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A125777 Moessner triangle based on A000217.

%I A125777 #26 Aug 01 2019 09:12:08
%S A125777 1,3,6,13,28,21,69,161,137,55,433,1078,1017,477,120,3141,8245,8437,
%T A125777 4460,1337,231,25873,71008,77620,45058,15415,3220,406,238629,680451,
%U A125777 786012,492264,186729,44955,6930,666,2436673,7184170,8699205,5804448,2394150
%N A125777 Moessner triangle based on A000217.
%C A125777 Begin with the triangular numbers A000217 and circle every T(k)-th term, getting the doubly triangular numbers, A002817. Per instructions shown in A125714, take partial sums of the uncircled terms in row 1, denoting this as row 2. Circle the row 2 terms which are one place to the left of row 1 terms. Take partial sums again in analogous operations for subsequent rows.
%C A125777 Left border = A104989: (1, 3, 13, 69, 433...). Right border = the doubly triangular numbers starting (1, 6, 21...): A002817.
%D A125777 J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.
%H A125777 Joshua Zucker, <a href="/A125777/b125777.txt">Table of n, a(n) for n = 1..55</a>
%H A125777 G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&amp;i=20">On a conjecture of Moessner and a general problem</a>, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
%H A125777 Dexter Kozen and Alexandra Silva, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.131">On Moessner's theorem</a>, Amer. Math. Monthly 120(2) (2013), 131-139.
%H A125777 Calvin T. Long, <a href="https://doi.org/10.2307/3615513">Strike it out--add it up</a>, Math. Gaz. 66 (438) (1982), 273-277.
%H A125777 Alfred Moessner, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1951_0029.pdf">Eine Bemerkung über die Potenzen der natürlichen Zahlen</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
%H A125777 M. Niqui and J. J. M. M. Rutten, <a href="https://doi.org/10.1007/s10990-012-9082-7">A proof of Moessner's theorem by coinduction</a>, High.-Order Symb. Comput. 24(3) (2011), 191-206.
%H A125777 Oskar Perron, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1951_0031-0034.pdf">Beweis des Moessnerschen Satzes</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.
%e A125777 First few rows of the triangle are as follows:
%e A125777     1;
%e A125777     3,    6;
%e A125777    13,   28,   21;
%e A125777    69,  161,  137,  55;
%e A125777   433, 1078, 1017, 477, 120;
%e A125777   ...
%Y A125777 Cf. A125714, A002817, A104989, A000217.
%K A125777 nonn,tabl
%O A125777 1,2
%A A125777 _Gary W. Adamson_, Dec 07 2006
%E A125777 More terms from _Joshua Zucker_, Jun 17 2007