A000931 -id:A000931 - cp's OEIS frontend

A012772 Take every 5th term of Padovan sequence A000931, beginning with the sixth term.

1, 3, 12, 49, 200, 816, 3329, 13581, 55405, 226030, 922111, 3761840, 15346786, 62608681, 255418101, 1042002567, 4250949112, 17342153393, 70748973084, 288627200960, 1177482265857, 4803651498529, 19596955630177

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -4, 1).

Cf. A012781 (partial sums).

I:=[1, 3, 12]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 18 2012

CoefficientList[Series[(1-x)^2/(1-5*x+4*x^2-x^3),{x,0,33}],x] (* Vincenzo Librandi, Apr 18 2012 *)
LinearRecurrence[{5,-4,1},{1,3,12},30] (* Harvey P. Dale, Aug 15 2024 *)

a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n).

G.f.: (1-x)^2/(1-5*x+4*x^2-x^3). - Colin Barker, Feb 02 2012

A012864 Take every 5th term of Padovan sequence A000931, beginning with the first term.

Original entry on oeis.org

1, 1, 3, 12, 49, 200, 816, 3329, 13581, 55405, 226030, 922111, 3761840, 15346786, 62608681, 255418101, 1042002567, 4250949112, 17342153393, 70748973084, 288627200960, 1177482265857, 4803651498529, 19596955630177

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -4, 1).

Cf. A012855.

I:=[1, 1, 3]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 18 2012

LinearRecurrence[{5,-4, 1},{1,1,3},30] (* Vincenzo Librandi, Apr 18 2012 *)

a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n).

O.g.f.: (1-4x+2x^2)/(1-5x+4x^2-x^3). a(n+1)=A012772(n). - R. J. Mathar, May 28 2008

A112882 Indices of prime Padovan numbers: values of k such that A000931(k+5) is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 14, 19, 30, 37, 84, 128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163, 5553, 6567, 8561, 11230, 18737, 35834, 44259, 536485, 727734

Offset: 1

Eric W. Weisstein, Oct 05 2005

Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Eric Weisstein's World of Mathematics, Padovan Sequence.

Cf. A000931, A100891, A152870.

a(n) = A152870(n) - 5. - Amiram Eldar, Nov 10 2024

a(27) from Eric W. Weisstein, May 22 2009

a(28) from Eric W. Weisstein, Apr 10 2011

A153462 Triangle read by rows, = A000931(n-k+3) * (A000073 * 0^(n-k)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 4, 2, 1, 1, 2, 0, 7, 2, 2, 1, 2, 4, 0, 13, 3, 2, 2, 2, 4, 7, 0, 24, 4, 3, 2, 4, 4, 7, 13, 0, 44, 5, 4, 3, 4, 8, 7, 13, 24, 0, 81, 7, 5, 4, 6, 8, 14, 13, 24, 44, 0, 149, 9, 7, 5, 8, 12, 14, 26, 24, 44, 81, 0, 274

Offset: 3

Gary W. Adamson, Dec 27 2008

An eigentriangle by rows, the Padovan sequence convolved with the tribonacci numbers.

Sum of n-th row terms = rightmost term of next row. Row sums = the tribonacci numbers, A000073.

			First few rows of the triangle =
   1;
   0, 1;
   1, 0, 1;
   1, 1, 0,  2;
   1, 1, 1,  0,  4;
   2, 1, 1,  2,  0,  7;
   2, 2, 1,  2,  4,  0, 13;
   3, 2, 2,  2,  4,  7,  0, 24;
   4, 3, 2,  4,  4,  7, 13,  0, 44;
   5, 4, 3,  4,  8,  7, 13, 24,  0, 81;
   7, 5, 4,  6,  8, 14, 13, 24, 44,  0, 149;
   9, 7, 5,  8, 12, 14, 26, 24, 44, 81,   0, 274;
  12, 9, 7, 10, 16, 21, 26, 48, 44, 81, 149,   0, 504;
  ...
Row 9 = (2, 2, 1, 2, 4, 0, 13) = termwise products of (1, 1, 1, 2, 4, 7, 13) and (2, 2, 1, 1, 1, 0, 1). Dot product = 24 = A000073(8).

Cf. A000073, A000931.

Triangle read by rows, = A000931(n-k+3) * (A000073 * 0^(n-k)).
Equals infinite lower triangular matrices P*M; where P = a matrix with the Padovan sequence in every column starting with offset 3: (1, 0, 1, 1, 1, 2, 2, 3, 4, 5, ...).
M = an infinite lower triangular matrix with the tribonacci sequence prefaced with a 1 as the main diagonal: (1, 1, 1, 2, 4, 7, 13, ...) and the rest zeros.

A216714 a(n) = 2^(n-5) - A000931(n).

Original entry on oeis.org

0, 1, 3, 6, 14, 29, 60, 123, 249, 503, 1012, 2032, 4075, 8164, 16347, 32719, 65471, 130986, 262030, 524137, 1048376, 2096887, 4193953, 8388143, 16776600, 33553616, 67107783, 134216296, 268433559, 536868399, 1073738495, 2147479238, 4294961454, 8589926853, 17179858932, 34359724787, 68719458745, 137438929639, 274877875372, 549755772064

Offset: 5

N. J. A. Sloane, Sep 14 2012

It is conjectured that this sequence (with a different offset) and A038360 are the same.

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
M. Waldschmidt, Lectures on Multiple Zeta Values (IMSC2011).
Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-2)

Cf. A000931, A038360.

I:=[0, 1, 3, 6]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-3)-2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 11 2013

CoefficientList[Series[-x (-1 - x + x^2)/((2 x - 1) (x^3 + x^2 - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 11 2013 *)
LinearRecurrence[{2,1,-1,-2},{0,1,3,6},40] (* Harvey P. Dale, Aug 22 2021 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -2,-1,1,2]^(n-5)*[0;1;3;6])[1,1] \\ Charles R Greathouse IV, Sep 09 2016

G.f.: -x^6*(-1-x+x^2) / ( (2*x-1)*(x^3+x^2-1) ). - R. J. Mathar, Sep 16 2012

a(n) = 2*a(n-1)+a(n-2)-a(n-3)-2*a(n-4). - Vincenzo Librandi, Mar 11 2013

A291289 The Padovan sequence A000931 doubled.

Original entry on oeis.org

2, 0, 0, 2, 0, 2, 2, 2, 4, 4, 6, 8, 10, 14, 18, 24, 32, 42, 56, 74, 98, 130, 172, 228, 302, 400, 530, 702, 930, 1232, 1632, 2162, 2864, 3794, 5026, 6658, 8820, 11684, 15478, 20504, 27162, 35982, 47666, 63144, 83648, 110810, 146792, 194458, 257602, 341250

Offset: 0

N. J. A. Sloane, Aug 29 2017

Like A000931, this sequence has the property that the largest of any four consecutive terms equals the sum of the two smallest.

Michael De Vlieger, Table of n, a(n) for n = 0..8192
David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Cf. A000931.

CoefficientList[Series[2*(1 - x^2)/(1 - x^2 - x^3), {x, 0, 49}], x] (* Michael De Vlieger, Mar 21 2023 *)

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

A012493 Take every 5th term of Padovan sequence A000931, beginning with the fifth term.

Original entry on oeis.org

0, 2, 9, 37, 151, 616, 2513, 10252, 41824, 170625, 696081, 2839729, 11584946, 47261895, 192809420, 786584466, 3208946545, 13091204281, 53406819691, 217878227876, 888855064897, 3626169232672, 14793304131648, 60350698792449, 246206446668325, 1004422742303477

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,1).

I:=[0, 2, 9]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 17 2012

CoefficientList[Series[x*(2-x)/(1-5*x+4*x^2-x^3),{x,0,30}],x] (* Vincenzo Librandi, Apr 17 2012 *)
LinearRecurrence[{5,-4,1},{0,2,9},30] (* Harvey P. Dale, Nov 24 2018 *)

a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n).

G.f.: x*(2-x)/(1-5*x+4*x^2-x^3). - Colin Barker, Feb 02 2012

First term corrected by Colin Barker, Feb 02 2012

A018243 Inverse Euler transform of A000931.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 21, 28, 34, 45, 56, 73, 92, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185, 160380, 208740, 271913, 354123, 461529, 601436, 784209, 1022505, 1333856

Offset: 1

N. J. A. Sloane, David Broadhurst

			x^3 + x^5 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + 3*x^13 + 3*x^14 + ...

Joerg Arndt, Table of n, a(n) for n = 1..1000
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, No.3-4, 403-412 (1997).
N. J. A. Sloane, Transforms

Cf. A000931, A113788.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000931):
seq(a(n), n = 1..65); # Peter Luschny, Nov 21 2022

a[n_] := (1/n)*Sum[ MoebiusMu[n/d]*Floor[ Re[ N[ RootSum[ -1-#+#^3&, #^d& ]]]] , {d, Divisors[n]}]; a[2]=0; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 05 2012, after Michael Somos *)

z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z**2 + z**3))/(1 - z**2)
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 25 2020

a(n) = A113788(n) unless n=2. - Michael Somos, Apr 06 2012
Reciprocal of g.f. of A000931 = (1 - x^2 - x^3) / (1 - x^2) = 1 - x^3 - x^5 - x^7 - x^9 - ... = Product_{k>0} (1 - x^k)^a(n). - Michael Somos, Jul 17 2012
a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019

More terms from Joerg Arndt, Jul 18 2012

A132347 Concatenation of first n members of the Padovan sequence A000931.

Original entry on oeis.org

1, 10, 100, 1001, 10010, 100101, 1001011, 10010111, 100101112, 1001011122, 10010111223, 100101112234, 1001011122345, 10010111223457, 100101112234579, 10010111223457912, 1001011122345791216, 100101112234579121621

Offset: 1

Omar E. Pol, Nov 10 2007

Cf. A000931, A007908, A019523, A102397. See A134732 for another version.

Module[{nn=20,padseq},padseq=LinearRecurrence[{0,1,1},{1,0,0},nn];Table[FromDigits[ Flatten[ IntegerDigits/@Take[padseq,n]]],{n,nn}]] (* Harvey P. Dale, Feb 18 2024 *)

A133039 a(n) = P(n)^3 - P(n)^2 where P(n) = A000931(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 18, 48, 100, 294, 648, 1584, 3840, 8820, 21168, 49284, 115248, 270400, 628660, 1468548, 3420150, 7960000, 18539400, 43120350, 100328400, 233365440, 542672640, 1262045880, 2934442944, 6822962664, 15863704528, 36881698048, 85746672900, 199347278724, 463445232298

Offset: 0

Omar E. Pol, Nov 02 2007

			a(10)=18 because Padovan(10)=3 and 3^3=27 and 3^2=9 and 27-9=18.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,1,-9,3,-9,3,-3,15,-9,9,-3,1,-2,1,-1).

Cf. A000290, A000578, A045991. Padovan sequence: A000931.

P[0] := 1; P[1] := 0; P[2] := 0; P[n_] := P[n] = P[n - 2] + P[n - 3]; Table[P[n]^3 - P[n]^2, {n, 0, 50}] (* G. C. Greubel, Oct 02 2017 *)

x='x+O('x^50); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(2*x^8*(x^7-x^6+2*x^5+x^2-2*x+2)/((x -1)*(x^3-2*x^2+3*x-1)*(x^3-x^2+2*x-1)*(x^3-x-1)*(x^6+3*x^5+5*x^4 +5*x^3 +5*x^2+3*x+1)))) \\ G. C. Greubel, Oct 02 2017

a(n) = P(n)^3 - P(n)^2 = A000931(n)^3 - A000931(n)^2.

G.f.: 2*x^8*(x^7-x^6+2*x^5+x^2-2*x+2) / ((x-1) * (x^3-2*x^2+3*x-1) * (x^3-x^2+2*x-1) * (x^3-x-1) * (x^6+3*x^5+5*x^4+5*x^3+5*x^2+3*x+1)). - Colin Barker, Sep 18 2013

Incorrect initial zero of the sequence deleted by Colin Barker, Sep 18 2013

Added more terms, Joerg Arndt, Sep 18 2013

cp's OEIS Frontend

A012772 Take every 5th term of Padovan sequence A000931, beginning with the sixth term.

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A012864 Take every 5th term of Padovan sequence A000931, beginning with the first term.

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A112882 Indices of prime Padovan numbers: values of k such that A000931(k+5) is prime.

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A153462 Triangle read by rows, = A000931(n-k+3) * (A000073 * 0^(n-k)).

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A216714 a(n) = 2^(n-5) - A000931(n).

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A291289 The Padovan sequence A000931 doubled.

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A012493 Take every 5th term of Padovan sequence A000931, beginning with the fifth term.

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A018243 Inverse Euler transform of A000931.

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