A270545 Number of equilateral triangle units forming perimeter of equilateral triangle.

1, 4, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315

Offset: 0

Author

Peter M. Chema, Mar 18 2016

Keywords

• nonn
• easy

Comments

This is the triangular equivalent of A008574 (square units forming perimeter of a square).

The height of each triangle is n+1 units.

Links

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

Crossrefs

Cf. A016945, A008574.

Programs

Mathematica

CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - x)^2, {x, 0, 53}], x] (* Michael De Vlieger, Mar 21 2016 *)

PARI

a(n)=if(n<2,3*n+1,6*n-3) \\ Charles R Greathouse IV, Mar 19 2016

PARI

Vec((1+x)*(1+x+x^2)/(1-x)^2 + O(x^50)) \\ Colin Barker, Mar 20 2016

Formula

a(0)=1 and a(1)=4; thereafter a(n) = (n+1)^2-(n-2)^2 = 6*n-3.

a(n) = 2*a(n-1)-a(n-2) for n>3. G.f.: (1+x)*(1+x+x^2) / (1-x)^2. - Colin Barker, Mar 20 2016

a(n) = A016945(n-1), n>1.