cp's OEIS Frontend

A319390 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=2, a(2)=3, a(3)=6, a(4)=8.

%I A319390 #30 Nov 13 2023 08:44:24
%S A319390 1,2,3,6,8,13,16,23,27,36,41,52,58,71,78,93,101,118,127,146,156,177,
%T A319390 188,211,223,248,261,288,302,331,346,377,393,426,443,478,496,533,552,
%U A319390 591,611,652,673,716,738,783,806,853,877,926,951,1002
%N A319390 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=2, a(2)=3, a(3)=6, a(4)=8.
%C A319390 The bisections A104249(n) = 1, 3, 8, ... and A143689(n+1) = 2, 6, 13, 23, ... are in the following hexagonal spiral:
%C A319390                       29--28--28--27--27
%C A319390                       /                 \
%C A319390                     29  17--17--16--16  26
%C A319390                     /   /             \   \
%C A319390                   30  18   9---8---8  15  26
%C A319390                   /   /   /         \   \   \
%C A319390                 30  18   9   3---3   7  15  25
%C A319390                 /   /   /   /     \   \   \   \
%C A319390               31  19  10   4   1   2   7  14  25
%C A319390                   /   /   /   /   /   /   /   /
%C A319390                 19  10   4   1---2   6  14  24
%C A319390                   \   \   \         /   /   /
%C A319390                   20  11   5---5---6  13  24
%C A319390                     \   \             /   /
%C A319390                     20  11--12--12--13  23
%C A319390                       \                 /
%C A319390                       21--21--22--22--23
%C A319390 .
%C A319390 a(n) mod 9 = A140265(n) mod 9.
%H A319390 Colin Barker, <a href="/A319390/b319390.txt">Table of n, a(n) for n = 0..1000</a>
%H A319390 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A319390 a(2n) = (3*n^2 + n + 2)/2. a(2n+1) = (3*n^2 + 5*n + 4)/2.
%F A319390 a(-n) = a(n).
%F A319390 a(n) = a(n-1) + A026741(n).
%F A319390 G.f.: (1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2). - _Colin Barker_, Jun 05 2019
%F A319390 a(n) = 1 + A001318(n). - _Peter Bala_, Feb 04 2021
%F A319390 E.g.f.: ((8 + 7*x + 3*x^2)*cosh(x) + (9 + 5*x + 3*x^2)*sinh(x))/8. - _Stefano Spezia_, Feb 05 2021
%t A319390 LinearRecurrence[{1,2,-2,-1,1},{1,2,3,6,8},100] (* _Paolo Xausa_, Nov 13 2023 *)
%o A319390 (PARI) Vec((1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Jun 05 2019
%Y A319390 Cf. A001318, A104249, A143689, A004526, A140265, A026741.
%K A319390 nonn,easy
%O A319390 0,2
%A A319390 _Paul Curtz_, Sep 18 2018