A100178 Structured hexagonal diamond numbers (vertex structure 5).

1, 8, 29, 72, 145, 256, 413, 624, 897, 1240, 1661, 2168, 2769, 3472, 4285, 5216, 6273, 7464, 8797, 10280, 11921, 13728, 15709, 17872, 20225, 22776, 25533, 28504, 31697, 35120, 38781, 42688, 46849, 51272, 55965, 60936, 66193, 71744, 77597, 83760, 90241, 97048, 104189

Offset: 1

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

• nonn
• easy
• changed

Comments

Row 1 of the convolution array A213838. - Clark Kimberling, Jul 05 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578 (alternate vertex), A000447 (structured diamonds) A100145 (for more on structured numbers).

Cf. A019558, A167471, A213838.

Programs

Magma

[(1/6)*(8*n^3-6*n^2+4*n): n in [1..40]]; // Vincenzo Librandi, Aug 03 2011

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 8, 29, 72}, 50] (* Paolo Xausa, Aug 06 2025 *)

Formula

a(n) = (1/6)*(8*n^3 - 6*n^2 + 4*n).

G.f.: x*(1+4*x+3*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012

From Elmo R. Oliveira, Aug 28 2025: (Start)

E.g.f.: exp(x)*x*(4*x^2 + 9*x + 3)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.

a(n) = A167471(n)/16 = A019558(n)/48. (End)