A134816 -id:A134816 - cp's OEIS frontend

A124745 Expansion of (1+x)/(1-x^2+x^3).

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, -97229, 128801

Offset: 0

Paul Barry, Nov 06 2006

Paolo Xausa, Table of n, a(n) for n = 0..1000
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).

Row sums of A124744.

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}, {1, 1, 1}, 100] (* Paolo Xausa, Aug 27 2024 *)

a(n) = Sum_{k=0..n} C(floor(k/2),n-k)*(-1)^(n-k) = (-1)^n*A078027(n).

a(n) = a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 1. - Taras Goy, Mar 24 2019

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

Original entry on oeis.org

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

A138890 Non-Padovan numbers.

Original entry on oeis.org

6, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Omar E. Pol, Apr 05 2008

Natural numbers that are not in the Padovan sequence A000931.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A000931, A001690, A062289, A092460, A134816, A138836, A138888, A138891.

Complement[Range[0, Max[#]], #] &@ Union@ LinearRecurrence[{0, 1, 1}, {1, 0, 0}, 23] (* Michael De Vlieger, Sep 17 2024 *)

def A138890(n):
    def f(x):
        if x<=1: return n+1
        a, b, c, d = 1, 1, 1, 0
        while c<=x:
            a, b, c = b, c, a+b
            d += 1
        return n+d-1
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

A172353 Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 4, 2, 1, 1, 3, 6, 12, 6, 3, 1, 1, 4, 12, 24, 24, 12, 4, 1, 1, 5, 20, 60, 60, 60, 20, 5, 1, 1, 7, 35, 140, 210, 210, 140, 35, 7, 1, 1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1

Offset: 0

Roger L. Bagula, Feb 01 2010

Start from the Padovan sequence A134816 and its partial products A126772, extended by A126772(0)=1. Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 4, 8, 14, 32, 82, 232, 786, 2981,..

			1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 4, 2, 1;
1, 3, 6, 12, 6, 3, 1;
1, 4, 12, 24, 24, 12, 4, 1;
1, 5, 20, 60, 60, 60, 20, 5, 1;
1, 7, 35, 140, 210, 210, 140, 35, 7, 1;
1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1;

Cf. A010048, A172355

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A054405 Row sums of array T as in A055215.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 63, 84, 112, 149, 198, 263, 349, 463, 614, 814, 1079, 1430, 1895, 2511, 3327, 4408, 5840, 7737, 10250, 13579, 17989, 23831, 31570, 41822, 55403, 73394, 97227, 128799, 170623, 226028, 299424

Offset: 0

Clark Kimberling, May 07 2000

nonn

Partial sums of A134816. [From Omar E. Pol, Jan 02 2009]

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

Cf. A134816. [From Omar E. Pol, Jan 02 2009]

If you add 2 to each term, you get the Padovan sequence A000931. - Gabriel Nivasch (gnivasch(AT)yahoo.com), Oct 15 2003

Partial sums of Padovan sequence A000931(n+5). G.f. : (1+x)/(1-x-x^2+x^4); a(n)=a(n-1)+a(n-2)-a(n-4). - Paul Barry, Jun 21 2004, sign corrected R. J. Mathar, Oct 16 2010

Signs in formula corrected by R. J. Mathar, Oct 16 2010

A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, 18560, 39865, 85626, 183916, 395033, 848491, 1822473, 3914488, 8407925, 18059374, 38789712, 83316385, 178955183, 384377665, 825604416, 1773314929, 3808901426

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Essentially the same as A141015. - R. J. Mathar, Sep 14 2008

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

Subsequence of A000930.

Cf. A000931, A001906, A078027, A134816, A141015, A182097.

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x^2-x^3)/(1-x-2*x^2-x^3))); // G. C. Greubel, Jun 05 2018

(* b = A000931 *)
b[0]=0; b[1]=1; b[2]=1; b[n_]:= b[n]= b[n-2] + b[n-3];
a[1]=1; a[n_]:= a[n]= Sum[b[k]*a[n-k], {k,n-1}];
Table[a[n], {n, 35}]
(* or *)
LinearRecurrence[{1, 2, 1}, {1, 1, 2, 4}, 31] (* Georg Fischer, Mar 23 2019 *)

x='x+O('x^35); Vec(x*(1-x^2-x^3)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Jun 05 2018

a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1. [That is, b(n) = A000931(n+4) = A078027(n+6) = A134816(n) = A182097(n+1). - Petros Hadjicostas, Aug 09 2020]
From Colin Barker, Feb 01 2012: (Start)
a(n) = a(n-1) + 2*a(n-2) + a(n-3), n > 4.
G.f.: x*(1 - x^2 - x^3)/(1 - x - 2*x^2 - x^3). (End)
a(n) = A000930(2*n - 3) for n >= 3. - Georg Fischer, Mar 23 2019

A228360 Table read by antidiagonals: T(l,L) is the number of all possible covers of L-length line segment by l-length line segments with allowed gaps < l.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 4, 3, 1, 0, 0, 0, 0, 1, 5, 3, 2, 0, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 0, 1, 9, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Philipp O. Tsvetkov, Aug 21 2013

Second row is A228361 which also corresponds to Padovan's spiral numbers A134816 for n>1.
Third row is A228362.
T(l,L) is also the number of compositions of L where parts do not exceeds l and where are no two adjacent parts less than l.
T(2,5) = 3: [2,2,1], [2,1,2], [1,2,2]
T(2,9) = 9: [2,2,2,2,1], [2,2,2,1,2], [2,2,1,2,2], [2,1,2,2,2], [1,2,2,2,2], [2,1,2,1,2,1], [1,2,2,1,2,1], [1,2,1,2,2,1], [1,2,1,2,1,2]
T(3,8) = 6: [3,3,2], [3,1,3,1], [3,2,3], [1,3,3,1], [1,3,1,3], [2,3,3]

			Table starts:
0, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1, ...
0, 0, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, ...
0, 0, 0, 1, 2, 3, 3, 4, 6, 8, 10, 13, 18, 24, 31, ...
0, 0, 0, 0, 1, 2, 3, 4, 4, 5,  7, 10, 13, 16, 20, ...
0, 0, 0, 0, 0, 1, 2, 3, 4, 5,  5,  6,  8, 11, 15, ...
0, 0, 0, 0, 0, 0, 1, 2, 3, 4,  5,  6,  6,  7,  9, ...
0, 0, 0, 0, 0, 0, 0, 1, 2, 3,  4,  5,  6,  7,  7, ...
0, 0, 0, 0, 0, 0, 0, 0, 1, 2,  3,  4,  5,  6,  7, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 1,  2,  3,  4,  5,  6, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1,  2,  3,  4,  5, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  1,  2,  3,  4, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  1,  2,  3, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  1,  2, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  1, ...
.....................................................

Gf[l_, z] := (1 - Sum[z^i, {i, l, 2 l - 1}])^-1*Sum[z^i, {i, 0, l - 1}]^2*z^l
T[l_, L_] := CoefficientList[Series[Gf[l, z], {z, 0, 100}], z][[L + 1]]
Table[T[n - b + 1, b - 1], {n, 1, 30}, {b, n, 1, -1}] // Flatten

For all l>=1:
G.f.: (1 - Sum[x^i, {i, l, 2 l - 1}])^-1*Sum[x^i, {i, 0, l - 1}]^2*x^l.
G.f. for l=1: x/(1-x).
G.f. for l=2: x^2*(1+x)^2/(1-x^2-x^3).
G.f. for l=3: x^3*(1 + x + x^2)^2/(1 - x^3 - x^4 - x^5).
For l>1, L>=0:
c[k, l, m] = Sum[(-1)^i binomial[k - 1 - i*l, m - 1] binomial[m, i], {i, 0, floor[(k - m)/l]}] // number of compositions of k into exactly m parts which do not exceed l.
a[L, l, m] = Sum[ binomial[m + 1, m + 1 - j]*c[L - l*m, l - 1, j], {j, 0, m + 1}] //the number of all possible covers of L-length line segment by m l-length line segments.
T[l, L] := Sum[a[L, l, j], {j, 1, ceiling[L/l]}].

A238389 Expansion of (1+x)/(1-x^2-3*x^3).

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 37, 67, 94, 178, 295, 460, 829, 1345, 2209, 3832, 6244, 10459, 17740, 29191, 49117, 82411, 136690, 229762, 383923, 639832, 1073209, 1791601, 2992705, 5011228, 8367508, 13989343, 23401192, 39091867, 65369221, 109295443

Offset: 0

Sergio Falcon, Feb 26 2014

			a(3) = 3*a(0)+a(1) = 4; a(4) = 3*a(1)+a(2) = 4; a(5) = 3*a(2)+a(3) = 7.

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

Cf. A006190, A134816, A159284, A213713.

[n le 3 select 1 else Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, May 09 2021

a:= n-> (<<0|1|0>, <0|0|1>, <3|1|0>>^n.<<(1$3)>>)[(1$2)]:
seq(a(n), n=0..44);  # Alois P. Heinz, May 09 2021

(* First program *)
For[j=0, j<3, j++, a[j] = 1]
For[j=3, j<51, j++, a[j] = 3a[j-3] + a[j-2]]
Table[a[j], {j, 0, 50}]
(* Second program *)
CoefficientList[Series[(1+x)/(1-x^2-3x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)
LinearRecurrence[{0,1,3},{1,1,1},40] (* Harvey P. Dale, Feb 28 2023 *)

Vec((1+x)/(1-x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

def A238389_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-x^2-3*x^3) ).list()
A238389_list(40) # G. C. Greubel, May 09 2021

a(0)=1, a(1)=1, a(2)=1; for n>2, a(n) = a(n-2) + 3*a(n-3).
a(2n) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-2)/3} binomial(n-1-j,2j+1)*3^(2j+1).
a(2n+1) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-1)/3} binomial(n-j,2j+1)*3^(2j+1).
a(n) = |A106855(n)| + |A106855(n-1)| . - R. J. Mathar, Mar 13 2014

Terms corrected by Charles R Greathouse IV, Mar 06 2014

cp's OEIS Frontend

A124745 Expansion of (1+x)/(1-x^2+x^3).

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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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A138890 Non-Padovan numbers.

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A172353 Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).

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A054405 Row sums of array T as in A055215.

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A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

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A228360 Table read by antidiagonals: T(l,L) is the number of all possible covers of L-length line segment by l-length line segments with allowed gaps < l.

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