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

%I A359626 #21 Aug 05 2023 21:21:50
%S A359626 1,4,9,15,21,27,34,43,54,66,78,90,103,118,135,153,171,189,208,229,252,
%T A359626 276,300,324,349,376,405,435,465,495,526,559,594,630,666,702,739,778,
%U A359626 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
%N A359626 a(n) is equal to the number of filled unit triangles in a regular triangle whose coloring scheme is given in the comments.
%C A359626 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.
%H A359626 Paolo Xausa, <a href="/A359626/b359626.txt">Table of n, a(n) for n = 1..10000</a>
%H A359626 Nicolay Avilov, <a href="/A359626/a359626.jpg">Triangle Coloring Scheme</a>
%H A359626 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,7,-4,1).
%F A359626 Let r = n (mod 6), then we get
%F A359626   a(n) = n*(n+3)/2 - 1 if r = 1 or r = 2;
%F A359626          n*(n+3)/2     if r = 0 or r = 3;
%F A359626          n*(n+3)/2 + 1 if r = 4 or r = 5.
%F A359626 From _Stefano Spezia_, Apr 20 2023: (Start)
%F A359626 O.g.f.: x/((1 - x)^3*(1 - x + x^2)).
%F A359626 E.g.f.: exp(x)*x*(4 + x)/2 - 2*exp(x/2)*sin(sqrt(3)*x/2)/sqrt(3). (End)
%F A359626 a(n) - a(n-1) = A077859(n-1). - _R. J. Mathar_, Apr 20 2023
%e A359626 a(7) = 7^2 - 4^2 + 1^2 = 34;
%e A359626 a(8) = 8^2 - 5^2 + 2^2 = 43;
%e A359626 a(9) = 9^2 - 6^2 + 3^2 = 54.
%t A359626 A359626list[nmax_]:=LinearRecurrence[{4,-7,7,-4,1},{1, 4, 9, 15, 21},nmax];A359626list[100] (* _Paolo Xausa_, Aug 05 2023 *)
%Y A359626 Cf. A000096, A077859 (first differences).
%K A359626 nonn,easy
%O A359626 1,2
%A A359626 _Nicolay Avilov_, Apr 20 2023