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

1,2

Comments

Right border of the triangle = A028387, left border = A007680.

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.

Example

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;
  ...

Crossrefs

Cf. A125714, A125751, A125752.

Sequence in context: A134522 A001445 A192860 * A018168 A320921 A084321

Adjacent sequences: A125747 A125748 A125749 * A125751 A125752 A125753

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 06 2006

Extensions

More terms from Joshua Zucker, Jun 17 2007

Status

approved