A045950 Number of upward triangles in a Star of David matchstick arrangement of size n.

0, 10, 59, 177, 394, 740, 1245, 1939, 2852, 4014, 5455, 7205, 9294, 11752, 14609, 17895, 21640, 25874, 30627, 35929, 41810, 48300, 55429, 63227, 71724, 80950, 90935, 101709, 113302, 125744, 139065, 153295, 168464, 184602, 201739, 219905, 239130, 259444

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Stefano Spezia, Illustration for n = 3.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A299965.

Programs

Mathematica

A045950[n_] := n*(n*(10*n + 9) + 1)/2; Array[A045950, 50, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 10, 59, 177}, 50] (* Paolo Xausa, Sep 18 2024 *)

PARI

concat(0, Vec(x*(x^2+19*x+10)/(x-1)^4 + O(x^100))) \\ Colin Barker, Dec 02 2014

Formula

a(n) = n*(10*n^2+9*n+1)/2.

From Colin Barker, Dec 02 2014: (Start)

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

G.f.: x*(x^2+19*x+10) / (x-1)^4. (End)

E.g.f.: exp(x)*x*(20 + 39*x + 10*x^2)/2. - Stefano Spezia, Sep 18 2024

Extensions

Name clarified by Paolo Xausa, Sep 19 2024