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a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.
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%I #50 May 01 2020 12:11:05

%S 0,3,13,30,53,83,120,163,213,270,333,403,480,563,653,750,853,963,1080,

%T 1203,1333,1470,1613,1763,1920,2083,2253,2430,2613,2803,3000,3203,

%U 3413,3630,3853,4083,4320,4563,4813,5070

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

%C Main N-S vertical in the pentagonal spiral for A002264:

%C 16

%C 16 10 10

%C 16 9 5 5 10

%C 15 9 4 1 2 5 11

%C 15 9 4 1 0 0 2 6 11

%C 15 8 4 1 0 2 6 11

%C 14 8 3 3 3 6 12

%C 14 8 7 7 7 12

%C 14 13 13 13 12

%C The main S-N vertical is A194275.

%H Colin Barker, <a href="/A330451/b330451.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F G.f.: x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)). - _Colin Barker_, Mar 02 2020

%F a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).

%F a(n) = (2/9)*(-1 + 15*n^2 + cos(2*n*Pi/3)). - _Stefano Spezia_, Mar 02 2020

%F a(3*n) = 30*n^2.

%t Table[2/9(-1+15n^2+Cos[2n*Pi/3]),{n,0,39}] (* _Stefano Spezia_, Mar 02 2020 *)

%o (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

%Y Cf. A000290, A002264, abs(A084103(n+3)), A194275, A244636.

%Y Cf. A049347.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Mar 01 2020