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

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

0,4

Comments

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, ... .

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

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

Maple

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

Mathematica

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

Prog

(PARI) concat(0, Vec(-x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Sep 14 2018
(GAP) Flat(List([0..30], n->[3*n^2-2*n, 3*n^2+n-1])); # Muniru A Asiru, Sep 19 2018

Crossrefs

Main diagonal of A318958.

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

Sequence in context: A322598 A363034 A317194 * A094110 A212649 A084535

Adjacent sequences: A319125 A319126 A319127 * A319129 A319130 A319131

Keyword

sign,easy

Author

Paul Curtz, Sep 11 2018

Status

approved