A125750 A Moessner triangle using (1, 3, 5, ...).
1, 3, 5, 10, 19, 11, 42, 89, 64, 19, 216, 498, 415, 160, 29, 1320, 3254, 3023, 1385, 335, 41, 9360, 24372, 24640, 12803, 3745, 623, 55, 75600, 206100, 223116, 127799, 42938, 8750, 1064, 71, 685440, 1943568, 2227276, 1380076, 516201, 122010, 18354, 1704
Offset: 1
Examples
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.
First few rows of the triangle are:
1;
3, 5;
10, 19, 11;
42, 89, 64, 19;
216, 498, 415, 160, 29;
...
References
- J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, pp. 63-64.
Links
- Joshua Zucker, Table of n, a(n) for n = 1..78
- G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
- Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
- R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
- Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
- Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
- Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
- Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
- Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
- Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
- Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.
Formula
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.
Extensions
More terms from Joshua Zucker, Jun 17 2007
Comments