A000931 -id:A000931 - cp's OEIS frontend

A220885 a(3)=5, a(4)=8, a(5)=12; thereafter a(n) = a(n-1) + A000931(n+7).

5, 8, 12, 19, 28, 40, 56, 77, 105, 142, 191, 256, 342, 456, 607, 807, 1072, 1423, 1888, 2504, 3320, 4401, 5833, 7730, 10243, 13572, 17982, 23824, 31563, 41815, 55396, 73387, 97220, 128792, 170616, 226021, 299417, 396646, 525447, 696072, 922102, 1221528, 1618183, 2143639, 2839720, 3761831, 4983368

Offset: 3

N. J. A. Sloane, Dec 29 2012, based on an email from Ludovic Mignot, Dec 27 2012

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
P. Caron, J.-M. Champarnaud and L. Mignot, Multi-tilde-bar expressions and their automata, Acta Informatica, September 2012, Volume 49, Issue 6, pp 413-436. DOI 10.1007/s00236-012-0167-x. See the sequence t(k).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

a220885 n = a220885_list !! (n-3)
a220885_list = 5 : 8 : zs where
   zs = 12 : zipWith (+) zs (drop 13 a000931_list)
-- Reinhard Zumkeller, Feb 19 2013

Join[{5,8},LinearRecurrence[{1,1,0,-1},{12,19,28,40},50]] (* Harvey P. Dale, May 30 2014 *)

a(3)=5, a(4)=8, a(5)=12; thereafter a(n) = a(n-1) + Pad(n) + 2*Pad(n+1) + 2*Pad(n+2), where Pad() = A000931().
a(n) = a(n-1)+a(n-2)-a(n-4) for n>8. G.f.: x^3*(x^5+2*x^4-x^3-x^2+3*x+5) / ((x-1)*(x^3+x^2-1)). [Colin Barker, Jan 04 2013]
a(n) = a(n-1) + A000931(n+7) for n > 5. - Reinhard Zumkeller, Feb 19 2013
a(n) = a(n-2) + a(n-3) + 9 for n >= 8. - Greg Dresden, May 18 2020

Simpler definition from Reinhard Zumkeller, Dec 30 2012

A329244 Sum of every third term of the Padovan sequence A000931.

Original entry on oeis.org

1, 2, 3, 5, 10, 22, 50, 115, 266, 617, 1433, 3330, 7740, 17992, 41825, 97230, 226031, 525457, 1221538, 2839730, 6601570, 15346787, 35676950, 82938845, 192809421, 448227522, 1042002568, 2422362080, 5631308625, 13091204282, 30433357675, 70748973085, 164471408186

Offset: 0

David Nacin, Nov 09 2019

			For n = 3, a(3) = 1+1+1+2 = 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,3,-1).

Partial sums of A034943.

Cf. A000931.

LinearRecurrence[{4, -5, 3, -1}, {1, 2, 3, 5}, 50] (* Paolo Xausa, Apr 08 2024 *)

Vec((1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)) + O(x^35)) \\ Colin Barker, Nov 09 2019

p = lambda x:[1, 0, 0][x] if x<3 else p(x-2)+p(x-3)
a = lambda x:sum(p(3*i) for i in range(x+1))

a(n) = Sum_{i=0..n} A000931(3*i).
a(n) = A000931(3n+2)+1.
From Colin Barker, Nov 09 2019: (Start)
G.f.: (1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)).
a(n) = 4*a(n-1) - 5*a(n-2) + 3*a(n-3) - a(n-4) for n>3. (End)

A108169 Alternating three ratio switched sequence based on characteristic root of A000931.

Original entry on oeis.org

1, 1, 1, 3, 18, 33, 111, 690, 1269, 4292, 26705, 49118, 166165, 1033922, 1901679, 6433333, 40029886, 73626439, 249076459, 1549819116, 2850561773, 9643381402, 60003650693, 110363919984, 373358466957, 2323134396636, 4272910326315

Offset: 0

Roger L. Bagula, Jun 13 2005

Limit[a(n)/a(n-1),n->Infinity]={1.83929, 3.38298, 6.22226}

Cf. A000931.

NSolve[x^3 - x^2 - x - 1 == 0, x] beta = 1.8392867552141612; F[1] = 1; F[2] = 1; F[n__] := F[n] = If[Mod[n, 3] == 0, Floor[beta*F[n - 1]], If[ Mod[n, 3] == 1, Floor[(beta^2)*F[n - 1]], Floor[(beta^3)*F[n - 1]]]] a = Table[F[n], {n, 1, 50}] an = Table[N[a[[n]]/a[[n - 1]]], {n, 6, 50}]

if Mod[n, 3]=0 then F[n] = Floor[beta*F[n-1]] if Mod[n, 3]=1 then F[n] = Floor[beta^2*F[n-1]] if Mod[n, 3]=2 then F[n] = Floor[beta^3*F[n-1]] a(n) = F[n]

A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)a(m + 1)Binomial[n, m].

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 6, 2, 3, 8, 24, 8, 3, 4, 15, 40, 40, 15, 4, 5, 24, 90, 80, 90, 24, 5, 7, 35, 168, 210, 210, 168, 35, 7, 9, 56, 280, 448, 630, 448, 280, 56, 9, 12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12, 16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 6, 16, 46, 118, 318, 840, 2216, 5898, 15584}

			{1},
{1, 1},
{2, 2, 2},
{2, 6, 6, 2},
{3, 8, 24, 8, 3},
{4, 15, 40, 40, 15, 4},
{5, 24, 90, 80, 90, 24, 5},
{7, 35, 168, 210, 210, 168, 35, 7},
{9, 56, 280, 448, 630, 448, 280, 56, 9},
{12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12},
{16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16}

Cf. A141611, A141617, A000931.

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

a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 6, 4, 6, 4, 5, 7, 5, 8, 6, 6, 8, 5, 7, 9, 7, 10, 8, 9, 8, 10, 7, 9, 12, 9, 14, 10, 12, 12, 10, 14, 9, 12, 16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 5, 8, 14, 22, 34, 52, 77, 114, 166}

			{1},
{1, 1},
{2, 1, 2},
{2, 2, 2, 2},
{3, 2, 4, 2, 3},
{4, 3, 4, 4, 3, 4},
{5, 4, 6, 4, 6, 4, 5},
{7, 5, 8, 6, 6, 8, 5, 7},
{9, 7, 10, 8, 9, 8, 10, 7, 9},
{12, 9, 14, 10, 12, 12, 10, 14, 9, 12},
{16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16}

Cf. A141611, A141617, A000931.

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

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 6, 4, 4, 6, 15, 6, 16, 6, 15, 32, 15, 24, 24, 15, 32, 65, 32, 60, 36, 60, 32, 65, 147, 65, 128, 90, 90, 128, 65, 147, 306, 147, 260, 192, 225, 192, 260, 147, 306, 660, 306, 588, 390, 480, 480, 390, 588, 306, 660, 1424, 660, 1224, 882, 975, 1024, 975, 882

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 9, 20, 58, 142, 350, 860, 2035, 4848, 11354}.

			{1},
{1, 1},
{4, 1, 4},
{6, 4, 4, 6},
{15, 6, 16, 6, 15},
{32, 15, 24, 24, 15, 32},
{65, 32, 60, 36, 60, 32, 65},
{147, 65, 128, 90, 90, 128, 65, 147},
{306, 147, 260, 192, 225, 192, 260, 147, 306},
{660, 306, 588, 390, 480, 480, 390, 588, 306, 660},
{1424, 660, 1224, 882, 975, 1024, 975, 882, 1224, 660, 1424}

Cf. A141611, A141617, A000931, A000045, A058071.

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

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

A116585 An interleaving of three sequences: a(3n) = A000045(3n) = A014445(n). a(3n+1) = A000931(3n+5) = A052921(n). a(3n+2) = A003269(3n-1).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 8, 4, 5, 34, 9, 14, 144, 21, 36, 610, 49, 95, 2584, 114, 250, 10946, 265, 657, 46368, 616, 1728, 196418, 1432, 4544, 832040, 3329, 11949, 3524578, 7739, 31422, 14930352, 17991, 82629, 63245986, 41824, 217286, 267914296, 97229, 571388, 1134903170

Offset: 0

Roger L. Bagula, Mar 26 2006

Eric Weisstein's World of Mathematics, Pisot Number.
Index entries for linear recurrences with constant coefficients, signature (0,0,8,0,0,-17,0,0,1,0,0,11,0,0,15,0,0,2,0,0,3,0,0,5,0,0,1).

Cf. A000045, A000073, A000931, A003269, A014445, A017898, A052921.

a[1, 0] = 0; a[1, 1] = 1; a[1, n_Integer?Positive] := a[1, n] = a[1, n - 1] + a[1, n - 2] a[2, 0] = 0; a[2, 1] = 1; a[2, 2] = 1; a[2, n_Integer?Positive] := a[2, n] = a[2, n - 2] + a[2, n - 3] a[3, 0] = 0; a[3, 1] = a[3, 2] = a[3, 3] = 1; a[3, n_Integer?Positive] := a[3, n] = a[3, n - 1] + a[3, n - 4] b = Table[a[1 + Mod[n, 3], n], {n, 0, 25}]

Edited by N. J. A. Sloane, Apr 09 2008

More terms from Amiram Eldar, Jun 09 2025

A129973 a(n) = A000045(n) - A000931(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 9, 16, 27, 46, 77, 128, 212, 349, 573, 938, 1532, 2498, 4067, 6614, 10746, 17446, 28306, 45903, 74409, 120577, 195337, 316379, 512332, 829527, 1342940, 2173899, 3518736, 5695148, 9217213, 14916771, 24139826, 39064336

Offset: 1

Roger L. Bagula, Jun 13 2007

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-2,-1).

Cf. A000045, A000931.

LinearRecurrence[{1,2,0,-2,-1},{0,0,0,1,1},40] (* Harvey P. Dale, Jan 31 2024 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,-2,0,2,1]^(n-1)*[0;0;0;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x^4/((1-x-x^2)(1-x^2-x^3)). a(n)=a(n-1)+2*a(n-2)-2*a(n-4)-a(n-5). - R. J. Mathar, Oct 30 2008

A133038 Cubes of A000931.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 8, 8, 27, 64, 125, 343, 729, 1728, 4096, 9261, 21952, 50653, 117649, 274625, 636056, 1481544, 3442951, 8000000, 18609625, 43243551, 100544625, 233744896, 543338496, 1263214441, 2936493568, 6826561273, 15870019697, 36892780289

Offset: 0

Omar E. Pol, Nov 02 2007

			a(10)=27 because Padovan(10)=3 and 3^3=27.

Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2,4,-4,-1,-1,0,-1).

Cf. A000578, A030078, A056570. Padovan sequence: A000931.

CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x]^3

a(n) = A000931(n)^3.
a(n) = a(n-1)+2*a(n-2)+3*a(n-3)-2*a(n-4)+4*a(n-5)-4*a(n-6)-a(n-7)-a(n-8)-a(n-10).
G.f.: (x^9-x^8+x^7+x^6-5*x^5+x^4-2*x^3-2*x^2-x+1) / ((x-1) * (x^3-2*x^2+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

A137298 Triangle read by rows: coefficients of a Hermite-like set of recursive polynomials that appear by integration to be orthogonal using the substitution on the Hermite recursion of n->f(n) where f(n)=A000931[n] is the Padovan sequence.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 2, 0, -4, 0, 1, 0, 6, 0, -6, 0, 1, -6, 0, 18, 0, -9, 0, 1, 0, -30, 0, 42, 0, -13, 0, 1, 30, 0, -120, 0, 87, 0, -18, 0, 1, 0, 240, 0, -414, 0, 178, 0, -25, 0, 1, -270, 0, 1320, 0, -1197, 0, 340, 0, -34, 0, 1

Offset: 1

Roger L. Bagula, Mar 14 2008

The number-like behavior of the Padovan sequence made me think that I might get a orthogonal polynomial set by this substitution:
Table[Integrate[Exp[ -x2/2]*P[x,n]*P[x, n + 1], {x, -Infinity, Infinity}], {n, 0, 10}];
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
The row sums are:
Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];
{1, 1, 0, -1, -1, 1, 4, 0, -20, -20, 160}
The tiling property of the Fibonacci and Padovan sequences makes me think that other sequence of fundamental number theory "beta Integer-like" sequences might give orthogonal polynomials as well.

			{1},
{0, 1},
{-1, 0, 1},
{0, -2, 0, 1},
{2, 0, -4, 0, 1},
{0, 6, 0, -6, 0, 1},
{-6, 0, 18, 0, -9, 0, 1},
{0, -30, 0, 42, 0, -13, 0, 1},
{30, 0, -120, 0, 87, 0, -18, 0, 1},
{0, 240, 0, -414, 0, 178, 0, -25, 0, 1},
{-270, 0, 1320, 0, -1197, 0, 340, 0, -34, 0, 1}

Cf. A000045, A000931.

f[0] = 0; f[1] = 1;f[2]=1; f[n_] := f[n] = f[n - 2] + f[n - 3]; P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - f[n]*P[x, n - 2]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}] Flatten[a]

a(n) = a(n-2)+a(n-3): A000931(n); p(x,0)=1;p(x,1)=x; p(x,n)=x*p(x,n-1)-a(n)*p(n,n-2)

cp's OEIS Frontend

A220885 a(3)=5, a(4)=8, a(5)=12; thereafter a(n) = a(n-1) + A000931(n+7).

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A329244 Sum of every third term of the Padovan sequence A000931.

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A108169 Alternating three ratio switched sequence based on characteristic root of A000931.

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A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

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A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

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A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

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A116585 An interleaving of three sequences: a(3n) = A000045(3n) = A014445(n). a(3n+1) = A000931(3n+5) = A052921(n). a(3n+2) = A003269(3n-1).

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A129973 a(n) = A000045(n) - A000931(n).

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A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)a(m + 1)Binomial[n, m].

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].