Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #29 Aug 10 2019 18:44:35
%S 1,2,1,4,5,2,10,18,9,2,38,78,53,15,1,186,422,344,129,23,1,1106,2704,
%T 2484,1123,268,32,2,7718,19998,20080,10342,2991,490,42,2,61662,167520,
%U 180466,102700,34211,6891,824,54,1,554330,1567518,1789474,1103206
%N A Moessner triangle using (1, 2, 1, 2, 1, 2, ...).
%C Circle terms n = 1, 3, 6, 10, ... in the sequence (1, 2, 1, 2, 1, 2, ...). Partial sums of the uncircled terms becomes row 2. Circle the terms in row 2 that are one place offset to the left of the circled row 1 terms. Take partial sums and continue with analogous operations. (Cf. A125714 and "The Book of Numbers", p. 64.)
%C Left border (1, 2, 4, 10, 38, 186, 1106, 7718, 61662, ...).
%D J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.
%H Joshua Zucker, <a href="/A125751/b125751.txt">Table of n, a(n) for n = 1..65</a>
%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&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 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:
%e 1;
%e 2, 1;
%e 4, 5, 2;
%e 10, 18, 9, 2;
%e 38, 78, 53, 15, 1;
%e ...
%Y Cf. A125714, A125750, A125752.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Dec 06 2006
%E More terms from _Joshua Zucker_, Jun 17 2007
%E Corrected the comment concerning the left border - _R. J. Mathar_, Sep 17 2009