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

0,2

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).

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A016945, A008574.

Sequence in context: A313295 A313296 A313297 * A359626 A358243 A099055

Adjacent sequences: A270542 A270543 A270544 * A270546 A270547 A270548

Keyword

nonn,easy

Author

Peter M. Chema, Mar 18 2016

Status

approved