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, 12019, 12562

Offset: 0

Author

Jonathan Vos Post, May 24 2007

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Programs

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

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 *)

PARI

a(n)=6*n^2+9*n+7 \\ Charles R Greathouse IV, Feb 20 2012

Formula

a(n) = H(n) + H(n+1) + H(n+2) where H(n) = A000384(n) = n*(2*n-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)

E.g.f.: (7 + 15*x + 6*x^2)*exp(x). - Elmo R. Oliveira, Nov 16 2024