A051936 Truncated triangular numbers: a(n) = n*(n+1)/2 - 9.

1, 6, 12, 19, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 369, 397, 426, 456, 487, 519, 552, 586, 621, 657, 694, 732, 771, 811, 852, 894, 937, 981, 1026, 1072, 1119, 1167, 1216, 1266, 1317, 1369, 1422, 1476

Offset: 4

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 21 1999

Keywords

• nonn
• nice
• easy

Comments

Equals binomial transform of [1, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008

Numbers m > 0 such that 8m+73 is a square. - Bruce J. Nicholson, Jul 29 2017

Examples

			Illustration of the initial terms:
                                                          .
                              .                         .   .
      .                     .   .                     o   o   o
    .   .                 o   o   o                 o   o   o   o
  .   o   .             .   o   o   .             .   o   o   o   .
.   .   .   .         .   .   o   .   .         .   .   o   o   .   .
----------------------------------------------------------------------
      1                       6                           12
----------------------------------------------------------------------
- _Bruno Berselli_, Oct 13 2016
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 4..10000
• Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes of Graphs, arXiv:2304.13675 [math.CO], 2023.
• Pratiksha Chauhan, Samir Shukla, and Kumar Vinayak, 3-Cut Complexes of Squared Cycle Graphs, arXiv:2406.01979 [math.CO], 2024. See p. 2.
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Wayback Machine copy]
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217.

Programs

Haskell

a051936 = (subtract 9) . a000217
a051936_list = scanl (+) 1 [5..]
-- Reinhard Zumkeller, Oct 25 2012

Mathematica

Table[n*(n + 1)/2 - 9, {n, 4, 60}] (* Stefan Steinerberger, Mar 25 2006 *)
k = 4; NestList[(k++; # + k) &, 1, 45] (* Robert G. Wilson v, Feb 02 2011 *)
Drop[Accumulate[Range[60]]-9,3] (* Harvey P. Dale, Jan 16 2012 *)

PARI

a(n)=n*(n+1)/2-9 \\ Charles R Greathouse IV, Oct 07 2015

Formula

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

a(n) = n + a(n-1) for n>4, a(4)=1. - Vincenzo Librandi, Aug 06 2010

a(n) = 2*A000217(n-3) - A000217(n-6), with A000217(-2)=1, A000217(-1)=0. - Bruno Berselli, Oct 13 2016

Sum_{n>=4} 1/a(n) = 53/72 + 2*Pi*tan(sqrt(73)*Pi/2)/sqrt(73). - Amiram Eldar, Dec 13 2022