A017817 -id:A017817 - cp's OEIS frontend

A099099 Quadrisection of a generalized Padovan sequence.

1, 1, 1, 2, 6, 16, 37, 80, 172, 377, 839, 1874, 4175, 9274, 20577, 45665, 101393, 225193, 500162, 1110790, 2466760, 5477917, 12164896, 27015092, 59993817, 133231279, 295872778, 657057431, 1459155634, 3240410561, 7196122817

Offset: 0

Paul Barry, Sep 29 2004

Quadrisection of sequence with g.f. 1/(1 - x^3 - x^4), or A017817.

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-1).

Join[{1},a=0;b=0;c=0;d=1;Table[a+=b;b+=c;c+=d;d+=a,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

G.f.: (1-x)^3/((1-x)^4-x^3).
a(n) = sum_{k=0..2n} binomial(k, 4n-3k).
a(n) = 4a(n-1) - 6a(n-2) + 5a(n-3) - a(n-4).
a(n) = A017817(4n).
a(n) = sum_{k=0..floor((n+1)/2)} binomial(n+k, 4k). - Paul Barry, May 09 2005

A110064 a(n+4) = a(n+1) - a(n), a(0) = 1, a(1) = -4, a(2) = 0, a(3) = 1.

Original entry on oeis.org

1, -4, 0, 1, -5, 4, 1, -6, 9, -3, -7, 15, -12, -4, 22, -27, 8, 26, -49, 35, 18, -75, 84, -17, -93, 159, -101, -76, 252, -260, 25, 328, -512, 285, 303, -840, 797, 18, -1143, 1637, -779, -1161, 2780, -2416, -382, 3941, -5196, 2034, 4323, -9137, 7230, 2289, -13460, 16367, -4941, -15749, 29827, -21308, -10808

Offset: 0

Creighton Dement, Jul 10 2005

One of several sequences, apparently all of the form a(n+4) = a(n+1) - a(n), which appear to "spiral outwards" when plotted against each other (see A110061-64). In reference to the FAMP program code, A017817 is also in this same batch of sequences and satisfies the same recurrence relation.

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

Cf. A110061, A110062, A110063, A017817.

seriestolist(series(-(-1+4*x)/(1-x^3+x^4), x=0,60)); -or- Floretion Algebra Multiplication Program, FAMP Code: 4kbaseseq[A*B] with A = + .5'i + .5'j + .5'k + .5e and B = - .5'i - .25'j + .25'k - .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e

LinearRecurrence[{0,0,1,-1},{1,-4,0,1},60] (* Harvey P. Dale, Oct 23 2016 *)

Expansion of (4*x-1)/(1-x^3+x^4)

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A123552 Expansion of 1/(1 - x - x^3 + x^5).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 6, 7, 8, 11, 14, 16, 20, 26, 31, 37, 47, 58, 69, 85, 106, 128, 155, 192, 235, 284, 348, 428, 520, 633, 777, 949, 1154, 1411, 1727, 2104, 2566, 3139, 3832, 4671, 5706, 6972, 8504, 10378, 12679, 15477, 18883, 23058, 28157, 34361, 41942, 51216, 62519

Offset: 0

Christian G. Bower, Nov 14 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1).

Number of nodes at n-th generation in A123015.

a=b=c=d=0;Table[e=a+b+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)
CoefficientList[Series[1/(1 - x - x^3 + x^5), {x, 0, 53}], x] (* Michael De Vlieger, Sep 15 2017 *)

x='x+O('x^99); Vec(1/(1-x-x^3+x^5)) \\ Altug Alkan, Sep 12 2017

G.f.: 1/((1-x)*(1-x^3-x^4)).
From Robert FERREOL, Sep 12 2017: (Start)
a(n) = a(n-1) + a(n-3) - a(n-5) for n >= 5, with a(0)=a(1)=a(2)=1, a(3)=2, a(4)=3.
a(n) = a(n-3) + a(n-4) + 1 for n >= 4, with a(0)=a(1)=a(2)=1, a(3)=2.
a(n) - a(n-1) = A017817(n). (End)

A001584 A generalized Fibonacci sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 8, 12, 12, 16, 21, 21, 31, 37, 38, 58, 65, 71, 106, 114, 135, 191, 201, 257, 341, 359, 485, 605, 652, 904, 1070, 1202, 1664, 1894, 2237, 3029, 3370, 4176, 5464, 6048, 7779, 9793, 10963, 14411, 17492, 20054, 26507, 31239, 36924, 48396

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
V. C. Harris and C. C. Styles, Generalized Fibonacci sequences associated with a generalized Pascal triangle, Fib. Quart., 4 (1966), 241-248.
V. C. Harris and C. C. Styles, Generalized Fibonacci sequences associated with a generalized Pascal triangle and accompanying letter (annotated scanned copy)
Alaa Ibrahim and Bruno Salvy, Positivity Proofs for Linear Recurrences through Contracted Cones, arXiv:2412.08576 [cs.SC], 2024. See p. 22.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 0, 0, -1, 0, 1).

Cf. A017817.

A001584:=(z-1)*(z**2+z+1)**2/(z**4-z**3+1)/(z**4+z**3-1); # Simon Plouffe in his 1992 dissertation

Vec((1+x+x^2-x^3-x^4-x^5)/(1-2*x^3+x^6-x^8) + O(x^80)) \\ Michel Marcus, Sep 07 2017

G.f.: (1 + x + x^2 - x^3 - x^4 - x^5)/(1 - 2*x^3 + x^6 - x^8).

More terms from David W. Wilson

A052697 Expansion of e.g.f. 1/(1-x^3-x^4).

Original entry on oeis.org

1, 0, 0, 6, 24, 0, 720, 10080, 40320, 362880, 10886400, 119750400, 958003200, 24908083200, 523069747200, 6538371840000, 125536739328000, 3556874280960000, 70426110763008000, 1338096104497152000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..430
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 646

Cf. A000142, A017817.

[Factorial(n)*(&+[Binomial(k,n-3*k): k in [0..Floor(n/3)]]): n in [0..30]]; // G. C. Greubel, May 31 2022

spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[1/(1-x^3-x^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 06 2016 *)

[factorial(n)*sum(binomial(k, n-3*k) for k in (0..n//3)) for n in (0..30)] # G. C. Greubel, May 31 2022

E.g.f.: 1/(1 - x^3 - x^4).
D-finite recurrence: a(0)=1, a(1)=0, a(2)=0, a(3)=6, a(n+4) = (24 + 26*n + 9*n^2 + n^3)*a(n+1) + (24 + 50*n + 35*n^2 + 10*n^3 + n^4)*a(n).
a(n) = (n!/283) * Sum_{alpha=RootOf(-1 + Z^3 + Z^4)} (- 16 - 73*alpha + 3*alpha^2 + 12*alpha^3)*alpha^(-1-n).
a(n) = n!*A017817(n). - R. J. Mathar, Nov 27 2011

A127838 a(1) = 1, a(2) = a(3) = a(4) = 0; a(n) = a(n-4) + a(n-3) for n > 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 4, 6, 5, 6, 10, 11, 11, 16, 21, 22, 27, 37, 43, 49, 64, 80, 92, 113, 144, 172, 205, 257, 316, 377, 462, 573, 693, 839, 1035, 1266, 1532, 1874, 2301, 2798, 3406, 4175, 5099, 6204, 7581

Offset: 1

Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007

Part of the phi_k family of sequences defined by a(1)=1, a(2)=...=a(k)=0, a(n) = a(n-k) + a(n-k+1) for n > k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.

The sequence can be interpreted as the top-left element of the n-th power of 6 different 4 X 4 (0,1) matrices. - R. J. Mathar, Mar 19 2014

G. Mantel, Resten van wederkeerige Reeksen (Remainders of the reciprocal series), Nieuw Archief v. Wiskunde, 2nd series, I (1894), 172-184. [From N. J. A. Sloane, Dec 17 2010]
S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007, apparently unpublished as of Mar 2014.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).

LinearRecurrence[{0,0,1,1},{1,0,0,0},60] (* Harvey P. Dale, Feb 15 2015 *)

Binet-like formula: a(n) = Sum_{i=1...4} (r_i^n)/(3(r_i)^2+4(r_i)) where r_i is a root of x^4=x+1.
From R. J. Mathar, Mar 06 2008: (Start)
a(n) = A017817(n-5) for n >= 5.
O.g.f.: x(x-1)(1+x+x^2)/(x^4+x^3-1). (End)

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

A373742 Expansion of e.g.f. exp(x^3/6 * (1 + x)).

Original entry on oeis.org

1, 0, 0, 1, 4, 0, 10, 140, 560, 280, 8400, 92400, 385000, 800800, 16816800, 169569400, 784784000, 3811808000, 68803134400, 673546473600, 3693641952000, 30454440016000, 507477434464000, 5002277568288000, 33870732912016000, 386622918281600000

Offset: 0

Seiichi Manyama, Jun 16 2024

nonn

Cf. A361568, A372743.

Cf. A017817.

a(n) = n!*sum(k=0, n\3, binomial(k, n-3*k)/(6^k*k!));

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(k,n-3*k)/(6^k * k!).

a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 4*(n-3)*a(n-4)).

A099101 Quintisection of 1/(1-x^3-x^4).

Original entry on oeis.org

1, 0, 3, 5, 16, 43, 113, 316, 839, 2301, 6204, 16855, 45665, 123800, 335659, 909845, 2466760, 6686979, 18128529, 49145300, 133231279, 361184653, 979156724, 2654456239, 7196122817, 19508406192, 52886508243, 143373224101

Offset: 0

Paul Barry, Sep 29 2004

Index entries for linear recurrences with constant coefficients, signature (0,5,6,1).

Cf. A099099.

a(n)=sum{k=0..floor(5n/3), binomial(k, 5n-3k)}.
a(n)=A017817(5n).
G.f.: (1+x)*(x^2+x-1) / ( -1+5*x^2+6*x^3+x^4 ). - R. J. Mathar, Feb 19 2015

cp's OEIS Frontend

A099099 Quadrisection of a generalized Padovan sequence.

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A110064 a(n+4) = a(n+1) - a(n), a(0) = 1, a(1) = -4, a(2) = 0, a(3) = 1.

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A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A123552 Expansion of 1/(1 - x - x^3 + x^5).

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PARI

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A001584 A generalized Fibonacci sequence.

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A052697 Expansion of e.g.f. 1/(1-x^3-x^4).

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A127838 a(1) = 1, a(2) = a(3) = a(4) = 0; a(n) = a(n-4) + a(n-3) for n > 4.

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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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