A318958 -id:A318958 - cp's OEIS frontend

0, -1, 1, 3, 8, 13, 21, 29, 40, 51, 65, 79, 96, 113, 133, 153, 176, 199, 225, 251, 280, 309, 341, 373, 408, 443, 481, 519, 560, 601, 645, 689, 736, 783, 833, 883, 936, 989, 1045, 1101, 1160, 1219, 1281, 1343, 1408, 1473, 1541, 1609, 1680, 1751

Offset: 0

Paul Curtz, Sep 11 2018

A144391(n) = -1, 3, 13, 29, 51, ... is in the hexagonal spiral begining with -1 (like from 0 in A000567):
.
                55--54--53--52--51
                /                 \
              56  32--31--30--29  50
              /   /             \   \
            57  33  15--14--13  28  49
            /   /   /         \   \   \
          58  34  16   4---3  12  27  48
          /   /   /   /     \   \   \   \
        59  35  17   5  -1   2  11  26  47
            /   /   /   /   /   /   /   /
          36  18   6   0---1  10  25  46
            \   \   \         /   /   /
            37  19   7---8---9  24  45
              \   \             /   /
              38  20--21--22--23  44
                \                 /
                39--40--41--42--43
.
A000567(n) = 0, 1, 8, 21, 40, ... is in the first hexagonal spiral.
The bisections 0, 1, 8, 21, ... and -1, 3, 13, 29, ...  are on the respective main antidiagonals.
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.
The bisections of a(n) are in the second spiral: ... 29, 13, 3, -1, 0, 1, 8, 21, ... .
The bisections of a(-n) are in the first and in the second spiral: ...  33, 16, 5, 0, 1, 9, 23, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Main diagonal of A318958.

Cf. A000567, A144391, A168236, A045944, A144390.

Flat(List([0..30],n->[3*n^2-2*n,3*n^2+n-1])); # Muniru A Asiru, Sep 19 2018

seq(op([3*n^2-2*n,3*n^2+n-1]),n=0..30); # Muniru A Asiru, Sep 19 2018

LinearRecurrence[{2, 0, -2, 1}, {0, -1, 1, 3 }, 40] (* Stefano Spezia, Sep 16 2018 *)

concat(0, Vec(-x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Sep 14 2018

a(n+1) = a(n) + (6*n^2 - 3*(-1)^n - 1)/4, n=0,1,2, ... , a(0) = 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>3.
From Colin Barker, Sep 14 2018: (Start)
G.f.: -x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)).
a(n) = (-4*n + 3*n^2) / 4 for n even.
a(n) = (-3 - 4*n + 3*n^2) / 4 for n odd.
(End)
a(n) = (-3 + 3*(-1)^n - 8*n + 6*n^2)/8. - Colin Barker, Sep 14 2018
E.g.f.: (x*(3*x - 1)*cosh(x) + (3*x^2 - x - 3)*sinh(x))/4. - Stefano Spezia, Mar 15 2020

cp's OEIS Frontend

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

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