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A059397 Triangle formed by right-bounded rhombus rule, read by rows. 3
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; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,n)=A128720(n). Mirror image of A132276. - Emeric Deutsch, Sep 03 2007

LINKS

Table of n, a(n) for n=0..62.

W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.

FORMULA

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, Sep 03 2007

EXAMPLE

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).

MAPLE

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, Sep 03 2007

MATHEMATICA

max = 10; g[z_] := (1 - z - z^2 - Sqrt[(1 + z - z^2)*(1 - 3*z - z^2)])/(2 z^2); s = Series[g[t*z]/(1 - z*g[t*z]), {z, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {z, 0, n}, {t, 0, k}]; t[0, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 16 2014, after Emeric Deutsch *)

CROSSREFS

A variation on A059317. Row sums give A059398.

Cf. A128720, A132276.

Sequence in context: A174014 A236376 A063967 * A209567 A208338 A236918

Adjacent sequences:  A059394 A059395 A059396 * A059398 A059399 A059400

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Jan 29 2001

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

STATUS

approved

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Last modified October 21 01:18 EDT 2019. Contains 328291 sequences. (Running on oeis4.)