 5^2 + 2^2 = 43;
a(9) = 9^2 - 6^2 + 3^2 = 54.
		

Crossrefs

Cf. A000096, A077859 (first differences).

Programs

  • Mathematica
    A359626list[nmax_]:=LinearRecurrence[{4,-7,7,-4,1},{1, 4, 9, 15, 21},nmax];A359626list[100] (* Paolo Xausa, Aug 05 2023 *)

Formula

Let r = n (mod 6), then we get
a(n) = n*(n+3)/2 - 1 if r = 1 or r = 2;
n*(n+3)/2 if r = 0 or r = 3;
n*(n+3)/2 + 1 if r = 4 or r = 5.
From Stefano Spezia, Apr 20 2023: (Start)
O.g.f.: x/((1 - x)^3*(1 - x + x^2)).
E.g.f.: exp(x)*x*(4 + x)/2 - 2*exp(x/2)*sin(sqrt(3)*x/2)/sqrt(3). (End)
a(n) - a(n-1) = A077859(n-1). - R. J. Mathar, Apr 20 2023