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Search: id:A125714
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| A125714 |
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Alfred Moessner's factorial triangle. |
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+0
12
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| 1, 2, 3, 6, 11, 6, 24, 50, 35, 10, 120, 274, 225, 85, 15, 720, 1764, 1624, 735, 175, 21, 5040, 13068, 13132, 6769, 1960, 322, 28, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 3628800
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums of the triangle = 1, 5, 23, 119, 719...(matching the terms 0, 0, 1, 5, 23, 119, 719...; of A033312).
The name of the triangle derives from the fact that A125714(A000124(n)) = A000142(n) for n > 0. Moessner's method uses only additions to compute the factorial n!. [From Peter Luschny (peter(AT)luschny.de), Jan 27 2009]
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REFERENCES
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John Horton Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.; (based on the work of Alfred Moessner).
Alfred Moessner, Eine Bemerkung ueber die Potenzen der natuerlichen Zahlen. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951. [From Peter Luschny (peter(AT)luschny.de), Jan 27 2009]
Oskar Perron, Beweis des Moessnerschen Satzes. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951. [From Peter Luschny (peter(AT)luschny.de), Jan 27 2009]
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LINKS
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Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jun 17 2007, Table of n, a(n) for n = 1..66
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FORMULA
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Starting with the natural numbers, circle each triangular number. Underneath, take partial sums of the uncircled terms and circle the terms in this row which are offset one place to the left of the circled 1, 3, 6, 10...in the first row. Repeat with analogous operations in succeeding rows. The circled terms in the infinite set become the triangle.
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EXAMPLE
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An "x" prefaced before each term will indicate the term following the x being circled.
x1 2 x3 4 5 x6 7 8 9 x10 11 12 13 14 x15...
__x2 6 x11 18 26 x35 46 58 71 x85...
_____________x6 24 x50 96 154 x225...
_________________________x24 120 x274...
___________________________________________x120...
...
i.e. circle the triangular terms in row 1. In row 2, take partial sums of the uncircled terms and circle the terms offset one place to the left of the triangular terms in row 1. Continue in subsequent rows with analogous operations. The triangle consists of the infinite set of terms prefaced with the x (circled on page 64 of "The Book of Numbers".).
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MAPLE
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Contribution from Peter Luschny (peter(AT)luschny.de), Jan 27 2009: (Start)
a := proc(n) local s, m, k, i; s := array(0..n); s[0] := 1;
for m from 1 to n do s[m] := 0; for k from m by - 1 to 1 do
for i from 1 to k do s[i] := s[i] + s[i - 1] od; lprint(s[k]);
if k = n then RETURN(s[n]) fi od; lprint("-") od end: (End)
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CROSSREFS
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Cf. A033312.
Sequence in context: A056606 A062527 A038752 this_sequence A004038 A152038 A105614
Adjacent sequences: A125711 A125712 A125713 this_sequence A125715 A125716 A125717
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2006
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EXTENSIONS
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More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jun 17 2007
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