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%I #35 Sep 08 2022 08:46:23
%S 1,5,9,21,29,49,61,89,105,141,161,205,229,281,309,369,401,469,505,581,
%T 621,705,749,841,889,989,1041,1149,1205,1321,1381,1505,1569,1701,1769,
%U 1909,1981,2129,2205,2361
%N a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.
%C The two bisections A136392(n+1)=1,9,29,61, ... and A201279(n)=5,21,49, ... are in the hexagonal spiral based on 2*n+1:
%C .
%C 67--65--63--61
%C / \
%C 69 33--31--29 59
%C / / \ \
%C 71 35 11---9 27 57
%C / / / \ \ \
%C 73 37 13 1 7 25 55
%C / / / / / /
%C 39 15 3---5 23 53
%C \ \ / /
%C 41 17--19--21 51
%C \ /
%C 43--45--47--49
%C .
%C A201279(n) - A136892(n) = 20*n.
%H Colin Barker, <a href="/A319384/b319384.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 (1,2,-2,-1,1).
%F a(2*n) = A136392(n+1), a(2*n+1) = A201279(n).
%F a(-n) = a(n).
%F a(2*n) + a(2*n+1) = 6*A001844(n).
%F a(n) = (6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4. - _Wesley Ivan Hurt_, Oct 04 2018
%F G.f.: (1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2). - _Colin Barker_, Jun 05 2019
%F a(n) = A104585(n) + A032766(n+1). - _Alex W. Nowak_, Jan 08 2021
%t Table[(6 n^2 + 6 n + 5 - (2 n + 1)*(-1)^n)/4, {n, 0, 80}] (* _Wesley Ivan Hurt_, Jan 07 2021 *)
%o (PARI) Vec((1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Jun 05 2019
%o (Magma) [(6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4 : n in [0..50]]; // _Wesley Ivan Hurt_, Jan 19 2021
%Y Cf. A001844.
%Y In the spiral: A003154(n+1), A080859, A126587, A136392, A201279, A227776.
%Y Cf. A032766, A104585.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Sep 18 2018
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