A160138 a(n) = number of solutions to the system: x + y + z + w = n, -2x - y + z + 2w = 5 with nonnegative x, y, z, w.

0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330, 340

Offset: 1

Author

Krishnan Sundararaman (krishnan.sundararaman(AT)enmu.edu), May 02 2009

Keywords

• nonn

Comments

Number of ways in which playing n one-card-poker games results in a payoff of $5.

x = # of games where player loses $2,

y = # of games where player loses $1,

z = # of games where player wins $1,

w = # of games where player wins $2.

The events i.e. winning $1, losing $2 etc. are mutually exclusive.

Hence in n games

x+y+z+w = n

-2x-y+z+2w = $5

Examples

			For n = 3, a(3) = 1, since the four-tuple <x=0, y=0, z=1, w=2> allows you to win $5 in 3 games. x + y + z + w = 1 + 2 =3, -2x - y + z + 2w = 1 + 2*2 = 5, as desired.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• One Card Poker
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Cf. A008806 is the number of ways in which playing n one-card-poker games results in a payoff of $0, i.e., the n-th term is the number of solutions to the system: x + y + z + w = n, -2x - y + z + 2w = 0, with nonnegative x, y, z, w.

Except for offset, same as A253186.

Programs

Maple

> fourples2 := proc (n) local i, c1, c2, c3, c4, c3positive, mylist, cash, k, howmanyways; cash := 2*n; for k from -cash to cash do i := 0; unassign(mylist); for c1 from 0 to n do c3positive := true; for c2 from 0 to n-c1 while c3positive do c3 := 2*n-4*c1-3*c2-k; if 0 <= c3 then c4 := n-c1-c2-c3; if 0 <= c4 then i := i+1; mylist[i] := [c1, c2, c3, c4] end if else c3positive := false end if end do end do; howmanyways[k] := [i, [seq(mylist[j], j = 1 .. i)]] end do; return howmanyways end proc; N := 20; for n to N do a := fourples2(n); points[n] := [n, a[5][1]] end do; seq(points[n], n = 1 .. N);

Mathematica

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 100] (* Jean-François Alcover, Apr 11 2020 *)

PARI

concat([0, 0], Vec((1 + x - x^3)/((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^60))) \\ Andrew Howroyd, Jan 12 2020

Formula

From Andrew Howroyd, Jan 12 2020: (Start)

a(n) = A253186(n-1).

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 6.

G.f.: x^3*(1 + x - x^3)/((1 - x)^3*(1 + x)*(1 + x + x^2)).

(End)

Extensions

Terms a(13) and beyond from Andrew Howroyd, Jan 12 2020