This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143689 #61 Apr 19 2025 11:54:00
%S A143689 1,2,6,13,23,36,52,71,93,118,146,177,211,248,288,331,377,426,478,533,
%T A143689 591,652,716,783,853,926,1002,1081,1163,1248,1336,1427,1521,1618,1718,
%U A143689 1821,1927,2036,2148,2263,2381,2502,2626,2753,2883,3016,3152,3291
%N A143689 a(n) = (3*n^2 - n + 2)/2.
%C A143689 Equals left border of triangle A033292.
%C A143689 Equals binomial transform of [1, 1, 3, 0, 0, 0, ...].
%C A143689 A242357(a(n)) = 1. - _Reinhard Zumkeller_, May 11 2014
%C A143689 These might be called "trisected pentagonal numbers": A figurate pentagonal number is composed of three triangles, of which the central one is the largest, and the removal of the triangular frame (3*n) of the central triangle trisects the figure. This is reflected in the formula a(n) = A000326(n+1) - 3*n. See illustration in links. - _John Elias_, May 27 2022
%H A143689 Michael De Vlieger, <a href="/A143689/b143689.txt">Table of n, a(n) for n = 0..10000</a>
%H A143689 D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, <a href="http://arxiv.org/abs/1510.08036">Pattern avoidance in forests of binary shrubs</a>, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
%H A143689 John Elias, <a href="/A143689/a143689.png">Trisected Pentagonal Numbers</a>
%H A143689 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H A143689 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A143689 a(n) = A000326(n+1) - 3*n. (A000326 are the pentagonal numbers.)
%F A143689 a(n) = (3*n^2 - n + 2)/2 = A027599(n+1)/2. - _R. J. Mathar_, Sep 03 2008
%F A143689 a(n) = a(n-1) + 3*n - 2 (with a(0)=1). - _Vincenzo Librandi_, Nov 25 2010
%F A143689 a(n) = 2*a(n-1) - a(n-2) + 3.
%F A143689 O.g.f.: (1-x+3*x^2)/((1-x)^3). - _Eric Werley_, Jun 27 2011
%F A143689 a(n) = A104249(-n). - _Bruno Berselli_, Jul 08 2015
%F A143689 a(n) = binomial(n,2) + n^2 + 1 = A152947(n+1) + A000290(n). - _Franck Maminirina Ramaharo_, Mar 01 2018
%F A143689 E.g.f.: exp(x)*(2 + 2*x + 3*x^2)/2. - _Stefano Spezia_, Apr 19 2025
%t A143689 Table[(3n^2-n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,6},50] (* _Harvey P. Dale_, May 05 2014 *)
%o A143689 (Haskell)
%o A143689 a143689 n = n*(3*n-1) `div` 2 + 1 -- _Reinhard Zumkeller_, May 11 2014
%o A143689 (PARI) a(n)=(3*n^2-n+2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A143689 (Maxima) makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* _Franck Maminirina Ramaharo_, Mar 01 2018 */
%Y A143689 a(n) = A000326(n+1) - 3n. Third column of A107111.
%Y A143689 Cf. A033292, A104249.
%K A143689 nonn,easy
%O A143689 0,2
%A A143689 _Gary W. Adamson_, Aug 29 2008
%E A143689 Index of A000326 in definition, formula and example corrected by _R. J. Mathar_, Sep 03 2008