A125752 - OEIS

%I #29 Aug 01 2019 09:10:48

%S 1,1,2,4,9,8,26,69,77,55,261,806,1088,920,610,4062,14362,22887,22856,

%T 17034,10946,98912,395253,728605,847832,721756,502606,317811,3809193,

%U 17008391,35644614,47557978,46166656,35655012,23828383,14930352

%N Moessner triangle using the Fibonacci terms.

%C 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.

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

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

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

%C 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.

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

%H Joshua Zucker, <a href="/A125752/b125752.txt">Table of n, a(n) for n = 1..55</a>

%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&amp;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 T(n,n) = A081667(n-1).

%e The upper left corner of the array M(n,c) is

%e 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

%e 1, 4, 9, 22, 43, 77, 166, 310, 543, 920, 1907, 3504, 6088, 10269, 17034, ...

%e 4, 26, 69, 235, 545, 1088, 2995, 6499, 12587, 22856, 57601, 121003, 230773, ...

%e 26, 261, 806, 3801, 10300, 22887, 80488, 201491, 432264, 847832, 2586423, ...

%e 261, 4062, 14362, 94850, 296341, 728605, 3315028, 9488917, 22445416, ...

%e 4062, 98912, 395253, 3710281, 13199198, 35644614, 213010460, 690899755, ...

%e and dropping the columns with column numbers in A014132, reading the remaining array by antidiagonals leads to the final triangle T(n,m):

%e     1;

%e     1,   2;

%e     4,   9,    8;

%e    26,  69,   77,  55;

%e   261, 806, 1088, 920, 610;

%e   ...

%Y Cf. A125714, A125750, A125751, A081667.

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, Dec 06 2006

%E More terms from _Joshua Zucker_, Jun 17 2007

%E Description of starting row corrected, comments detailed with formulas by _R. J. Mathar_, Sep 17 2009