A059397 Triangle formed by right-bounded rhombus rule, read by rows.

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

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-François 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