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Search: id:A059397
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| A059397 |
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Triangle formed by right-bounded rhombus rule, read by rows. |
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+0
4
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| 1, 1, 1, 1, 2, 3, 1, 3, 7, 6, 1, 4, 12, 18, 16, 1, 5, 18, 37, 53, 40, 1, 6, 25, 64, 120, 148, 109, 1, 7, 33, 100, 227, 369, 430, 297, 1, 8, 42, 146, 385, 760, 1146, 1244, 836, 1, 9, 52, 203, 606, 1391, 2518, 3519, 3656, 2377, 1, 10, 63, 272, 903, 2346, 4900, 8188
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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T(n,n)=A128720(n). Mirror image of A132276. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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REFERENCES
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W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.
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FORMULA
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Each entry is sum of 3 entries above it in previous row and the entry directly above two rows back (provided the entries are properly aligned).
G.f.=G(t,z)=g(tz)/(1-zg(tz)), where g(z)=(1-z-z^2-sqrt((1+z-z^2)(1-3z-z^2)))/(2z^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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EXAMPLE
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If triangle is reflected in the vertical axis it looks like this:
1
1 1
3 2 1
6 7 3 1
16 18 12 4 1
and now the rhombus rule is clearly visible (e.g. 18 = 6 + 7 + 3 + 2).
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MAPLE
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g:=proc(z) options operator, arrow: (1/2-(1/2)*z-(1/2)*z^2-(1/2)*sqrt((1+z-z^2)*(1-3*z-z^2)))/z^2 end proc: G:=simplify(g(t*z)/(1-z*g(t*z))): Gser:=simplify(series(G, z=0, 13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j=0..n) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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CROSSREFS
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A variation on A059317. Row sums give A059398.
Cf. A128720, A132276.
Sequence in context: A113592 A136555 A063967 this_sequence A152821 A071943 A062869
Adjacent sequences: A059394 A059395 A059396 this_sequence A059398 A059399 A059400
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jan 29 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001
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