A125752 Moessner triangle using the Fibonacci terms.

1, 1, 2, 4, 9, 8, 26, 69, 77, 55, 261, 806, 1088, 920, 610, 4062, 14362, 22887, 22856, 17034, 10946, 98912, 395253, 728605, 847832, 721756, 502606, 317811, 3809193, 17008391, 35644614, 47557978, 46166656, 35655012, 23828383, 14930352

Offset: 1

Author

Gary W. Adamson, Dec 06 2006

Keywords

• nonn
• tabl

Comments

A Moessner triangle is generated with the recurrence described in A125714, starting from a first row M(1,c) filled with the Fibonacci numbers M(1,c) = A000045(c), c >= 1.

Subsequent rows n are generated from the numbers in their previous rows with the rule:

Mark/circle all elements M(n-1, A000217(t)) of the previous row n-1, t >= 1.

Define the elements M(n,.) as the partial sums of the M(n-1,.) that have not been marked:

M(n,c) = Sum_{j=1..c} M(n-1,A014132(j)), c >= 1. The T(n,m) are then defined by reading the marked/circled terms "along antidiagonals": T(n,m) = M(n+m-1, A000217(m)), n >= 1, 1 <= m <= n.

Examples

			The upper left corner of the array M(n,c) is
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...
1, 4, 9, 22, 43, 77, 166, 310, 543, 920, 1907, 3504, 6088, 10269, 17034, ...
4, 26, 69, 235, 545, 1088, 2995, 6499, 12587, 22856, 57601, 121003, 230773, ...
26, 261, 806, 3801, 10300, 22887, 80488, 201491, 432264, 847832, 2586423, ...
261, 4062, 14362, 94850, 296341, 728605, 3315028, 9488917, 22445416, ...
4062, 98912, 395253, 3710281, 13199198, 35644614, 213010460, 690899755, ...
and dropping the columns with column numbers in A014132, reading the remaining array by antidiagonals leads to the final triangle T(n,m):
    1;
    1,   2;
    4,   9,    8;
   26,  69,   77,  55;
  261, 806, 1088, 920, 610;
  ...
		

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.
• 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.

Crossrefs

Cf. A125714, A125750, A125751, A081667.

Formula

T(n,n) = A081667(n-1).

Extensions

More terms from Joshua Zucker, Jun 17 2007

Description of starting row corrected, comments detailed with formulas by R. J. Mathar, Sep 17 2009