A086570 -id:A086570 - cp's OEIS frontend

A115291 Expansion of (1+x)^3/(1-x).

1, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Paul Barry, Jan 19 2006

Partial sums are A086570. Partial sums of squares are A115295. Correlation triangle is A115292.
Let m=4. We observe that a(n) = Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A113311 and A040000: it is the same formula with respectively m=3 and m=2. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
Also continued fraction expansion of (132-sqrt(17))/103. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 1331/9000. - Vincenzo Librandi, Sep 23 2011

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

Cf. A040000, A113311, A171418, A171440, A171441, A171442, A171443.

CoefficientList[Series[(1+x)^3/(1-x),{x,0,100}],x] (* or *) PadRight[ {1,4,7},120,{8}] (* Harvey P. Dale, May 23 2016 *)

a(n) = 8 - C(2, n) - 2*C(1, n) - 4*C(0, n).
a(n) = Sum_{k=0..n} C(3, k).
a(n) = A004070(n, 3).
From Elmo R. Oliveira, Aug 09 2024: (Start)
E.g.f.: 8*exp(x) - 7 - 4*x - x^2/2.
a(n) = 8, n > 2. (End)

A188553 T(n,k) = Number of n X k binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 7, 5, 6, 12, 12, 9, 6, 7, 17, 20, 16, 11, 7, 8, 23, 32, 28, 20, 13, 8, 9, 30, 49, 48, 36, 24, 15, 9, 10, 38, 72, 80, 64, 44, 28, 17, 10, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12, 13, 68, 187, 303, 321, 256, 176

Offset: 1

R. H. Hardin, Apr 04 2011

From Miquel A. Fiol, Feb 06 2024: (Start)
Also, T(n,k) is the number of words of length k, x(1)x(2)...x(k), on the alphabet {0,1,...,n}, such that, for i=2,...,k, x(i)=either x(i-1) or x(i)=x(i-1)-1.
For the bijection between arrays and sequences, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to n of 1's.
Such a bijection implies that all the empirical/conjectured formulas in A188554, A188555, A188556, A188557, A188558, and A188559 become correct.
(End)

			Table starts
..2..3..4..5...6...7...8...9...10...11...12....13....14....15....16.....17
..3..5..8.12..17..23..30..38...47...57...68....80....93...107...122....138
..4..7.12.20..32..49..72.102..140..187..244...312...392...485...592....714
..5..9.16.28..48..80.129.201..303..443..630...874..1186..1578..2063...2655
..6.11.20.36..64.112.192.321..522..825.1268..1898..2772..3958..5536...7599
..7.13.24.44..80.144.256.448..769.1291.2116..3384..5282..8054.12012..17548
..8.15.28.52..96.176.320.576.1024.1793.3084..5200..8584.13866.21920..33932
..9.17.32.60.112.208.384.704.1280.2304.4097..7181.12381.20965.34831..56751
.10.19.36.68.128.240.448.832.1536.2816.5120..9217.16398.28779.49744..84575
.11.21.40.76.144.272.512.960.1792.3328.6144.11264.20481.36879.65658.115402
Some solutions for 5 X 3:
  1 1 1   1 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 0 0   1 1 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 0   0 0 0   1 0 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 0 0   0 0 0   0 0 0   1 1 0
Some solutions for T(5,3): By taking the sums of the columns in the above arrays we get 555, 100, 000, 543, 322, 432, 554. - _Miquel A. Fiol_, Feb 04 2024

R. H. Hardin, Table of n, a(n) for n = 1..9934

Diagonal is A045623.
Column 4 is A086570.
Rows 2..8 are A022856(n+4), A188554, A188555, A188556, A188557, A188558, A188559.
Upper diagonals T(n,n+i) for i=1..8 give: A001792, A001787(n+1), A000337(n+1), A045618, A045889, A034009, A055250, A055251.
Lower diagonals T(n+i,n) for i=1..7 give: A045891(n+1), A034007(n+2), A111297(n+1), A159694(n-1), A159695(n-1), A159696(n-1), A159697(n-1).
Antidiagonal sums give A065220(n+5).

T:= (n,k)-> `if`(k<=n+1, (2*n+3-k)*2^(k-2), (n+1-k)*binomial(k-1, n) * add(binomial(n, j-1)/(k-j)*T(n, j)*(-1)^(n-j), j=1..n+1)): seq(seq(T(n, 1+d-n), n=1..d), d=1..15); #Alois P. Heinz in the Sequence Fans Mailing List, Apr 04 2011 [We do not permit programs based on conjectures, but this program is now justified by Fiol's comment. - N. J. A. Sloane, Mar 09 2024]

Empirical: T(n,k) = (n+1)*2^(k-1) + (1-k)*2^(k-2) for k < n+3, and then the entire row n is a polynomial of degree n in k.
From Miquel A. Fiol, Feb 06 2024: (Start)
The above empirical formula is correct.
It can be proved that T(n,k) satisfies the recurrence
  T(n,k) = Sum_{r=1..n+1} (-1)^(r+1)*binomial(n+1,r)*T(n,k-r)
with initial values
  T(n,k) = Sum_{r=0..k-1} (n+1-r)*binomial(k-1,r) for k = 1..n+1. (End)

A113128 A simple 4-diagonal matrix based on (1+x)^3.

Original entry on oeis.org

1, 3, 2, 3, 6, 3, 1, 6, 9, 4, 0, 2, 9, 12, 5, 0, 0, 3, 12, 15, 6, 0, 0, 0, 4, 15, 18, 7, 0, 0, 0, 0, 5, 18, 21, 8, 0, 0, 0, 0, 0, 6, 21, 24, 9, 0, 0, 0, 0, 0, 0, 7, 24, 27, 10, 0, 0, 0, 0, 0, 0, 0, 8, 27, 30, 11, 0, 0, 0, 0, 0, 0, 0, 0, 9, 30, 33, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 33, 36, 13, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 14 2005

Row sums are A086570. Diagonal sums are the odd numbers A005408.

			Triangle begins
1;
3, 2;
3, 6, 3;
1, 6, 9, 4;
0, 2, 9, 12, 5;
0, 0, 3, 12, 15, 6;
0, 0, 0, 4, 15, 18, 7;
0, 0, 0, 0, 5, 18, 21, 8;

Number triangle where column k has g.f. (1+x)^3*(k+1)x^k.

A281813 a(0) = 3, a(n) = 8*n + 4 for n > 0.

Original entry on oeis.org

3, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404

Offset: 0

Rayan Ivaturi, Jan 30 2017

Consider a 1 X S rectangle on an infinite grid and surround it successively with the minimum number of 1 X 1 tiles: the initial S on step 0, 2S + 6 on step 1, 2S + 14 on step 2, and so on. This sequence is case S = 3. See Ivaturi link for a connection to sieving for primes.

Rayan Ivaturi, Ripple Numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017113.

Other 'ripple sequences': A022144 (s=1), A017089 (s=2).

a(n)=if(n>0, 8*n+4, 3) \\ Charles R Greathouse IV, Feb 07 2017

G.f.: (3 + 6*x - x^2)/(1 - x)^2.
a(n) = A017113(n) for n>0, a(0) = 3.
a(n) = A086570(n+1) for n>=1. - R. J. Mathar, Jun 21 2025

Entry revised by Editors of OEIS, Feb 09 2017

cp's OEIS Frontend

A115291 Expansion of (1+x)^3/(1-x).

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A188553 T(n,k) = Number of n X k binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

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A113128 A simple 4-diagonal matrix based on (1+x)^3.

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A281813 a(0) = 3, a(n) = 8*n + 4 for n > 0.

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A090829 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A085163/A085164.

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