A125777 - OEIS

%I #26 Aug 01 2019 09:12:08

%S 1,3,6,13,28,21,69,161,137,55,433,1078,1017,477,120,3141,8245,8437,

%T 4460,1337,231,25873,71008,77620,45058,15415,3220,406,238629,680451,

%U 786012,492264,186729,44955,6930,666,2436673,7184170,8699205,5804448,2394150

%N Moessner triangle based on A000217.

%C 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 Left border = A104989: (1, 3, 13, 69, 433...). Right border = the doubly triangular numbers starting (1, 6, 21...): A002817.

%D J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.

%H Joshua Zucker, <a href="/A125777/b125777.txt">Table of n, a(n) for n = 1..55</a>

%H 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 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 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 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 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 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 First few rows of the triangle are as follows:

%e     1;

%e     3,    6;

%e    13,   28,   21;

%e    69,  161,  137,  55;

%e   433, 1078, 1017, 477, 120;

%e   ...

%Y Cf. A125714, A002817, A104989, A000217.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Dec 07 2006

%E More terms from _Joshua Zucker_, Jun 17 2007