A005253 -id:A005253 - 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).

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)

A005687 Number of Twopins positions.

Original entry on oeis.org

1, 2, 4, 6, 9, 14, 22, 36, 57, 90, 139, 214, 329, 506, 780, 1200, 1845, 2830, 4337, 6642, 10170, 15572, 23838, 36486, 55828, 85408, 130641, 199814, 305599, 467366, 714735, 1092980, 1671335, 2555650, 3907781, 5975202, 9136288, 13969560, 21359528

Offset: 7

N. J. A. Sloane

R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
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 (2, 0, -2, 1, 2, -2, 0, 0, 0, -1).

a:= n-> (Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [2,0,-2,1,2,-2,0,0,0,-1][i] else 0 fi)^n)[1,8]: seq(a(n), n=7..70); # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{2, 0, -2, 1, 2, -2, 0, 0, 0, -1}, {1, 2, 4, 6, 9, 14, 22, 36, 57, 90}, 40] (* Jean-François Alcover, Nov 12 2015 *)

G.f.: x^7/((1-x^2-x^5)*(1-2*x+x^2-x^5)). - Simon Plouffe in his 1992 dissertation.

2*a(n) = A005253(n-2) - A005686(n). - R. J. Mathar, May 29 2019

More terms from Alois P. Heinz, Aug 14 2008

A098577 a(n) = Sum_{k=0..floor(n/5)} C(n-3k,2k) * 2^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 13, 21, 31, 47, 77, 133, 231, 391, 645, 1053, 1727, 2863, 4781, 7989, 13303, 22071, 36565, 60621, 100655, 167295, 278077, 461989, 767143, 1273607, 2114661, 3511869, 5833055, 9688527, 16091213, 26723221, 44378967, 73700823

Offset: 0

Paul Barry, Sep 16 2004

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

Cf. A098575, A005253.

I:=[1,1,1,1,1,]; [n le 5 select I[n] else 2*Self(n-1) -Self(n-2) + 2*Self(n-5): n in [1..30]]; // G. C. Greubel, Feb 03 2018

LinearRecurrence[{2,-1,0,0,2},{1,1,1,1,1},40] (* Harvey P. Dale, Feb 11 2015 *)
CoefficientList[Series[(1-x)/((1-x)^2-2*x^5), {x,0,50}], x] (* G. C. Greubel, Feb 03 2018 *)

x='x+O('x^30); Vec((1-x)/((1-x)^2-2*x^5)) \\ G. C. Greubel, Feb 03 2018

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

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

A113032 a(n) = Sum_{k=0..floor(n/8)} binomial(n-5k, 3k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 121, 167, 228, 315, 449, 666, 1023, 1605, 2533, 3974, 6156, 9394, 14137, 21051, 31159, 46066, 68305, 101850, 152857, 230720, 349576, 530476, 804579, 1217951, 1838897, 2769267, 4161918, 6247570, 9375799

Offset: 0

Alexey Kistanov (plast(AT)solid.ru), Jan 05 2006

			a(10+1)=11 because C(10,0) + C(5,3) = 1+10 = 11.

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1).

Cf. A005251, A005253, A000045.

[(&+[Binomial(n-5*k, 3*k): k in [0..Floor(n/8)]]): n in [0..50]]; // G. C. Greubel, Apr 09 2018

Table[Sum[Binomial[n - 5*k, 3*k], {k, 0, Floor[n/8]}], {n, 0, 50}] (* G. C. Greubel, Apr 09 2018 *)

a(n) = sum(k=0, n\8, binomial(n-5*k, 3*k)); \\ Michel Marcus, Sep 05 2013

lista(nn) = {my(x = xx + O(xx^nn)); gf = (1-x)^2/(1-3*x+3*x^2-x^3-x^8); for (i=0, nn-1, print1(polcoeff(gf, i, xx), ", "));} \\ Michel Marcus, Sep 05 2013

((1-x)^2/(1-3*x+3*x^2-x^3-x^8)).series(x, 44).coefficients(x, sparse=False) # Stefano Spezia, Aug 19 2023

G.f.: (1-x)^2/(1-3*x+3*x^2-x^3-x^8). [corrected by Georg Fischer, Apr 17 2020]

Corrected by T. D. Noe, Nov 01 2006

More terms from Michel Marcus, Sep 05 2013

cp's OEIS Frontend

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

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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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A005687 Number of Twopins positions.

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A098577 a(n) = Sum_{k=0..floor(n/5)} C(n-3*k,2*k) * 2^k.

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Magma

Mathematica

PARI

Formula

A113032 a(n) = Sum_{k=0..floor(n/8)} binomial(n-5*k, 3*k).

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Magma

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Formula

Extensions

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

A098577 a(n) = Sum_{k=0..floor(n/5)} C(n-3k,2k) * 2^k.

A113032 a(n) = Sum_{k=0..floor(n/8)} binomial(n-5k, 3k).