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
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).
Programs
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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
Comments