A140063 Binomial transform of [1, 3, 7, 0, 0, 0, ...].

1, 4, 14, 31, 55, 86, 124, 169, 221, 280, 346, 419, 499, 586, 680, 781, 889, 1004, 1126, 1255, 1391, 1534, 1684, 1841, 2005, 2176, 2354, 2539, 2731, 2930, 3136, 3349, 3569, 3796, 4030, 4271, 4519, 4774, 5036, 5305, 5581, 5864, 6154, 6451, 6755, 7066, 7384, 7709, 8041, 8380, 8726, 9079, 9439, 9806, 10180, 10561, 10949, 11344, 11746, 12155

Offset

0,2

Comments

Section 8.3 of the "Cutting a Pancake" paper shows that nthis sequence arises both from cutting up a pancake with an H-shaped cookie-cutter and a phi-shaped cookie-cutter. - N. J. A. Sloane, Dec 27 2025

Links

Table of n, a(n) for n=0..59.

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.

Derek Kinsella, Plane division by lines and circles. [Wayback Machine link]

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

A007318 * [1, 3, 7, 0, 0, 0, ...].

O.g.f.: (1 + x + 5x^2)/(1-x)^3. - R. J. Mathar, May 06 2008

a(n) = 7*n^2/2 - n/2 + 1 = 3*a(n-1) - 3*a(n-2) + a(n-3). - R. J. Mathar, Dec 08 2025

a(n) = a(n-1) + 7*n - 4 (with a(0)=1). - Vincenzo Librandi, Nov 24 2010, 2025

a(n) = A022264(n) + 1. - Hugo Pfoertner, Dec 08 2025

Example

a(4) = 31 = (1, 3, 3, 1) dot (1, 3, 7, 0) = (1 + 9 + 21 + 0).

Mathematica

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 265, 7}]; lst (* Zerinvary Lajos, Jul 11 2009 *)

Prog

(PARI) a(n)=n*(7*n-1)/2+1 \\ Charles R Greathouse IV, Jun 17 2017, 2025

Crossrefs

Cf. A022264.

Main diagonal of both A393442 and A393443.

Sequence in context: A079776 A117109 A317031 * A051409 A072475 A001740

Adjacent sequences: A140060 A140061 A140062 * A140064 A140065 A140066

Keyword

nonn,easy

Author

Gary W. Adamson, May 03 2008

Extensions

Offset set to 0; formulas adjusted - R. J. Mathar, Dec 08 2025

Status

approved