A125777 Moessner triangle based on A000217.

1, 3, 6, 13, 28, 21, 69, 161, 137, 55, 433, 1078, 1017, 477, 120, 3141, 8245, 8437, 4460, 1337, 231, 25873, 71008, 77620, 45058, 15415, 3220, 406, 238629, 680451, 786012, 492264, 186729, 44955, 6930, 666, 2436673, 7184170, 8699205, 5804448, 2394150

Offset

1,2

Comments

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.

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

References

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

Links

Joshua Zucker, Table of n, a(n) for n = 1..55

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.

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.

M. Niqui and J. J. M. M. Rutten, A proof of Moessner's theorem by coinduction, High.-Order Symb. Comput. 24(3) (2011), 191-206.

Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Example

First few rows of the triangle are as follows:
    1;
    3,    6;
   13,   28,   21;
   69,  161,  137,  55;
  433, 1078, 1017, 477, 120;
  ...

Crossrefs

Cf. A125714, A002817, A104989, A000217.

Sequence in context: A036886 A052251 A032253 * A103788 A371564 A182137

Adjacent sequences: A125774 A125775 A125776 * A125778 A125779 A125780

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 07 2006

Extensions

More terms from Joshua Zucker, Jun 17 2007

Status

approved