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A140229 Binomial transform of [1, 3, 3, 1, -2, 3, -4, 5, ...].

%I A140229 #37 Sep 07 2024 21:10:35
%S A140229 1,4,10,20,33,49,68,90,115,143,174,208,245,285,328,374,423,475,530,
%T A140229 588,649,713,780,850,923,999,1078,1160,1245,1333,1424,1518,1615,1715,
%U A140229 1818,1924,2033,2145,2260,2378,2499,2623,2750,2880,3013,3149,3288,3430,3575
%N A140229 Binomial transform of [1, 3, 3, 1, -2, 3, -4, 5, ...].
%C A140229 The falling diagonal starting with T(1,4) in A049777 (as a square array) gives the terms of this sequence for n >=3. - _Bob Selcoe_, Oct 27 2014
%C A140229 From _Fatma E. Kurnaz_, _Salih E. Ariturk_, and _Arzu Caglar_, Aug 08 2024: (Start)
%C A140229 Suppose a regular hexagon is divided into equilateral triangles. When we number the corners of the triangles in a spiral manner starting from the center, if the numbers increase counterclockwise, let's define them as "right triangle" and the remaining ones as "left triangle".
%C A140229  The number of  downward-pointing left triangles, constitute this sequence for n>=2.
%C A140229 For n=3 a(3)=20, A hexagon with a side of 3 units consists of 54 equilateral triangles. The number of upward right triangles is 12, the number of upward left triangles  is 15, the number of downward right triangles  is 7. The number of downward left triangles is equal to 20. (End)
%H A140229 İ. H. Karakaş and İ. Aliyev, <a href="https://mathematicalolympiads.wordpress.com/wp-content/uploads/2012/08/analiz-ve-cebirde-ilginc3a7-olimpiyat-problemleri-ve-c3a7c3b6zc3bcmleri.pdf">Problem 8.36</a>, Analiz ve Cebirde İlginç OLİMPİYAT PROBLEMLERİ VE ÇÖZÜMLERİ, TÜBİTAK Publications, 5 (2003), p. 298.
%H A140229 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A140229 A007318 * [1, 3, 3, 1, -2, 3, -4, 5,...].
%F A140229 a(n) = (n+1)(3n-4)/2, for n>=3. - _Emeric Deutsch_, May 18 2008
%F A140229 G.f.: x(1+x+x^2+x^3-x^4)/(1-x)^3. a(n) = 3*a(n-1) -3*a(n-2) + a(n-3), n>5. a(n+1)-a(n) = A016777(n), n>3. - _R. J. Mathar_, Nov 25 2008
%e A140229 a(5) = 33 = (1, 4, 6, 4, 1) dot (1, 3, 3, 1, -2) = (1 + 12 + 18 + 4 - 2).
%p A140229 1,4,seq((1/2)*(n+1)*(3*n-4), n=3..40); # _Emeric Deutsch_, May 18 2008
%t A140229 s=-2;lst={1, 4};Do[s+=n+1;If[n>3, AppendTo[lst, s]], {n, 0, 6!, 3}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 25 2008 *)
%o A140229 (Magma) [1,4] cat [(n+1)*(3*n-4)/2: n in [3..50]]; // _Vincenzo Librandi_, Oct 27 2014
%Y A140229 Cf. A049777
%K A140229 nonn,easy
%O A140229 1,2
%A A140229 _Gary W. Adamson_, May 13 2008
%E A140229 More terms from _Emeric Deutsch_, May 18 2008
%E A140229 More terms from _Vladimir Joseph Stephan Orlovsky_, Oct 25 2008