A330082 - OEIS

A330082

a(n) = 5*A064038(n+1).

%I #34 Dec 05 2023 03:25:35

%S 0,5,15,15,25,75,105,70,90,225,275,165,195,455,525,300,340,765,855,

%T 475,525,1155,1265,690,750,1625,1755,945,1015,2175,2325,1240,1320,

%U 2805,2975,1575,1665,3515,3705,1950,2050,4305,4515,2365,2475,5175,5405,2820,2940

%N a(n) = 5*A064038(n+1).

%C Main column of a pentagonal spiral for A026741:

%C (25)

%C 49 (15) 31

%C 24 29 (15) 8 16

%C 47 14 7 ( 5) 3 17 33

%C 23 27 13 2 ( 0) 1 7 9 17

%C 45 13 6 3 1 4 19 35

%C 22 25 11 5 9 10 18

%C 43 12 23 11 21 37

%C 21 41 20 39 19

%C a(n) = 5 * A064038(n+1) from a pentagonal spiral.

%C Compare to A319127 = 6 * A002620 in the hexagonal spiral:

%C 22 23 23 22 (24)

%C 20 12 13 13 (12) 25

%C 21 10 5 4 ( 6) 14 25

%C 21 11 5 1 ( 0) 7 15 24

%C 20 11 4 1 ( 0) 2 7 15 26

%C 18 10 2 3 3 6 14 27

%C 19 8 9 9 8 16 27

%C 19 18 16 17 17 26

%C 30 28 29 29 28

%H Colin Barker, <a href="/A330082/b330082.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).

%F a(n) = A026741(A028895(n)).

%F From _Colin Barker_, Dec 08 2019: (Start)

%F G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).

%F a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.

%F a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).

%F (End)

%t A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* _Paolo Xausa_, Dec 04 2023 *)

%o (PARI) concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ _Colin Barker_, Dec 08 2019

%Y Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Dec 01 2019

%E More terms from _Colin Barker_, Dec 22 2019

%E Name corrected by _Paolo Xausa_, Dec 04 2023