A005689 -id:A005689 - 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)

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 122, 173, 249, 371, 575, 918, 1485, 2398, 3830, 6030, 9369, 14422, 22107, 33909, 52226, 80888, 125925, 196706, 307653, 480873, 750275, 1168085, 1815191, 2817518, 4371772, 6785606, 10539893, 16384908, 25488736

Offset: 0

Seiichi Manyama, Feb 28 2024

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

Cf. A003522, A100134, A137356.

Cf. A003520, A005689, A348289.

LinearRecurrence[{3, -3, 1, 0, 0, 0, 1}, Table[1, 7], 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\7, binomial(n-4*k, 3*k));

my(N=50, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3-x^7))

G.f.: (1-x)^2/((1-x)^3 - x^7).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7).

cp's OEIS Frontend

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

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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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Formula

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).

Views

Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

PARI

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4*k,3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).