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A319128 Interleave n*(3*n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .

%I A319128 #36 Mar 15 2020 18:02:39
%S A319128 0,-1,1,3,8,13,21,29,40,51,65,79,96,113,133,153,176,199,225,251,280,
%T A319128 309,341,373,408,443,481,519,560,601,645,689,736,783,833,883,936,989,
%U A319128 1045,1101,1160,1219,1281,1343,1408,1473,1541,1609,1680,1751
%N A319128 Interleave n*(3*n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .
%C A319128 A144391(n) = -1, 3, 13, 29, 51, ... is in the hexagonal spiral begining with -1 (like from 0 in A000567):
%C A319128 .
%C A319128                 55--54--53--52--51
%C A319128                 /                 \
%C A319128               56  32--31--30--29  50
%C A319128               /   /             \   \
%C A319128             57  33  15--14--13  28  49
%C A319128             /   /   /         \   \   \
%C A319128           58  34  16   4---3  12  27  48
%C A319128           /   /   /   /     \   \   \   \
%C A319128         59  35  17   5  -1   2  11  26  47
%C A319128             /   /   /   /   /   /   /   /
%C A319128           36  18   6   0---1  10  25  46
%C A319128             \   \   \         /   /   /
%C A319128             37  19   7---8---9  24  45
%C A319128               \   \             /   /
%C A319128               38  20--21--22--23  44
%C A319128                 \                 /
%C A319128                 39--40--41--42--43
%C A319128 .
%C A319128 A000567(n) = 0, 1, 8, 21, 40, ... is in the first hexagonal spiral.
%C A319128 The bisections 0, 1, 8, 21, ... and -1, 3, 13, 29, ...  are on the respective main antidiagonals.
%C A319128 a(-n) = 0, 1, 5, 9, 16, 23, ... . The bisections n*(3*n + 2) and 3*n^2 - n - 1 are in both spirals on main diagonals.
%C A319128 The bisections of a(n) are in the second spiral: ... 29, 13, 3, -1, 0, 1, 8, 21, ... .
%C A319128 The bisections of a(-n) are in the first and in the second spiral: ...  33, 16, 5, 0, 1, 9, 23, ... .
%H A319128 Colin Barker, <a href="/A319128/b319128.txt">Table of n, a(n) for n = 0..1000</a>
%H A319128 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A319128 a(n+1) = a(n) + (6*n^2 - 3*(-1)^n - 1)/4, n=0,1,2, ... , a(0) = 0.
%F A319128 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>3.
%F A319128 From _Colin Barker_, Sep 14 2018: (Start)
%F A319128 G.f.: -x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)).
%F A319128 a(n) = (-4*n + 3*n^2) / 4 for n even.
%F A319128 a(n) = (-3 - 4*n + 3*n^2) / 4 for n odd.
%F A319128 (End)
%F A319128 a(n) = (-3 + 3*(-1)^n - 8*n + 6*n^2)/8. - _Colin Barker_, Sep 14 2018
%F A319128 E.g.f.: (x*(3*x - 1)*cosh(x) + (3*x^2 - x - 3)*sinh(x))/4. - _Stefano Spezia_, Mar 15 2020
%p A319128 seq(op([3*n^2-2*n,3*n^2+n-1]),n=0..30); # _Muniru A Asiru_, Sep 19 2018
%t A319128 LinearRecurrence[{2, 0, -2, 1}, {0, -1, 1, 3 }, 40] (* _Stefano Spezia_, Sep 16 2018 *)
%o A319128 (PARI) concat(0, Vec(-x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ _Colin Barker_, Sep 14 2018
%o A319128 (GAP) Flat(List([0..30],n->[3*n^2-2*n,3*n^2+n-1])); # _Muniru A Asiru_, Sep 19 2018
%Y A319128 Main diagonal of A318958.
%Y A319128 Cf. A000567, A144391, A168236, A045944, A144390.
%K A319128 sign,easy
%O A319128 0,4
%A A319128 _Paul Curtz_, Sep 11 2018