A129109 Sums of three consecutive hexagonal numbers.

7, 22, 49, 88, 139, 202, 277, 364, 463, 574, 697, 832, 979, 1138, 1309, 1492, 1687, 1894, 2113, 2344, 2587, 2842, 3109, 3388, 3679, 3982, 4297, 4624, 4963, 5314, 5677, 6052, 6439, 6838, 7249, 7672, 8107, 8554, 9013, 9484, 9967, 10462, 10969, 11488

Offset

0,1

Comments

Arises in hexagonal number analog to A129803 Triangular numbers which are the sum of three consecutive triangular numbers. What are the hexagonal numbers which are the sum of three consecutive hexagonal numbers? Prime for a(0) = 7, a(4) = 139, a(6) = 277, a(8) = 463, a(18) = 2113, a(22) = 3109, a(26) = 4297, a(38) = 9013, a(40) = 9967.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = H(n) + H(n+1) + H(n+2) where H(n) = A000384(n) = n(2n-1). a(n) = 6*n^2 + 9*n + 7.

From Colin Barker, Feb 20 2012: (Start)

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

G.f.: (7+x+4*x^2)/(1-x)^3. (End)

Example

a(0) = H(0) + H(1) + H(2) = 0 + 1 + 6 = 7 = 6*0^2 + 9*0 + 7.
a(1) = H(1) + H(2) + H(3) = 1 + 6 + 15 = 22 = 6*1^2 + 9*1 + 7.
a(2) = H(2) + H(3) + H(4) = 6 + 15 + 28 = 49 = 6*2^2 + 9*2 + 7.

Mathematica

LinearRecurrence[{3, -3, 1}, {7, 22, 49}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
Total/@Partition[PolygonalNumber[6, Range[0, 50]], 3, 1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 14 2020 *)

Prog

(Magma) I:=[7, 22, 49]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012
(PARI) a(n)=6*n^2+9*n+7 \\ Charles R Greathouse IV, Feb 20 2012

Crossrefs

Cf. A000384, A007667, A034961, A129803, A129863.

Sequence in context: A014073 A288114 A369549 * A224141 A002412 A211652

Adjacent sequences: A129106 A129107 A129108 * A129110 A129111 A129112

Keyword

easy,nonn

Author

Jonathan Vos Post, May 24 2007

Status

approved