A000931 -id:A000931 - cp's OEIS frontend

A134727 Successive digits of members of the Padovan sequence A000931(n).

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

Offset: 1

Omar E. Pol, Nov 10 2007

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000931.

Flatten[IntegerDigits/@ LinearRecurrence[{0,1,1},{1,0,0},50]] (* Harvey P. Dale, Dec 31 2022 *)

A140514 a(n) = A000931(n+4) - A010060(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 7, 9, 11, 16, 20, 27, 37, 48, 65, 86, 113, 151, 199, 264, 351, 465, 615, 815, 1081, 1431, 1897, 2513, 3328, 4409, 5842, 7739, 10251, 13581, 17990, 23832, 31572, 41824, 55404, 73395, 97229, 128800, 170625, 226030, 299425, 396655

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Thue-Morse sequence.

Cf. A000931, A010060.

A000931 := LinearRecurrence[{0, 1, 1}, {1, 0, 0}, 500]; Table[A000931[[n + 5]] - ThueMorse[n], {n, 0, 100}] (* G. C. Greubel, Jan 14 2018 *)

a(n) = A000931(n+4) - A010060(n).

Definition simplified, indices clarified, references converted to links, Nov 16 2010

A144401 Padovan ( A000931) version of A038137: expansion of polynomials as antidiagonal: p(x,n)=1/(1-x-x^3)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 3, 1, 5, 10, 13, 11, 4, 1, 6, 15, 24, 27, 18, 6, 1, 7, 21, 40, 55, 51, 30, 9, 1, 8, 28, 62, 100, 116, 94, 50, 13, 1, 9, 36, 91, 168, 231, 234, 171, 81, 19, 1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28, 1, 11, 55, 174, 402, 714, 987, 1065

Offset: 1

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

Row sums are: 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332 (cf. A008998).
These polynomials are sort of pseudo-combinations with the last element Padovan instead of one.
If you subtract the binomial triangle sequence you get:
{0},
{0, 0},
{0, 0, 0},
{0, 0, 0, 1},
{0, 0, 0, 2, 2},
{0, 0, 0, 3, 6, 3},
{0, 0, 0, 4, 12, 12, 5},
{0, 0, 0, 5, 20, 30, 23, 8},
{0, 0, 0, 6, 30, 60, 66, 42, 12}

			{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 2},
{1, 4, 6, 6, 3},
{1, 5, 10, 13, 11, 4},
{1, 6, 15, 24, 27, 18, 6},
{1, 7, 21, 40, 55, 51, 30, 9},
{1, 8, 28, 62, 100, 116, 94, 50, 13},
{1, 9, 36, 91, 168, 231, 234, 171, 81, 19},
{1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28},
{1, 11, 55, 174, 402, 714, 987, 1065, 879, 527, 208, 41},
{1, 12, 66, 230, 585, 1152, 1792, 2220, 2175, 1640, 906, 330, 60},
{1, 13, 78, 297, 825, 1782, 3072, 4278, 4815, 4320, 3006, 1539, 520, 88},
{1, 14, 91, 376, 1133, 2662, 5028, 7752, 9807, 10122, 8391, 5424, 2586, 816, 129}

Cf. A000931, A038137.

Clear[f, b, a, g, h, n, t]; f[t_, n_] = 1/(1 - t - t^3)^n; a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b]

p(x,n)=1/(1-x-x^3)^n; t(n,m)=anti_diagonal_expansion(p(x,n)).

A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).

Original entry on oeis.org

0, 1, -1, 1, 0, -3, 10, -24, 49, -89, 145, -208, 245, -174, -176, 1121, -3185, 7137, -13920, 24301, -37926, 51256, -53615, 20407, 97265, -386224, 984549, -2083934, 3896480, -6537023, 9734175, -12231999, 10690624, 2126301, -39992150, 126414472, -297132815, 598577351, -1075051951, 1730868336, -2443923755

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (-3,-2,1).

Cf. A000073, A000931, A073358.

I:=[0,1,1]; [n le 3 select I[n] else -3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // G. C. Greubel, Mar 27 2021

(* First program *)
b[n_]:= b[n]= If[n==0, 0, If[n<3, 1, b[n-2] +b[n-3]]];
a[n_]:= Sum[(-1)^m*Binomial[n, m]*b[n-m], {m,0,n}];
Table[a[n], {n, 0, 50}]
(* Second program *)
LinearRecurrence[{-3,-2,1}, {0,1,-1}, 51] (* G. C. Greubel, Mar 27 2021 *)

@CachedFunction
def A000931(n): return 1 if n==0 else sum( binomial(k, n-2*k-3) for k in (0..floor((n-3)/2)))
def A144413(n): return sum( (-1)^k*binomial(n, k)*A000931(n-k+4) for k in (0..n))
[A144413(n) for n in (0..50)] # G. C. Greubel, Mar 27 2021

From R. J. Mathar, Jan 21 2009: (Start)
a(n) = -3*a(n-1) - 2*a(n-2) + a(n-3).
G.f.: x*(1 +2*x)/(1 +3*x +2*x^2 -x^3). (End)

Terms a(30) onward added and edited by G. C. Greubel, Mar 27 2021

A145462 Eigentriangle, row sums = the Padovan sequence, A000931.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, 0, -1, 2, 2, 1, 0, -2, 2, 3, -1, 1, 0, -2, 3, 4, 0, -1, 2, 0, -3, 4, 5, 1, 0, -2, 2, 0, -4, 5, 7, -1, 1, 0, -2, 3, 0, -5, 7, 9, 0, -1, 2, 0, -3, 4, 0, -7, 9, 12, 1, 0, -2, 2, 0, -4, 5, 0, -9, 12, 16, -1, 1, 0, -2, 3, 0, -5, 7, 0, -12, 16, 21

Offset: 6

Gary W. Adamson, Oct 10 2008

Right border = Padovan sequence starting with offset 6.
Row sums = Padovan sequence starting with offset 7.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
-1, 1, 2;
0, -1, 2, 2;
1, 0, -2, 2, 3;
-1, 1, 0, -2, 3, 4;
0, -1, 2, 0, -3, 4, 5;
1, 0, -2, 2, 0, -4, 5, 7;
-1, 1, 0, -2, 3, 0, -5, 7, 9;
0, -1, 2, 0, -3, 4, 0, -7, 9, 12;
1, 0, -2, 2, 0, -4, 5, 0, -9, 12, 16;
...
Example: Row 10 = (1, 0, -2, 2, 3) with A000931(10) = 3, rightmost term. This row = the termwise products of (1, 0, -1, 1, 1) and (1, 1, 2, 2, 3); where the Padovan sequence starting with offset 6 = (1, 1, 2, 2, 3, 4, 5, 7, 9,...).

A000931, A106510

Triangle read by rows, T(n,k) = M * (A000931 * 0^(n-k)). M = an infinite lower triangular matrix with A106510 in every column: (1, 1, -1, 0, 1, -1, 0, 1, -1,...); and A000931 is a diagonalized infinite lower triangular matrix with the Padovan sequence starting with offset 6: (1, 1, 2, 2, 3, 4, 5, 7, 9,...) as the main diagonal and the rest zeros.

A146973 Eigentriangle, row sums = A000931 starting with offset 3.

Original entry on oeis.org

1, -1, 1, 2, -1, 0, -2, 2, 0, 1, 3, -2, 0, -1, 1, -3, 3, 0, 2, -1, 1, 4, -3, 0, -2, 2, -1, 2, -4, 4, 0, 3, -2, 2, -2, 2, 5, -4, 0, -3, 3, -2, 4, -2, 3, -5, 5, 0, 4, -3, 3, -4, 4, -3, 4, 6, -5, 0, -4, 4, -3, 6, -4, 5, -6, 6, 0, 5, -4, 4, -6, 6, -6, 8, -5, 7

Offset: 3

Gary W. Adamson, Nov 03 2008

Row sums and right border = the Padovan sequence, A000931 starting with offset 3: (1, 1, 0, 1, 1, 1, 2, 2, 3,...).

Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  -1, 1;
  2, -1, 0;
  -2, 2, 0, 1;
  3, -2, 0, -1, 1;
  -3, 3, 0, 2, -1, 1;
  4, -3, 0, -2, 2, -1, 2;
  -4, 4, 0, 3, -2, 2, -2, 2;
  5, -4, 0, -3, 3, -2, 4, -2, 3;
  -5, 5, 0, 4, -3, 3, -4, 4, -3, 4;
  6, -5, 0, -4, 4, -3, 6, -4, 6, -4, 5;
  -6, 6, 0, 5, -4, 4, -6, 6, -6, 8, -5, 7;
  7, -6, 0, -5, 5, -4, 8, -6, 9, -8, 10, -7, 9
  -7, 7, 0, 6, -5, 5, -8, 8, -9, 12, -10, 14, -9, 12;
  ...
Row 6 = (-2, 2, 0, 1) = termwise products of (-2, 2, 0, 1) and (1, 1, 0, 1).

Cf. A000931.

Triangle read by rows, T * Q, where T = an infinite lower triangular matrix with (1, -1, 2, -2, 3, -3,...) in every column and Q = an infinite lower triangular matrix with the Padovan sequence, A000931 as the main diagonal starting with offset 3: (1, 1, 0, 1, 1, 1, 2, 2, 3,...). The rest of triangle Q = all zeros. This triangle = T * Q.

A152870 Indices of primes in the Padovan sequence A000931.

Original entry on oeis.org

8, 9, 10, 12, 13, 19, 24, 35, 42, 89, 133, 474, 671, 1267, 1578, 2008, 2215, 2294, 4168, 5558, 6572, 8566, 11235, 18742, 35839, 44264, 536490, 727739

Offset: 1

Roger L. Bagula, Dec 14 2008

Cf. A000931, A001605, A100891, A112882.

a[0] = 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
Flatten[Table[If[PrimeQ[a[n]], n, {}], {n, 0, 10000}]]

v=[1,1,1];for(n=8,1e4,v=[v[2],v[3],v[1]+v[2]];if(ispseudoprime(v[3]),print1(n", "))) \\ Charles R Greathouse IV, Nov 07 2011

a(n) = A112882(n) + 5. - Amiram Eldar, Nov 10 2024

a(23)-a(26) from Charles R Greathouse IV, Nov 07 2011

a(27)-a(28) calculated from the data at A112882 by Amiram Eldar, Nov 10 2024

A167385 a(n)= sum_{i=7..n+6} A000931(i).

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 24, 33, 45, 61, 82, 110, 147, 196, 261, 347, 461, 612, 812, 1077, 1428, 1893, 2509, 3325, 4406, 5838, 7735, 10248, 13577, 17987, 23829, 31568, 41820, 55401, 73392, 97225, 128797, 170621, 226026, 299422, 396651, 525452, 696077, 922107, 1221533

Offset: 0

Roger L. Bagula, Nov 02 2009

Cf. A018917.

Clear[f, g, n]
f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[n - 2] + f[n - 3];
g[n_] := Sum[f[i + 3], {i, 0, n}]
Table[g[n], {n, 0, 30}]

a(n+1)/a(n)-> A060005 as n->infinity.
G.f.: (1+x)^2/((x-1)*(x^3+x^2-1)). a(n)= +a(n-1) +a(n-2) -a(n-4). [Nov 05 2009]
a(n) = A000931(n+12)-4. [Nov 05 2009]

Notation normalized, definition corrected, g.f. added - The Assoc. Editors of the OEIS, Nov 05 2009

A173692 a(n) = ceiling(A000931(n)/2).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 11, 14, 19, 25, 33, 43, 57, 76, 100, 133, 176, 233, 308, 408, 541, 716, 949, 1257, 1665, 2205, 2921, 3870, 5126, 6791, 8996, 11917, 15786, 20912, 27703, 36698, 48615, 64401, 85313, 113015, 149713, 198328, 262728, 348041, 461056

Offset: 0

Roger L. Bagula, Nov 25 2010

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

Cf. A000931.

a[0] = 0; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 2] + a[n - 3]
Table[a[n] - Floor[a[n]/2], {n, 0, 30}]

concat(0, Vec(x*(1 + x)*(1 - x^3 - x^7) / ((1 - x)*(1 - x^2 - x^3)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^40))) \\ Colin Barker, Feb 26 2020

a(n) = A000931(n) - floor(A000931(n)/2).
a(n) = a(n-2) + a(n-3) + a(n-7) - a(n-9) - a(n-10). - R. J. Mathar, Mar 11 2012
G.f.: x*(1 + x)*(1 - x^3 - x^7) / ((1 - x)*(1 - x^2 - x^3)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, Feb 26 2020

More terms from Jinyuan Wang, Feb 26 2020

A176505 Triangle T(n,m) read by rows: T(n,m) = a(n) - a(m) - a(n-m) + 1, where a(n) = A000931(n+4).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 1, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 1, 2, 4, 5, 5, 6, 5, 5, 4, 2, 1

Offset: 0

Roger L. Bagula, Apr 19 2010

			Triangle:
{1},
{1, 1},
{1, 0, 1},
{1, 0, 0, 1},
{1, 1, 1, 1, 1},
{1, 0, 1, 1, 0, 1},
{1, 1, 1, 2, 1, 1, 1},
{1, 1, 2, 2, 2, 2, 1, 1},
{1, 1, 2, 3, 2, 3, 2, 1, 1},
{1, 2, 3, 4, 4, 4, 4, 3, 2, 1},
{1, 2, 4, 5, 5, 6, 5, 5, 4, 2, 1},
...
T(6,3) = a(6) - a(3) - a(6-3) + 1 = a(6) - 2 * a(3) + 1 = 3 - (2 * 1) + 1 = 2. - _Indranil Ghosh_, Feb 17 2017

Indranil Ghosh, Rows 0..120, flattened
Indranil Ghosh, Python Program to generate the b-file

Cf. A000931.

a[0] := 0; a[1] := 1; a[2] := 1;
a[n_] := a[n] = a[n - 2] + a[n - 3];
t[n_, m_] := t[n, m] = a[n] - a[m] - a[n - m] + 1;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

T(n,m) = a(n) - a(m) - a(n-m) + 1, a(n) = A000931(n+4).

Name and formula sections corrected by Indranil Ghosh, Feb 17 2017

cp's OEIS Frontend

A134727 Successive digits of members of the Padovan sequence A000931(n).

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A140514 a(n) = A000931(n+4) - A010060(n).

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A144401 Padovan ( A000931) version of A038137: expansion of polynomials as antidiagonal: p(x,n)=1/(1-x-x^3)^n.

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A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).

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Sage

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A145462 Eigentriangle, row sums = the Padovan sequence, A000931.

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A146973 Eigentriangle, row sums = A000931 starting with offset 3.

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A152870 Indices of primes in the Padovan sequence A000931.

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PARI

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A167385 a(n)= sum_{i=7..n+6} A000931(i).

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A173692 a(n) = ceiling(A000931(n)/2).

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