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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. 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 2 09:50 EST 2023. Contains 367517 sequences. (Running on oeis4.)