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A330451 a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.

%I A330451 #54 Aug 13 2025 10:17:37
%S A330451 0,3,13,30,53,83,120,163,213,270,333,403,480,563,653,750,853,963,1080,
%T A330451 1203,1333,1470,1613,1763,1920,2083,2253,2430,2613,2803,3000,3203,
%U A330451 3413,3630,3853,4083,4320,4563,4813,5070
%N A330451 a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.
%C A330451 Main N-S vertical in the pentagonal spiral for A002264:
%C A330451                       16
%C A330451                   16  10  10
%C A330451               16   9   5   5  10
%C A330451           15   9   4   1   2   5  11
%C A330451       15   9   4   1   0   0   2   6  11
%C A330451         15   8   4   1   0   2   6  11
%C A330451           14   8   3   3   3   6  12
%C A330451             14   8   7   7   7  12
%C A330451               14  13  13  13  12
%C A330451 The main S-N vertical is A194275.
%H A330451 Colin Barker, <a href="/A330451/b330451.txt">Table of n, a(n) for n = 0..1000</a>
%H A330451 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A330451 G.f.: x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)). - _Colin Barker_, Mar 02 2020
%F A330451 a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
%F A330451 a(n) = (2/9)*(-1 + 15*n^2 + cos(2*n*Pi/3)). - _Stefano Spezia_, Mar 02 2020
%F A330451 a(3*n) = 30*n^2.
%F A330451 a(n) = floor(10*n^2/3). - _Chai Wah Wu_, Aug 12 2025
%t A330451 Table[2/9(-1+15n^2+Cos[2n*Pi/3]),{n,0,39}] (* _Stefano Spezia_, Mar 02 2020 *)
%o A330451 (PARI) concat(0, Vec(x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^40))) \\ _Colin Barker_, Mar 02 2020
%o A330451 (Python)
%o A330451 def A330451(n): return 10*n**2//3 # _Chai Wah Wu_, Aug 12 2025
%Y A330451 Cf. A000290, A002264, abs(A084103(n+3)), A194275, A244636.
%Y A330451 Cf. A049347.
%K A330451 nonn,easy
%O A330451 0,2
%A A330451 _Paul Curtz_, Mar 01 2020