A329482 Interleave 1 - n + 3*n^2, 1 + 3*n*(1+n) for n >= 0.

1, 1, 3, 7, 11, 19, 25, 37, 45, 61, 71, 91, 103, 127, 141, 169, 185, 217, 235, 271, 291, 331, 353, 397, 421, 469, 495, 547, 575, 631, 661, 721, 753, 817, 851, 919, 955, 1027, 1065, 1141, 1181, 1261

Offset: 0

Author

Paul Curtz, Nov 14 2019

Keywords

• nonn
• easy

Comments

a(n+1) - 2*a(n) = -1, 1, 1, -3, -3, -13, -13, -29, -29, ...

Hexagonal spiral for A000265:

.

17--35---9--37

/

33 17---9--19---5

/ / \

1 1 3---7---1 21

/ / / \ \

31 15 5 1---1 9 11

\ \ \ / / /

15 7 1---3 5 23

\ \ / /

29 13---3--11 3

\ /

7--27--13--25

.

The two sequences are perpendicular.

a(n+1) - a(n) = 0, 2, 4, 4, 8, 6, 12, ... = 2*A029578(n+2).

A003215 is a bisection of 1, 1, 13, 7, 49, 19, 109, 37, ... .

Links

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

Crossrefs

Cf. A003215, A029578, A056106, A056108.

Programs

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 1, 3, 7, 11}, 42] (* Amiram Eldar, Nov 23 2019 *)
Module[{nn=20,a,b},a=Table[1-n+3 n^2,{n,0,nn}];b=Table[1+3n(1+n),{n,0,nn}];Riffle[a,b]] (* Harvey P. Dale, Apr 30 2023 *)

PARI

Vec((1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Nov 15 2019

Formula

From Colin Barker, Nov 14 2019: (Start)

G.f.: (1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.

a(n) = (5 + 3*(-1)^n - 2*(1 + (-1)^n)*n + 6*n^2) / 8.

(End)

E.g.f.: (1/8)*exp(-x)*(3 + 2*x + exp(2*x)*(5 + 4*x + 6*x^2)). - Stefano Spezia, Nov 14 2019 after Colin Barker

a(-n) = 1, 1, 5, 7, 15, 19, ... = interleave 1 + n + 3*n^2, 1 + 3*n*(1+n), both in the spiral.