A017818 -id:A017818 - cp's OEIS frontend

A228362 The number of all possible covers of L-length line segment by 3-length line segments with allowed gaps < 3.

0, 0, 0, 1, 2, 3, 3, 4, 6, 8, 10, 13, 18, 24, 31, 41, 55, 73, 96, 127, 169, 224, 296, 392, 520, 689, 912, 1208, 1601, 2121, 2809, 3721, 4930, 6531, 8651, 11460, 15182, 20112, 26642, 35293, 46754, 61936, 82047, 108689, 143983, 190737

Offset: 0

Philipp O. Tsvetkov, Aug 21 2013

nonn

For n>2, a(n) = A017818(n+3).

Third row of A228360.

CoefficientList[Series[(1 - x^3 - x^4 - x^5)^-1*(1 + x + x^2)^2*x^3 , {x,0, 100}],x]

G.f.: x^3*(1 + x + x^2)^2 / (1 - x^3 - x^4 - x^5).

a(0)=a(1)=a(2)=0; a(3)=1; a(4)=2; a(5)= a(6)=3; a(7)=4; a(n)= a(n-3) + a(n-4) + a(n-5).

A245367 Compositions of n into parts 3, 5 and 7.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 8, 10, 11, 17, 18, 25, 32, 37, 52, 61, 79, 102, 123, 163, 200, 254, 326, 402, 519, 649, 819, 1045, 1305, 1664, 2096, 2643, 3358, 4220, 5352, 6759, 8527, 10806, 13622, 17237, 21785, 27501, 34802, 43934, 55544, 70209, 88672, 112131, 141644, 179018, 226274, 285860, 361358

Offset: 0

David Neil McGrath, Aug 20 2014

			a(16) = 10: the compositions are the permutations of [5533] (there are 4!/2!2!=6 of them) and the permutations of [7333] (there are 4!/3!=4).

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,1).

Cf. A000073, A013979, A017818, A060945, A079956, A079957, A079971, A079973, A120400, A121833.

LinearRecurrence[{0,0,1,0,1,0,1},{1,0,0,1,0,1,1},70] (* Harvey P. Dale, Jan 27 2017 *)

Vec(1/(1-x^3-x^5-x^7) +O(x^66)) \\ Joerg Arndt, Aug 20 2014

G.f: 1/(1-x^3-x^5-x^7).

a(n) = a(n-3) + a(n-5) + a(n-7).

A245368 Compositions of n into parts 3, 4 and 7.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 3, 1, 1, 5, 5, 2, 7, 13, 8, 10, 25, 26, 20, 42, 64, 54, 72, 131, 144, 146, 245, 339, 344, 463, 715, 827, 953, 1423, 1881, 2124, 2839, 4019, 4832, 5916, 8281, 10732, 12872, 17036, 23032, 28436, 35824, 48349, 62200, 77132, 101209, 133581

Offset: 0

David Neil McGrath, Aug 20 2014

			a(14) = 13. The compositions (ordered partitions) of 14 into parts 3, 4 and 7 are the permutations of (7,7) (there is only one), the permutations of (7,4,3) (there are 3!=6 of these) and the permutations of (4,4,3,3) (there are 4!/2!2!=6 of these).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,1).

Cf. A017818, A079956.

I:=[1,0,0,1,1,0,1]; [n le 7 select I[n] else Self(n-3)+Self(n-4)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Jan 08 2016

a:= proc(n) option remember; `if`(n=0, 1,
      `if`(n<0, 0, add(a(n-j), j=[3, 4, 7])))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 21 2014

LinearRecurrence[{0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 1, 0, 1}, 60] (* Jean-François Alcover, Jan 08 2016 *)

G.f: 1/(1-x^3-x^4-x^7).

a(n) = a(n-3) + a(n-4) + a(n-7).

A193771 Expansion of 1 / (1 - x - x^3 + x^6) in powers of x.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 23, 31, 41, 54, 72, 96, 127, 168, 223, 296, 392, 519, 688, 912, 1208, 1600, 2120, 2809, 3721, 4929, 6530, 8651, 11460, 15181, 20111, 26642, 35293, 46753, 61935, 82047, 108689, 143982, 190736, 252672, 334719, 443408, 587391

Offset: 0

Michael Somos, Jan 01 2013

nonn

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 0, -1).

Cf. A000931, A008621, A017818.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^3+x^6)));  // G. C. Greubel, Aug 10 2018

CoefficientList[Series[1/(1-x-x^3+x^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,1,0,0,-1},{1,1,1,2,3,4},50] (* Harvey P. Dale, Jul 25 2017 *)

{a(n) = if( n<0, n = -n; polcoeff( - x^6 / (1 - x^3 - x^5 + x^6) + x * O(x^n), n), polcoeff( 1 / (1 - x - x^3 + x^6) + x * O(x^n), n))};

G.f.: 1 / (1 - x - x^3 + x^6) = 1 / (1 - x / (1 - x^2 / (1 + x^2 / (1 - x / (1 + x / (1 + x^2 / (1 - x^2))))))).
a(n) = a(n-1) + a(n-3) - a(n-6) for all n in Z.
Convolution of A008621 and A000931. PSUM transform of A017818.

A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.

Original entry on oeis.org

6, 0, 0, 12, 12, 12, 20, 30, 50, 60, 80, 120, 162, 225, 305, 401, 560, 763, 1017, 1365, 1834, 2484, 3328, 4420, 5936, 7943, 10593, 14148, 18828, 25092, 33468, 44517, 59214, 78734, 104698, 139232, 184889, 245532, 326177, 433052, 574841, 762856, 1012219, 1343160

Offset: 12

David Neil McGrath, Jul 30 2014

Compositions of n from the set {3,4,5} that can be partitioned into the equivalence classes [345][34][45][35][3][4][5], where each class is defined by the relation "all elements are present".

			a(24) = 162 = 42 + 90 + 30: the tuples are (5433333) -> 7!/5! = 42, (554433) -> 6!/2!2!2! = 90, (544443) -> 6!/4! = 30.

Robert Israel, Table of n, a(n) for n = 12..7366
Index entries for linear recurrences with constant coefficients, signature (-2, -2, 2, 9, 16, 14, -2, -29, -52, -52, -20, 34, 82, 97, 67, 7, -53, -84, -77, -43, -4, 22, 29, 23, 13, 5, 1).

Cf. A245492, A245487, A245527, A022003, A011765, A112765, A017818.

N:= 100;
C34:= Vector(N):
C35:= Vector(N):
C45:= Vector(N):
C345:= Vector(N):
C1:= Vector(N,i -> numboccur([i mod 3, i mod 4, i mod 5],0)):
C34[3]:= 1: C34[4]:= 1:
C35[3]:= 1: C35[5]:= 1:
C45[4]:= 1: C45[5]:= 1:
C345[3]:= 1: C345[4]:= 1: C345[5]:= 1:
for n from 6 to N do
  C34[n]:= C34[n-3] + C34[n-4];
  C35[n]:= C35[n-3] + C35[n-5];
  C45[n]:= C45[n-4] + C45[n-5];
  C345[n]:= C345[n-3]+C345[n-4]+C345[n-5];
od:
A:= C345 - C34 - C35 - C45 + C1:
convert(A[12..N],list); # Robert Israel, Aug 18 2014

CoefficientList[Series[x^12*(x^15 + 5*x^14 + 13*x^13 + 24*x^12 + 34*x^11 + 36*x^10 + 24*x^9 - 26*x^7 - 40*x^6 - 36*x^5 - 18*x^4 + 12*x^2 + 12*x +6)/((1 - x)*(x + 1)*(x^2 + 1)*(x^3 + x^2 - 1)*(x^4 + x^3 - 1)*(x^5 + x^3 - 1)*(x^2 + x + 1)*(x^5 + x^4 - 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Aug 02 2014 *)
Drop[LinearRecurrence[{-2,-2,2,9,16,14,-2,-29,-52,-52,-20,34,82,97,67,7,-53,-84,-77,-43,-4,22,29,23,13,5,1},{0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,12,12,20,30,50,60,80,120,162,225,305,401},60],12] (* Harvey P. Dale, Jun 06 2025 *)

a(n) = A017818(n-1) -A245492(n) -A245487(n) -A245527(n) -A022003(n) -A011765(n) -A112765(n).
G.f.: -(x^15 +5*x^14 +13*x^13 +24*x^12 +34*x^11 +36*x^10 +24*x^9-26*x^7 -40*x^6 -36*x^5 -18*x^4 +12*x^2 +12*x +6) *x^12 /((x-1) *(x+1) *(x^2+1) *(x^3+x^2-1) *(x^4+x^3-1) *(x^5+x^3-1) *(x^2+x+1) *(x^5+x^4-1) *(x^4+x^3+x^2+x+1)). - Alois P. Heinz, Jul 30 2014
a(n) = A017818(n) - A017817(n) - A052920(n) - A017827(n) + A079978(n) + A121262(n) + A079998(n). - Robert Israel, Aug 18 2014

A247920 Expansion of 1 / (1 + x + x^2 - x^5) in powers of x.

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 0, 2, -3, 2, 0, -2, 4, -5, 3, 2, -7, 9, -7, 1, 8, -16, 17, -8, -8, 24, -32, 25, -1, -32, 57, -57, 25, 31, -88, 114, -83, -6, 120, -202, 196, -77, -125, 322, -399, 273, 49, -447, 720, -672, 225, 496, -1168, 1392, -896, -271, 1663, -2560

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 - x + x^3 - x^4 + x^5 - x^6 + 2*x^8 - 3*x^9 + 2*x^10 - 2*x^12 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eunmi Choi, Yuna Oh, Diagonal sums in negative trinomial table, Korean J. Math (2019) Vol. 27, No. 3, 723-734.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,0,1).

Cf. A017818.

a:=[1,-1,0,1,-1];; for n in [6..60] do a[n]:=-(a[n-1]+a[n-2]-a[n-5]); od; a; # G. C. Greubel, Dec 29 2019

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1+x^2)*(1+x-x^3)))); // G. C. Greubel, Aug 04 2018

seq(coeff(series(1/(1+x+x^2-x^5), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 29 2019

CoefficientList[Series[1/(1+x+x^2-x^5), {x, 0, 60}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-5-n; polcoeff( 1 / (1 - x^3 - x^4 - x^5) + x * O(x^n), n), polcoeff( 1 / (1 + x + x^2 - x^5) + x * O(x^n), n))};

def A247920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1+x+x^2-x^5) ).list()
A247920_list(60) # G. C. Greubel, Dec 29 2019

G.f.: 1 / ((1 + x^2) * (1 + x - x^3)).
a(n) = A017818(-5-n) for all n in Z.
0 = a(n) - a(n+3) - a(n+4) - a(n+5) for all n in Z.
0 = a(n) - a(n+2) - a(n+3) + (-1)^floor(n/2) * mod(n,2) for all n in Z.

cp's OEIS Frontend

A228362 The number of all possible covers of L-length line segment by 3-length line segments with allowed gaps < 3.

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A245367 Compositions of n into parts 3, 5 and 7.

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A245368 Compositions of n into parts 3, 4 and 7.

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A193771 Expansion of 1 / (1 - x - x^3 + x^6) in powers of x.

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A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.

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A247920 Expansion of 1 / (1 + x + x^2 - x^5) in powers of x.

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