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%I #29 Aug 01 2019 07:26:38
%S 1,3,5,10,19,11,42,89,64,19,216,498,415,160,29,1320,3254,3023,1385,
%T 335,41,9360,24372,24640,12803,3745,623,55,75600,206100,223116,127799,
%U 42938,8750,1064,71,685440,1943568,2227276,1380076,516201,122010,18354,1704
%N A Moessner triangle using (1, 3, 5, ...).
%C Right border of the triangle = A028387, left border = A007680.
%D J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, pp. 63-64.
%H Joshua Zucker, <a href="/A125750/b125750.txt">Table of n, a(n) for n = 1..78</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 R. Krebbers, L. Parlant, and A. Silva, <a href="https://doi.org/10.1007/978-3-319-30734-3_21">Moessner's theorem: an exercise in coinductive reasoning in Coq</a>, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
%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 Ivan Paasche, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1952_0001-0005.pdf">Ein neuer Beweis des Moessnerschen Satzes</a> S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
%H Ivan Paasche, <a href="https://doi.org/10.1007/BF01900739">Beweis des Moessnerschen Satzes mittels linearer Transformationen</a>, Arch. Math. (Basel) 6 (1955), 194-199.
%H Ivan Paasche, <a href="http://www.numdam.org/article/CM_1954-1956__12__263_0.pdf">Eine Verallgemeinerung des Moessnerschen Satzes</a>, Compositio Math. 12 (1956), 263-270.
%H Hans Salié, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1952_0007-0011.pdf">Bemerkung zu einem Satz von A. Moessner</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
%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.
%F Using "Moessner's Magic" (Conway and Guy, pp. 63-64; cf. A125714), we circle the 1, 3, 6, 10, ...(-th) terms in the sequence (1, 3, 5, 7, ...) and take partial sums of the remaining terms, making row 2. Circle the terms in row 2 one place offset to the left of row 1 terms, then take partial sums. Continue with analogous operations for succeeding rows. The triangle = leftmost circled terms in each row.
%e Circling the 1, 3, 6, ...(-th) terms in the sequence (1, 3, 5, 7, ...), we get A018387: (1, 5, 11, 19, 29, ...). Taking partial sums of the remaining terms, we get (3, 10, 19, 32, ...) in row 2 and we circle 3 and 19. In row 3 we circle the 10.
%e First few rows of the triangle are:
%e 1;
%e 3, 5;
%e 10, 19, 11;
%e 42, 89, 64, 19;
%e 216, 498, 415, 160, 29;
%e ...
%Y Cf. A125714, A125751, A125752.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Dec 06 2006
%E More terms from _Joshua Zucker_, Jun 17 2007
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