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%I #30 Sep 08 2022 08:46:24
%S 0,1,3,7,13,20,28,38,50,63,77,93,111,130,150,172,196,221,247,275,305,
%T 336,368,402,438,475,513,553,595,638,682,728,776,825,875,927,981,1036,
%U 1092,1150,1210,1271,1333,1397,1463
%N a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.
%C Essentially four odds followed by four evens.
%C Last digit is neither 4 nor 9.
%C Essentially twice or twin sequences in the hexagonal spiral from A002265.
%C 21 21 21 22 22 22 22
%C 21 14 14 14 14 15 15 23
%C 20 13 8 8 8 9 9 15 23
%C 20 13 8 4 4 4 4 9 15 23
%C 20 13 7 3 1 1 1 5 9 16 23
%C 20 13 7 3 1 0 0 2 5 10 16 24
%C 19 12 7 3 0 0 2 5 10 16 24
%C 19 12 7 3 2 2 5 10 16 24
%C 19 12 6 6 6 6 10 17 24
%C 19 12 11 11 11 11 17 25
%C 18 18 18 18 17 17 25
%C .
%C There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.
%C 0, 1, 3, 7, 13, 20, 28, 38, 50, ...
%C 1, 2, 4, 6, 7, 8, 10, 12, 13, ...
%C 1, 2, 2, 1, 1, 2, 2, 1, 1, ... period 4. See A014695.
%H Colin Barker, <a href="/A330707/b330707.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 (3,-4,4,-3,1).
%F a(n) = A231559(-n).
%F a(1+2*n) + a(2+2*n) = A033579(n+1).
%F a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).
%F From _Colin Barker_, Dec 27 2019: (Start)
%F G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)).
%F a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
%F (End)
%F E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - _Stefano Spezia_, Dec 27 2019
%F a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - _G. C. Greubel_, Dec 30 2019
%p seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # _G. C. Greubel_, Dec 30 2019
%t LinearRecurrence[{3,-4,4,-3,1}, {0,1,3,7,13}, 60] (* _Amiram Eldar_, Dec 27 2019 *)
%o (PARI) concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ _Colin Barker_, Dec 27 2019
%o (Magma) [(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // _G. C. Greubel_, Dec 30 2019
%o (Sage) [(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # _G. C. Greubel_, Dec 30 2019
%Y Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.
%K nonn,easy
%O 0,3
%A _Paul Curtz_, Dec 27 2019
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