A298125 The hex numbers (A003215) together with 3.

1, 3, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487

Offset: 1

Author

Brian Watson, Jan 13 2018

Keywords

• nonn
• easy

Comments

Apart from the term 3, these are precisely the sizes of hexagonal clusters of circles in the A_2 (or hexagonal) lattice.

These numbers include many common numbers of strands used in cables.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
• Wikipedia, Wire: Number of strands
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A003215.

Programs

PARI

Vec(x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Feb 05 2018

Formula

From Colin Barker, Feb 05 2018: (Start)

G.f.: x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3.

a(n) = 7 - 9*n + 3*n^2 for n>2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.

(End)

Extensions

Entry revised by N. J. A. Sloane, Jan 22 2018