A245332 -id:A245332 - cp's OEIS frontend

A245487 Number of compositions of n into parts 3,4 where both parts are always present.

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 3, 0, 4, 6, 4, 5, 10, 10, 11, 15, 20, 22, 27, 35, 43, 49, 63, 79, 92, 112, 144, 171, 204, 257, 316, 375, 462, 573, 692, 838, 1035, 1265, 1532, 1873, 2300, 2798, 3406, 4173, 5099, 6204, 7580, 9273, 11303, 13784, 16855, 20576

Offset: 0

David Neil McGrath, Jul 23 2014

			a(16)=5, the compositions being 43333, 34333, 33433, 33343, 33334.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, -1, 1, 3, 3, 2, -1, -2, -2, -1).

Cf. A245332.

CoefficientList[Series[x^7 (x^3 + 2 x^2 + 2 x + 2)/((x - 1) (x + 1) (x^2 + 1) (x^2 + x + 1) * (x^4 + x^3 - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a=[0,0,0,0,0,0,2,0,0,3]; b=[1,0,1,0,1,1,0,0,2,0,0,1]; k=1; for(n=11, 100, a=concat(a, a[n-3]+a[n-4]+b[k]); if(k==#b, k=1, k++)); a \\ Colin Barker, Jul 24 2014

a(n) = a(n-3)+a(n-4)+b(n) where b(n) is the 12-cycle (1,0,1,0,1,1,0,0,2,0,0,1) starting with initial value b(11)=1 and b(n)=b(n-12) e.g. b(23)=b(11). The initial values for a(n) are a(7)=2,a(8)=0,a(9)=0,a(10)=3.

G.f.: x^7*(x^3+2*x^2+2*x+2) / ((x-1)*(x+1)*(x^2+1)*(x^2+x+1)*(x^4+x^3-1)). - Colin Barker, Jul 24 2014

More terms from Colin Barker, Jul 24 2014

A245492 Number of compositions of n into parts 3 and 5 with at least one 3 and one 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 3, 4, 0, 6, 5, 4, 10, 6, 10, 15, 12, 20, 21, 23, 35, 34, 44, 56, 57, 80, 91, 101, 137, 148, 181, 230, 249, 318, 379, 430, 549, 629, 748, 928, 1060, 1298, 1557, 1809, 2226, 2617, 3109, 3783, 4426, 5336, 6400, 7536, 9120

Offset: 0

David Neil McGrath, Jul 24 2014

			a(20)=6, the tuples being: (533333),(353333),(335333),(333533),(333353),(333335).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, -1, 1, 1, 3, 2, 2, -1, -1, -2, -1, -1).

Cf. A245332, A245487.

a245492 n = a245492_list !! (n-1)
a245492_list = [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0] ++
               zipWith3 (((+) .) . (+))
               (drop 8 a245492_list) (drop 10 a245492_list)
               (cycle [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0])
-- Reinhard Zumkeller, Jul 28 2014

CoefficientList[Series[x^8*(x^4 + x^3 + 2*x^2 + 2*x + 2)/((x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^5 + x^3 - 1)), {x, 0, 60}], x] (* Wesley Ivan Hurt, Jul 24 2014 *)

a=[0,0,0,0,0,0,0,2,0,0,3,0]; b=[1,1,0,0,1,1,0,1,0,0,2,0,0,1,0]; k=1; for(n=13, 100, a=concat(a, a[n-3]+a[n-5]+b[k]); if(k==#b, k=1, k++)); a \\ Colin Barker, Jul 24 2014

a(n) = a(n-3)+a(n-5)+b(n) where b(n) is the 15-cycle: (1,1,0,0,1,1,0,1,0,0,2,0,0,1,0) with b(n)=b(n-15) starting at b(13)=1. e.g. b(28)=b(13). The initial values for a(n) are: a(8)=2, a(9)=0, a(10)=0, a(11)=3, a(12)=0.

G.f.: x^8*(x^4+x^3+2*x^2+2*x+2) / ((x-1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)*(x^5+x^3-1)). - Colin Barker, Jul 24 2014

More terms from Colin Barker, Jul 24 2014

A245738 Number of compositions of n into parts 1 and 2 with both parts present.

Original entry on oeis.org

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

David Neil McGrath, Jul 31 2014

			a(9) = 54. The tuples are (22221) = 5!/4! = 5, (222111) = 6!/3!/3! = 20, (2211111) = 7!/5!/2! = 21, (21111111) = 8!/7! = 8.

Colin Barker, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A245332, A245492, A245487, A245527.

Column k=2 of A373118.

LinearRecurrence[{1,2,-1,-1},{2,3,7,11},50] (* Harvey P. Dale, Dec 20 2014 *)

Vec(1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2)+O(x^66)) \\ Joerg Arndt, Aug 04 2014

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

A245527 Number of compositions of n into parts 4 and 5 with at least one 4 and one 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 3, 0, 0, 4, 6, 4, 0, 5, 10, 10, 5, 6, 15, 20, 15, 13, 21, 35, 35, 29, 35, 56, 70, 65, 64, 92, 126, 136, 129, 156, 219, 263, 265, 285, 375, 484, 528, 550, 660, 860, 1013, 1078, 1210, 1521, 1873, 2092, 2288, 2732, 3394

Offset: 0

David Neil McGrath, Jul 25 2014

			a(22)=10 The tuples are (55444)(54544)(54454)... where a(22)=5!/3!2!=10.

Index entries for linear recurrences with constant coefficients, signature (-1, -1, -1, 1, 3, 3, 3, 2, -1, -2, -2, -2, -1).

Cf. A245332, A245492, A245487.

CoefficientList[Series[1 + 1/(1 - x^5 - x^4) - 1/(1 - x^5) - 1/(1 - x^4), {x, 0, 60}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)

a(n) = a(n-4)+a(n-5)+b(n) where b(n) is the 20-cycle (1,0,0,1,0,1,0,1,0,0,1,1,0,0,0,2,0,0,0,1) and b(n)=b(n-20). Initial values are b(14)=1, a(9)=2, a(10)=0, a(11)=0, a(12)=0, a(13)=3.

G.f.: 1+1/(1-x^5-x^4)-1/(1-x^5)-1/(1-x^4) (see comment A245332). - courtesy of Joerg Arndt

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A245487 Number of compositions of n into parts 3,4 where both parts are always present.

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A245492 Number of compositions of n into parts 3 and 5 with at least one 3 and one 5.

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A245738 Number of compositions of n into parts 1 and 2 with both parts present.

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A245527 Number of compositions of n into parts 4 and 5 with at least one 4 and one 5.

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