A334694 a(n) = (n/4)*(n^3+2*n^2+5*n+8).

0, 4, 17, 51, 124, 260, 489, 847, 1376, 2124, 3145, 4499, 6252, 8476, 11249, 14655, 18784, 23732, 29601, 36499, 44540, 53844, 64537, 76751, 90624, 106300, 123929, 143667, 165676, 190124, 217185, 247039, 279872, 315876, 355249, 398195, 444924, 495652, 550601, 609999, 674080, 743084, 817257, 896851, 982124, 1073340

Offset

0,2

Comments

Consider a figure made up of a row of n >= 1 adjacent congruent rectangles in which all possible diagonals of the rectangles have been drawn. The number of regions formed is A306302. If we distort all these diagonals very slightly so that no three lines meet at a point, the number of regions changes to a(n).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to stained glass windows

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

Satisfies the identity a(n) = A306302(n) + Sum_{k=3..(n+1)} binomial(k-1,2)*A333275(n,2*k). E.g. for n=4 we have a(4) = 104 + 8*1 + 2*3 + 1*6 = 124.

From Colin Barker, May 27 2020: (Start)

G.f.: x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.

(End)

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 4, 17, 51, 124}, 50] (* or *)
A334694[n_]:=n/4(n^3+2n^2+5n+8); Array[A334694, 50, 0] (* Paolo Xausa, Nov 08 2023 *)

Prog

(PARI) concat(0, Vec(x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, May 27 2020

Crossrefs

Cf. A306302, A331452, A333275.

Sequence in context: A173704 A297817 A184445 * A228960 A370212 A131339

Adjacent sequences: A334691 A334692 A334693 * A334695 A334696 A334697

Keyword

nonn,easy

Author

Scott R. Shannon and N. J. A. Sloane, May 19 2020

Status

approved