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

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

Author

Nicolay Avilov, Apr 20 2023

Keywords

• nonn
• easy

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.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Nicolay Avilov, Triangle Coloring Scheme
• Index entries for linear recurrences with constant coefficients, signature (4,-7,7,-4,1).

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