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A245738
Number of compositions of n into parts 1 and 2 with both parts present.
3
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
3,1
FORMULA
G.f.: 1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2).
a(n) = A052952(n-4)+2*A052952(n-3). - R. J. Mathar, Aug 05 2014
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
a(n) = A000045(n+1) - A000034(n+1). - J. M. Bergot and Robert Israel, Oct 11 2021
EXAMPLE
a(9) = 54. The tuples are (22221) = 5!/4! = 5, (222111) = 6!/3!/3! = 20, (2211111) = 7!/5!/2! = 21, (21111111) = 8!/7! = 8.
MATHEMATICA
LinearRecurrence[{1, 2, -1, -1}, {2, 3, 7, 11}, 50] (* Harvey P. Dale, Dec 20 2014 *)
PROG
(PARI) Vec(1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2)+O(x^66)) \\ Joerg Arndt, Aug 04 2014
CROSSREFS
Column k=2 of A373118.
Sequence in context: A159262 A160434 A139630 * A368032 A265093 A133044
KEYWORD
nonn,easy
AUTHOR
David Neil McGrath, Jul 31 2014
STATUS
approved