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 A213772 #45 Aug 09 2025 10:04:56
%S A213772 1,11,42,106,215,381,616,932,1341,1855,2486,3246,4147,5201,6420,7816,
%T A213772 9401,11187,13186,15410,17871,20581,23552,26796,30325,34151,38286,
%U A213772 42742,47531,52665,58156,64016,70257,76891,83930,91386,99271,107597,116376,125620,135341
%N A213772 Principal diagonal of the convolution array A213771.
%C A213772 Zhu Shijie gives in his Magnus Opus "Jade Mirror of the Four Unknowns" the problem: "The total number of apples in a pile in the form of a cone is 932, and the number of layers is an odd number." Zhu Shijie assumed the rational sequence s(k) = (k*(k+1)*(2*k+1)+k+1)/8 for the total number of apples in k layers, with n = (k+1)/2 is the solution 932 = a((15+1)/2) with k = 15. Zhu Shijie gave the solution polynomial: "Let the element tian be the number of layers. From the statement we have 7455 for the negative shi, 2 for the positive fang, 3 for the positive first lian, and 2 for the positive yu." This translates into the polynomial equation: 2*x^3 + 3*x^2 + 2*x - 7455 = 0. - _Thomas Scheuerle_, Feb 10 2025
%D A213772 Zhu Shijie, Jade Mirror of the Four Unknowns (Siyuan yujian), Book III Guo Duo Die Gang (Piles of Fruit), Problem number 7, (1303).
%H A213772 Clark Kimberling, <a href="/A213772/b213772.txt">Table of n, a(n) for n = 1..1000</a>
%H A213772 Zhu Shijie, <a href="https://archive.org/details/jademirrorofthefourunknowns2/">Jade Mirror of the Four Unknowns 2</a>, Translation by Library of Chinese classics, original from 1303.
%H A213772 Wikipedia, <a href="https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns">Jade Mirror of the Four Unknowns</a>.
%H A213772 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A213772 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A213772 G.f.: x*(1 + 7*x + 4*x^2)/(1 - x)^4.
%F A213772 a(n) = (4*n^2 - 3*n + 1)*n/2 = n*A002411(n) - (n-1)*A002411(n-1). - _Bruno Berselli_, Dec 11 2012
%F A213772 a(n) = n*A000326(n) + Sum_{i=0..n-1} A000326(i). - _Bruno Berselli_, Dec 18 2013
%F A213772 a(n) - a(n-1) = A085473(n-1). - _R. J. Mathar_, Mar 02 2025
%F A213772 E.g.f.: exp(x)*x*(1 + 4*x)*(2 + x)/2. - _Elmo R. Oliveira_, Aug 08 2025
%t A213772 (See A213771.)
%t A213772 LinearRecurrence[{4,-6,4,-1},{1,11,42,106},70] (* _Harvey P. Dale_, Mar 29 2025 *)
%o A213772 (PARI) a(n) = (4*n^3-3*n^2+n)/2; \\ _Altug Alkan_, Dec 16 2017
%Y A213772 Cf. A000326, A002411, A085473, A213771, A220084 (for a list of numbers of the form n*P(k,n) - (n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).
%Y A213772 Cf. A260260 (comment). [_Bruno Berselli_, Jul 22 2015]
%K A213772 nonn,easy
%O A213772 1,2
%A A213772 _Clark Kimberling_, Jul 04 2012