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

1,2

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.

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).

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

Example

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.

Mathematica

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

Crossrefs

Cf. A000096, A077859 (first differences).

Sequence in context: A313296 A313297 A270545 * A358243 A099055 A162801

Adjacent sequences: A359623 A359624 A359625 * A359627 A359628 A359629

Keyword

nonn,easy

Author

Nicolay Avilov, Apr 20 2023

Status

approved