A245738 Number of compositions of n into parts 1 and 2 with both parts present.
2, 3, 7, 11, 20, 32, 54, 87, 143, 231, 376, 608, 986, 1595, 2583, 4179, 6764, 10944, 17710, 28655, 46367, 75023, 121392, 196416, 317810, 514227, 832039, 1346267, 2178308, 3524576, 5702886, 9227463, 14930351, 24157815, 39088168, 63245984, 102334154, 165580139, 267914295, 433494435, 701408732, 1134903168, 1836311902
Offset: 3
Examples
a(9) = 54. The tuples are (22221) = 5!/4! = 5, (222111) = 6!/3!/3! = 20, (2211111) = 7!/5!/2! = 21, (21111111) = 8!/7! = 8.
Links
- Colin Barker, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
Programs
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Mathematica
LinearRecurrence[{1,2,-1,-1},{2,3,7,11},50] (* Harvey P. Dale, Dec 20 2014 *) -
PARI
Vec(1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2)+O(x^66)) \\ Joerg Arndt, Aug 04 2014
Formula
G.f.: 1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2).
From Colin Barker, Jul 13 2017: (Start)
a(n) = (-20 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n even.
a(n) = (-10 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>6. (End)
a(n) = Sum_{i=1..floor((n-1)/2)} C(n-i,i). - Wesley Ivan Hurt, Sep 19 2017