

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



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
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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



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.
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)


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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



