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A358994 The sum of the numbers that are inside the contour of an n-story Christmas tree drawn at the top of the numerical pyramid containing the positive integers in natural order.

%I A358994 #64 Jun 14 2025 23:12:58
%S A358994 21,151,561,1503,3310,6396,11256,18466,28683,42645,61171,85161,115596,
%T A358994 153538,200130,256596,324241,404451,498693,608515,735546,881496,
%U A358994 1048156,1237398,1451175,1691521,1960551,2260461,2593528,2962110,3368646,3815656,4305741,4841583,5425945
%N A358994 The sum of the numbers that are inside the contour of an n-story Christmas tree drawn at the top of the numerical pyramid containing the positive integers in natural order.
%C A358994 The numbers of the natural series are written line by line in the form of a numerical pyramid: the first line contains the number 1, the second line contains the next two numbers 2 and 3, the third line contains the next three numbers 4, 5 and 6, etc.; that is, the line starting with the number k contains the k following numbers. In this numerical pyramid, the contour of a "multi-story Christmas tree" is distinguished, each floor of which occupies three lines. The numbers of the sequence are the sum of all the numbers that fall into the contour of the Christmas tree, which has n floors.
%H A358994 Nicolay Avilov, <a href="https://www.diofant.ru/problem/3908/">Problem 2128</a> (in Russian).
%H A358994 Nicolay Avilov, <a href="/A358994/a358994.jpg">Explanatory drawing</a>
%H A358994 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A358994 a(n) = n*(27*n^3 + 66*n^2 + 49*n + 26) / 8.
%F A358994 G.f.: x*(21 + 46*x + 16*x^2 - 2*x^3)/(1 - x)^5. - _Stefano Spezia_, Dec 25 2022
%F A358994 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Jun 14 2025
%e A358994 a(1) = 1 + 2 + 3 + 4 + 5 + 6 = 21;
%e A358994 a(2) = a(1) + (8 + 9 + 12 + 13 + 14 + 17 +18 + 19 + 20) = 151.
%o A358994 (Python)
%o A358994 def a(n): return n*(27*n**3 + 66*n**2 + 49*n + 26) // 8
%o A358994 print([a(n) for n in range(1, 36)]) # _Michael S. Branicky_, Dec 25 2022
%o A358994 (Magma) [n*(27*n^3 + 66*n^2 + 49*n + 26)/8 : n in [1..60]]; // _Wesley Ivan Hurt_, Jun 14 2025
%Y A358994 Cf. A001844, A022266, A006137.
%K A358994 nonn,easy
%O A358994 1,1
%A A358994 _Nicolay Avilov_, Dec 25 2022