A005252 -id:A005252 - cp's OEIS frontend

A098574 a(n) = Sum_{k=0..floor(n/7)} C(n-5k,2k).

1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 38, 51, 71, 102, 149, 218, 316, 452, 639, 897, 1257, 1766, 2493, 3536, 5031, 7165, 10196, 14484, 20538, 29085, 41168, 58282, 82561, 117036, 165995, 235492, 334074, 473824, 671856, 952449, 1350078, 1913702

Offset: 0

Paul Barry, Sep 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,5,2).
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1).

Cf. A005251, A005252, A005253, A005689.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)/(1-2*x+x^2-x^7))) // G. C. Greubel, Feb 03 2018

CoefficientList[Series[(1-x)/(1-2*x+x^2-x^7), {x,0,50}], x] (* G. C. Greubel, Feb 03 2018 *)

a(n) = sum(k=0, n\7, binomial(n-5*k, 2*k)); \\ Michel Marcus, Sep 06 2017

x='x+O('x^30); Vec((1-x)/(1-2*x+x^2-x^7)) \\ G. C. Greubel, Feb 03 2018

G.f.: (1-x)/(1-2*x+x^2-x^7).

A106511 Expansion of g.f. (1+x)^2/((1 + x + x^2)*(1 + x - x^2)).

Original entry on oeis.org

1, 0, 0, 0, 1, -2, 3, -4, 6, -10, 17, -28, 45, -72, 116, -188, 305, -494, 799, -1292, 2090, -3382, 5473, -8856, 14329, -23184, 37512, -60696, 98209, -158906, 257115, -416020, 673134, -1089154, 1762289, -2851444, 4613733, -7465176, 12078908, -19544084, 31622993

Offset: 0

Paul Barry, May 04 2005

Diagonal sums of the Riordan array ((1+x)/(1+x+x^2), x/(1+x)), A106509.

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

Cf. A000045, A005252, A011646, A024490, A039834, A106509.

I:=[1,0,0,0]; [n le 4 select I[n] else -2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 12 2014

CoefficientList[Series[(1 + x)^2/((1 + x + x^2)(1 + x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jan 12 2014 *)

a(n) = (fibonacci(1-n) + 1 - n%3) >> 1; \\ Kevin Ryde, Apr 29 2021

def A005252(n): return sum( binomial(n-2*k, 2*k) for k in (0..n//4) )
def A106511(n): return (-1)^n*( fibonacci(n-1) - A005252(n-2) )
[A106511(n) for n in (0..45)] # G. C. Greubel, Apr 29 2021

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} (-1)^j*binomial(2n-3k-j, j).
a(n) = (1/2)*((-1)^n*Fibonacci(n) + Kronecker(-3,n)). - Ralf Stephan, Jun 02 2007
a(n) = -2*a(n-1) - a(n-2) + a(n-4), a(0)=1, a(1)=a(2)=a(3)=0. - Philippe Deléham, Jan 12 2014
a(n) = (-1)^n*(Fibonacci(n-1) - A005252(n-2)), n>=2. - Katharine Ahrens, May 05 2019
E.g.f.: exp(-x/2)*(15*cos(sqrt(3)*x/2) + 15*cosh(sqrt(5)*x/2) + 5*sqrt(3)*sin(sqrt(3)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - Stefano Spezia, Oct 19 2023

A348290 a(n) = Sum_{k=0..floor(n/10)} binomial(n-5k,5k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 253, 463, 793, 1288, 2003, 3005, 4380, 6255, 8855, 12630, 18508, 28358, 45783, 77408, 134883, 237888, 418513, 727513, 1243163, 2083888, 3426771, 5535911, 8808206, 13850761, 21615771, 33638409, 52455339, 82332229, 130506914, 209273284

Offset: 0

Seiichi Manyama, Oct 10 2021

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1,0,0,0,0,1).

Cf. A000045, A005252, A100134, A348289.

a(n) = sum(k=0, n\10, binomial(n-5*k, 5*k));

my(N=66, x='x+O('x^N)); Vec((1-x)^4/((1-x)^5-x^10))

G.f.: (1-x)^4/((1-x)^5 - x^10).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + a(n-10).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 85, 212, 476, 1016, 2172, 4825, 11213, 26763, 64095, 151851, 354737, 820328, 1889968, 4361521, 10106859, 23509678, 54793282, 127709888, 297336790, 691382201, 1606284377, 3731020629, 8668253125, 20146856893, 46840732201, 108918637566, 253262275888

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-14,6,-1).

Cf. A003269, A003522, A005252, A038503, A099131.

Cf. A107025, A373904, A373905.

a(n) = sum(k=0, n\4, binomial(n+2*k, n-4*k));

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6).

G.f.: 1/(1 - x - x^4/(1 - x)^5).

A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 10, 11, 17, 17, 28, 27, 45, 44, 72, 72, 116, 117, 188, 189, 305, 305, 494, 493, 799, 798, 1292, 1292, 2090, 2091, 3382, 3383, 5473, 5473, 8856, 8855, 14329, 14328, 23184, 23184, 37512, 37513, 60696

Offset: 1

Mark Dols, Mar 23 2010

Combination a(2n)=A005252(n-1) and a(2n+1)=A024490(n). Consecutive pairs add up to A000045 and subtract to A010892. If a(1)= 1 formula gives: A103609.

			As consecutive pairs: (0,1),(0,1),(1,1),(2,1),(3,2),(4,4),...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,1).

Cf. A000045, A005252, A010892, A024490, A079977, A110161.

R:=PowerSeriesRing(Integers(), 70);
[0] cat Coefficients(R!( x^2*(1-x^2+x^3)/((1-x^2+x^4)*(1-x^2-x^4)) )); // G. C. Greubel, Oct 23 2024

nxt[{n_,a_,b_,c_,d_,e_}]:={n+1,b,c,d,e,If[EvenQ[n],d+c,d+a]}; NestList[nxt,{5,0,1,0,1,1},50][[All,2]] (* or *) LinearRecurrence[ {0,2,0,-1,0,0,0,1},{0,1,0,1,1,1,2,1},60] (* Harvey P. Dale, Nov 15 2019 *)

def A174618(n): return (kronecker(12,n-3) - kronecker(12,n-2) + ((n+1)%2)*fibonacci(n//2) + (n%2)*fibonacci((n+1)//2))//2
[A174618(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

G.f.: x^2*(1-x^2+x^3) / ( (1-x^2+x^4)*(1-x^2-x^4) ). - R. J. Mathar, Jan 27 2011

a(n) = (1/2)*(A110161(n-3) - A110161(n-2) + A079977(n-2) + A079977(n-1)). - G. C. Greubel, Oct 23 2024

A217839 T(n,k)=Number of n element 0..1 arrays with each element the minimum of k adjacent elements of a random 0..1 array of n+k-1 elements.

Original entry on oeis.org

2, 2, 4, 2, 4, 8, 2, 4, 7, 16, 2, 4, 7, 12, 32, 2, 4, 7, 11, 21, 64, 2, 4, 7, 11, 17, 37, 128, 2, 4, 7, 11, 16, 27, 65, 256, 2, 4, 7, 11, 16, 23, 44, 114, 512, 2, 4, 7, 11, 16, 22, 34, 72, 200, 1024, 2, 4, 7, 11, 16, 22, 30, 52, 117, 351, 2048, 2, 4, 7, 11, 16, 22, 29, 42, 81, 189, 616

Offset: 1

R. H. Hardin Oct 12 2012

Table starts
.....2....2....2....2...2...2...2...2...2...2...2...2...2...2
.....4....4....4....4...4...4...4...4...4...4...4...4...4...4
.....8....7....7....7...7...7...7...7...7...7...7...7...7...7
....16...12...11...11..11..11..11..11..11..11..11..11..11..11
....32...21...17...16..16..16..16..16..16..16..16..16..16..16
....64...37...27...23..22..22..22..22..22..22..22..22..22..22
...128...65...44...34..30..29..29..29..29..29..29..29..29..29
...256..114...72...52..42..38..37..37..37..37..37..37..37..37
...512..200..117...81..61..51..47..46..46..46..46..46..46..46
..1024..351..189..126..91..71..61..57..56..56..56..56..56..56
..2048..616..305..194.137.102..82..72..68..67..67..67..67..67
..4096.1081..493..296.205.149.114..94..84..80..79..79..79..79
..8192.1897..798..450.303.218.162.127.107..97..93..92..92..92
.16384.3329.1292..685.443.316.232.176.141.121.111.107.106.106
.32768.5842.2091.1046.644.452.331.247.191.156.136.126.122.121

			Some solutions for n=8 k=4
..0....0....1....0....1....0....1....1....0....1....0....0....0....0....0....0
..1....1....1....1....0....0....1....1....1....0....0....1....0....1....0....0
..1....1....1....1....0....1....0....1....1....0....1....0....1....0....0....0
..1....1....0....0....0....0....0....0....0....0....1....0....1....0....1....0
..1....1....0....0....0....0....0....0....0....0....1....0....1....0....1....1
..1....1....0....0....0....0....0....0....0....1....1....0....1....0....1....0
..1....0....0....0....0....0....0....0....0....1....1....0....0....1....1....0
..0....0....1....0....1....0....1....0....1....1....1....1....0....1....0....0

R. H. Hardin, Table of n, a(n) for n = 1..435

Column 2 is A005251(n+3)
Column 3 is A005252(n+3)
Column 4 is A005253(n+3)
Column 5 is A005689(n+6)
Column 6 is A098574(n+6)
Diagonal is A000124

Empirical for columns 1-7: a(n) = 2*a(n-1) -a(n-2) +a(n-k-1)

A110317 Triangle read by rows: T(n,k) (k>=0) is the number of RNA secondary structures of size n (i.e., with n nodes) having k arcs that are covered by other arcs.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 1, 12, 5, 21, 15, 1, 37, 37, 8, 65, 84, 35, 1, 114, 182, 115, 12, 200, 381, 323, 73, 1, 351, 777, 825, 313, 17, 616, 1554, 1977, 1087, 138, 1, 1081, 3062, 4524, 3291, 754, 23, 1897, 5962, 9999, 9063, 3209, 241, 1, 3329, 11496, 21515, 23300

Offset: 0

Emeric Deutsch, Jul 19 2005

Rows 0,1,2 have one term each; row n >= 3 has ceiling(n/2) - 1 terms.
Rows sums yield A004148.
T(n,0) = A005251(n+1).
Sum_{k>=0} k*T(n,k) = A110318(n-5).

			T(6,1)=5 because we have 15/(24)/3/6, 16/(24)/3/5, 16/(25)/3/4, 16/2/(35)/4 and 1/26/(35)/4 (the covered arcs are shown between parentheses).
Triangle begins
   1;
   1;
   1;
   2;
   4;
   7,  1;
  12,  5;
  21, 15,  1;
  37, 37,  8;

W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A005252, A110318.

G:=2*t/(2*t-2*z*t-1+z+t*z^2+sqrt(1-2*z-2*t*z^2+z^2-2*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 17 do seq(coeff(t*P[n],t^k),k=1..ceil(n/2)-1) od; # yields sequence in triangular form

G.f.: 2t/(2t - 2tz - 1 + z + tz^2 + sqrt(1 - 2z - 2tz^2 + z^2 - 2tz^3 + t^2*z^4)).

A259964 A specially constructed B_2 sequence with sum of reciprocals greater than that of the Mian-Chowla sequence A005282.

Original entry on oeis.org

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 229, 257, 290, 312, 381, 419, 467, 507, 621, 721, 770, 864, 927, 1050, 1178, 1289, 1457, 1561, 1615, 1774, 1907, 2090, 2164, 2263, 2309, 2539, 2800, 2938, 3035, 3310, 3380, 3738, 4043, 4239, 4439, 4726, 4851, 5016, 5169, 5289, 5490, 5760, 6646, 6843, 7015, 7442, 7674, 7986, 8284, 8506, 8772, 9240, 9778, 9996, 10344, 10431, 11614, 12263

Offset: 1

N. J. A. Sloane, Jul 11 2015

nonn

Constructed using the greedy algorithm (as for A005252) except at 15th term, which is 229.

It had been conjectured that the sum of reciprocals for B_2 sequences was maximized by A005282. This sequence gives a counterexample.

Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.

Cf. A005282.

cp's OEIS Frontend

A098574 a(n) = Sum_{k=0..floor(n/7)} C(n-5*k,2*k).

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A106511 Expansion of g.f. (1+x)^2/((1 + x + x^2)*(1 + x - x^2)).

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A348290 a(n) = Sum_{k=0..floor(n/10)} binomial(n-5*k,5*k).

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A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2*k,n-4*k).

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A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

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A217839 T(n,k)=Number of n element 0..1 arrays with each element the minimum of k adjacent elements of a random 0..1 array of n+k-1 elements.

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A110317 Triangle read by rows: T(n,k) (k>=0) is the number of RNA secondary structures of size n (i.e., with n nodes) having k arcs that are covered by other arcs.

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A259964 A specially constructed B_2 sequence with sum of reciprocals greater than that of the Mian-Chowla sequence A005282.

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A098574 a(n) = Sum_{k=0..floor(n/7)} C(n-5k,2k).

A348290 a(n) = Sum_{k=0..floor(n/10)} binomial(n-5k,5k).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).