A182097 -id:A182097 - cp's OEIS frontend

A228361 The number of all possible covers of L-length line segment by 2-length line segments with allowed gaps < 2.

0, 0, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456

Offset: 0

Philipp O. Tsvetkov, Aug 21 2013

Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Second row of A228360.

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

CoefficientList[Series[(1 - x^2 - x^3)^-1 (1 + x)^2 x^2 , {x, 0, 100}], x]

For n>1, a(n) = A134816(n).
G.f.: x^2*(1+x)^2/(1-x^2-x^3).
a(n) = a(n-2) +a(n-3) for n >= 5.
a(n) = A000931(n+5), n>1. - R. J. Mathar, Sep 02 2013

A020720 Pisot sequences E(7,9), P(7,9).

Original entry on oeis.org

7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
S. B. Ekhad, N. J. A. Sloane, and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

A subsequence of A000931.
See A008776 for definitions of Pisot sequences.
The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50] (* Jean-François Alcover, Aug 31 2018 *)
CoefficientList[Series[(7 + 9 x + 5 x^2)/(1 - x^2 - x^3), {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)

a(n) = a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

G.f.: (7+9*x+5*x^2) / (1-x^2-x^3). - Colin Barker, Jun 05 2016

A133034 First differences of Padovan sequence A000931.

Original entry on oeis.org

-1, 0, 1, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396

Offset: 0

Omar E. Pol, Nov 05 2007

Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A002026.

LinearRecurrence[{0,1,1},{-1,0,1},60] (* Harvey P. Dale, Dec 14 2013 *)

a(n+4) = A000931(n).
G.f.: ( 1-2*x^2 ) / ( -1+x^2+x^3 ). - R. J. Mathar, Sep 11 2011
a(n) = a(n-2) + a(n-3) with a(0) = -1, a(1) = 0, a(2) = 1. - Taras Goy, Mar 24 2019

A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233

Offset: 0

Seiichi Manyama, Mar 05 2019

A(n,k) is the number of compositions of n into parts k and k+1.

			Square array begins:
    1, 1, 1, 1, 1, 1, 1, 1, 1, ...
    1, 0, 0, 0, 0, 0, 0, 0, 0, ...
    2, 1, 0, 0, 0, 0, 0, 0, 0, ...
    3, 1, 1, 0, 0, 0, 0, 0, 0, ...
    5, 1, 1, 1, 0, 0, 0, 0, 0, ...
    8, 2, 0, 1, 1, 0, 0, 0, 0, ...
   13, 2, 1, 0, 1, 1, 0, 0, 0, ...
   21, 3, 2, 0, 0, 1, 1, 0, 0, ...
   34, 4, 1, 1, 0, 0, 1, 1, 0, ...
   55, 5, 1, 2, 0, 0, 0, 1, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.

Cf. A306646, A306680.

T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).

A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595

Offset: 0

Roger L. Bagula, Apr 29 2010

Except for signs the sequence is the essentially same as A078013, A050935 and A104769.

Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017

			G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Wikipedia, Padovan sequence.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1). [From _R. J. Mathar_, Apr 30 2010]

Cf. A000931, A005251, A050935, A078013, A104769, A182097.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018

a[0] := 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
b = Table[a[n], {n, 0, 50}];
Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, Abs@n}]]; (* Michael Somos, Dec 13 2013 *)

{a(n) = if( n>=0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Dec 13 2013 */

a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).
a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010
a(n) = -A104769(n) - A104769(n+1). - Ralf Stephan, Aug 18 2013
G.f.: 1 / (1 - x^3 / (1 + x)). - Michael Somos, Dec 13 2013
a(n) = A182097(-n) for all n in Z. - Michael Somos, Dec 13 2013
A000931(n) = a(n)^2 - a(n-1) * a(n+1). - Michael Somos, Dec 13 2013
Binomial transform is A005251(n+1). - Michael Somos, Dec 13 2013

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

At positions A000225 seems to occur the record values of this sequence: 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, ... which seem to match with A000931 (Padovan sequence), or more exactly, with A182097 (Number of compositions (ordered partitions) into parts 2 and 3). Note that these values give also the number of predecessors for each "repunit-pattern" (2^n)-1 in Rule 110 cellular automaton, as rules 110 and 124 are mirror images of each other.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000225, A000931, A182097, A269174.

Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).

(definec (A269175 n) (let loop ((p 0) (s 0)) (cond ((> p n) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0))))))) ;; Very straightforward and very slow.
;; Somewhat optimized version:
(definec (A269175 n) (if (zero? n) 1 (let ((nwid-1 (- (A000523 n) 1))) (let loop ((p (if (< n 2) 0 (A000079 nwid-1))) (s 0)) (cond ((> (A000523 p) nwid-1) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0)))))))))

A288025 Array read by antidiagonals: T(m,n) = number of minimal edge covers in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 6, 6, 1, 2, 17, 38, 17, 2, 2, 45, 190, 190, 45, 2, 3, 120, 1021, 1834, 1021, 120, 3, 4, 324, 5494, 19988, 19988, 5494, 324, 4, 5, 873, 29042, 208186, 419710, 208186, 29042, 873, 5, 7, 2349, 154772, 2177591, 8704085, 8704085, 2177591, 154772, 2349, 7

Offset: 1

Andrew Howroyd, Jun 04 2017

A minimal edge cover is an edge cover such that the removal of any edge in the cover destroys the covering property. Equivalently, these are the edge covers whose connected components are stars. A minimal edge cover is not the same as a minimum edge cover.

			Table starts:
================================================================
m\n| 1   2     3       4         5           6             7
---|------------------------------------------------------------
1  | 0   1     1       1         2           2             3 ...
2  | 1   2     6      17        45         120           324 ...
3  | 1   6    38     190      1021        5494         29042 ...
4  | 1  17   190    1834     19988      208186       2177591 ...
5  | 2  45  1021   19988    419710     8704085     179649371 ...
6  | 2 120  5494  208186   8704085   356269056   14484264119 ...
7  | 3 324 29042 2177591 179649371 14484264119 1163645044100 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Minimal Edge Cover

Main diagonal is A288027.
Rows 1-3 are A182097, A288029, A288030.
Cf. A286912.

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)binomial(n, j)x^(j - 1)*(1 - x)^(n - j).

Original entry on oeis.org

1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

			Triangle begins:
    1;
    2,  -1;
    3,  -3,   1;
    4,  -6,   4, 0;
    5, -10,  10, 0,   -3;
    6, -15,  20, 0,  -18,   10;
    7, -21,  35, 0,  -63,   70,   -24;
    8, -28,  56, 0, -168,  280,  -192,   49;
    9, -36,  84, 0, -378,  840,  -864,  441,  -89;
   10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Row sums: subsequence of A000931, A078027, A182097.

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.

a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten

@CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018

Edited, and new name by Franck Maminirina Ramaharo, Oct 22 2018

A288026 Array read by antidiagonals: T(m,n) = number of maximal matchings in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 5, 5, 2, 3, 11, 22, 11, 3, 4, 24, 75, 75, 24, 4, 5, 51, 264, 400, 264, 51, 5, 7, 109, 941, 2357, 2357, 941, 109, 7, 9, 234, 3286, 13407, 22228, 13407, 3286, 234, 9, 12, 503, 11623, 76667, 207423, 207423, 76667, 11623, 503, 12

Offset: 1

Andrew Howroyd, Jun 04 2017

			Table starts:
=====================================================
m\n| 1   2    3     4       5        6          7
---|-------------------------------------------------
1  | 1   1    2     2       3        4          5 ...
2  | 1   2    5    11      24       51        109 ...
3  | 2   5   22    75     264      941       3286 ...
4  | 2  11   75   400    2357    13407      76667 ...
5  | 3  24  264  2357   22228   207423    1922112 ...
6  | 4  51  941 13407  207423  3136370   47256485 ...
7  | 5 109 3286 76667 1922112 47256485 1158560776 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..276
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set

Main diagonal is A287595.
Rows 1-3 are A182097(n+2), A286945, A288028.
Cf. A210662, A099390, A286912, A288025.

A098523 Expansion of (1+x^2)/(1-x-x^5) = (1+x^2)/((1-x+x^2)*(1-x^2-x^3)).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 6, 8, 10, 13, 17, 23, 31, 41, 54, 71, 94, 125, 166, 220, 291, 385, 510, 676, 896, 1187, 1572, 2082, 2758, 3654, 4841, 6413, 8495, 11253, 14907, 19748, 26161, 34656, 45909, 60816, 80564, 106725, 141381, 187290, 248106

Offset: 0

Paul Barry, Sep 12 2004

The expansion of (1+kx^2)/(1-x-k^2*x^5) satisfies the recurrence a(n)=a(n-1)+k^2*a(n-5),a(0)=1,a(1)=1,a(2)=k+1,a(3)=k+1,a(4)=k+1, with a(n)=sum{k=0..floor(n/2), binomial(n-2k,floor(k/2))r^k}.

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

Cf. A097333.

CoefficientList[Series[(1+x^2)/(1-x-x^5),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,0,0,1},{1,1,2,2,2},50] (* Harvey P. Dale, Mar 05 2014 *)

a(n)=a(n-1)+a(n-5); a(n)=sum{k=0..floor(n/2), binomial(n-2k, floor(k/2))}.

7*a(n) = 8*A182097(n) +5*A182097(n-1) +3*A182097(n-2) - A010892(n) +3*A010892(n-1). - R. J. Mathar, Jul 07 2023

cp's OEIS Frontend

A228361 The number of all possible covers of L-length line segment by 2-length line segments with allowed gaps < 2.

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A020720 Pisot sequences E(7,9), P(7,9).

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A133034 First differences of Padovan sequence A000931.

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A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

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A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

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Magma

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PARI

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A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

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A288025 Array read by antidiagonals: T(m,n) = number of minimal edge covers in the grid graph P_m X P_n.

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A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j).

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Sage

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)binomial(n, j)x^(j - 1)*(1 - x)^(n - j).