A267489 a(n) = n^2 - 4*floor(n^2/6).

0, 1, 4, 5, 8, 9, 12, 17, 24, 29, 36, 41, 48, 57, 68, 77, 88, 97, 108, 121, 136, 149, 164, 177, 192, 209, 228, 245, 264, 281, 300, 321, 344, 365, 388, 409, 432, 457, 484, 509, 536, 561, 588, 617, 648, 677, 708, 737, 768, 801, 836, 869, 904

Offset: 0

Author

Kival Ngaokrajang, Jan 16 2016

Keywords

• nonn
• easy

Comments

Inspired by A137932 and A042948.

The pattern is generated by adding subdiagonals parallel to principal diagonals at a spacing of at least 1 box in any direction from the previous generation.

Conjectures:

(i) a(n) is the total number of boxes (or 1's) at the n-th iteration.

(ii) The total number of left boxes (or 0's) is 4*A056827.

Links

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

Crossrefs

Cf. A042948, A056827, A137932.

Programs

Magma

[0] cat [n^2-4*Floor(n^2/6): n in [1..70]]; // Vincenzo Librandi, Jan 16 2016

Maple

A267489:=n->n^2-4*floor(n^2/6): seq(A267489(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2017

Mathematica

Table[n^2 - 4 Floor[n^2 / 6], {n, 0, 70}] (* Vincenzo Librandi, Jan 16 2016 *)

PARI

for (n = 0, 100, a = n^2-4*floor(n^2/6); print1(a, ", "))

PARI

concat(0, Vec(x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6)/((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 16 2016

PARI

a(n)=n^2 - n^2\6*4 \\ Charles R Greathouse IV, Mar 22 2017

Formula

a(n) = n^2 - 4*floor(n^2/6) for n >= 0.

From Colin Barker, Jan 16 2016: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>7.

G.f.: x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).

(End)