A254006 -id:A254006 - cp's OEIS frontend

A047081 a(n) = Sum_{k=0..n} T(n, k), array T as in A047080.

1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423, 778, 1431, 2632, 4841, 8904, 16377, 30122, 55403, 101902, 187427, 344732, 634061, 1166220, 2145013, 3945294, 7256527, 13346834, 24548655, 45152016, 83047505, 152748176, 280947697, 516743378, 950439251, 1748130326, 3215312955, 5913882532, 10877325813

Offset: 0

Clark Kimberling

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

Cf. A047080, A047082, A047083, A047084, A047085, A047086, A047087, A047088.

Cf. A000073, A254006.

[n le 3 select n else Self(n-1) +Self(n-2) +Self(n-3): n in [1..60]]; // G. C. Greubel, Oct 31 2022

LinearRecurrence[{1,1,1}, {1,2,3}, 61] (* G. C. Greubel, Oct 31 2022 *)

@CachedFunction
def a(n): return (n+1) if (n<3) else a(n-1) +a(n-2) +a(n-3) # a = A047081
[a(n) for n in (0..60)] # G. C. Greubel, Oct 31 2022

From G. C. Greubel, Oct 31 2022: (Start)
G.f.: (1 + x)/(1 - x - x^2 - x^3).
a(n) = A000073(n+1) + A000073(n+2). (End)
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-3*r^2+5*r+2). - Fabian Pereyra, Nov 23 2024
a(n) = A001590(n+3). - R. J. Mathar, Mar 28 2025

Data corrected by G. C. Greubel, Oct 31 2022

A163325 Pick digits at the even distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 10, 11, 9, 10, 11, 9, 10, 11, 12, 13, 14, 12, 13, 14

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '102' (1*9 + 0*3 + 2), from which we pick the "zeroth" and 2nd digits from the right, giving '12' = 1*3 + 2 = 5, thus a(11) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..728
Kevin Ryde, Plot2 of X=A163325,Y=A163326, illustrating the ternary Z-order curve.
Index entries for sequences related to coordinates of 2D curves

A059905 is an analogous sequence for binary.

Cf. A007089, A163327, A163328, A163329.

a(n) = fromdigits(digits(n,9)%3,3); \\ Kevin Ryde, May 14 2020

a(0) = 0, a(n) = (n mod 3) + 3*a(floor(n/9)).
a(n) = Sum_{k>=0} {A030341(n,k)*b(k)} where b is the sequence (1,0,3,0,9,0,27,0,81,0,243,0... = A254006): powers of 3 alternating with zeros. - Philippe Deléham, Oct 22 2011
A037314(a(n)) + 3*A037314(A163326(n)) = n for all n.

Edited by Charles R Greathouse IV, Nov 01 2009

A052993 a(n) = a(n-1) + 3a(n-2) - 3a(n-3), with a(0)=a(1)=1, a(2)=4.

Original entry on oeis.org

1, 1, 4, 4, 13, 13, 40, 40, 121, 121, 364, 364, 1093, 1093, 3280, 3280, 9841, 9841, 29524, 29524, 88573, 88573, 265720, 265720, 797161, 797161, 2391484, 2391484, 7174453, 7174453, 21523360, 21523360, 64570081, 64570081, 193710244

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1069
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A062318.

I:=[1,1,4]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 21 2018

spec := [S,{S=Prod(Sequence(Prod(Union(Z,Z,Z),Z)),Sequence(Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

(3^(1+Floor[(Range@40-1)/2])-1)/2 (* Federico Provvedi, Nov 22 2018 *)
LinearRecurrence[{1,3,-3}, {1,1,4}, 30] (* or *) RecurrenceTable[{a[n + 2] == 3*a[n] + 1, a[0] == 1, a[1] == 1}, a, {n,0,30}] (* G. C. Greubel, Nov 21 2018 *)

x='x+O('x^30); Vec(1/((1-3*x^2)*(1-x))) \\ G. C. Greubel, Nov 21 2018

s=(1/((1-3*x^2)*(1-x))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 21 2018

G.f.: 1/((1-3*x^2)*(1-x)).
a(n+2) = 3*a(n) + 1, where a(0) = a(1) = 1.
a(n) = -1/2 + Sum((1/4)*(1+3*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1 + 3*_Z^2)).
a(n) = Sum{k=0..n} 3^(k/2)*(1-(-1)^k)/(2*sqrt(3)). - Paul Barry, Jul 28 2004
a(n) = (3^(1+floor((n-1)/2)) - 1)/2. - Federico Provvedi, Nov 22 2018
a(n)-a(n-1) = A254006(n). - R. J. Mathar, Feb 27 2019

More terms from James Sellers, Jun 06 2000

A124812 Number of 4-ary Lyndon words of length n with exactly four 1s.

Original entry on oeis.org

3, 21, 135, 702, 3402, 15282, 65610, 270540, 1082565, 4221639, 16120377, 60450138, 223205220, 813100356, 2927177028, 10428053400, 36804946455, 128817263385, 447470664795, 1543773631158, 5292938720718, 18044108743734, 61193066237550

Offset: 5

Mike Zabrocki, Nov 08 2006

nonn

			a(6) = 21 because 1111ab, 1111ba, 111a1b, 111b1a, 11a11b for ab = 23, 24, 34 (accounting for 15 words) and 1111aa, 111a1a for a=2,3,4 (accounting for 6 words) are all Lyndon of length 6

G. C. Greubel, Table of n, a(n) for n = 5..1005
Index entries for linear recurrences with constant coefficients, signature (12,-48,36,234,-540,0,972,-729).

Cf. A006918, A124722, A124810, A124811, A124813, A124814, A254006.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 3*(1-5*x+9*x^2-6*x^3)/((1-3*x)^4*(1-3*x^2)^2) )); // G. C. Greubel, Aug 09 2023

3*(1-5*x+9*x^2-6*x^3)/((1-3*x)^4*(1-3*x^2)^2) + O[x]^23 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2017 *)
LinearRecurrence[{12,-48,36,234,-540,0,972,-729}, {3,21,135,702,3402, 15282,65610,270540}, 41] (* G. C. Greubel, Aug 09 2023 *)

def A124812(n): return (3/4)*(3^(n-5)*binomial(n-1,3) - ((n-2)//2)*3^((n-6)//2)*((n-5)%2))
[A124812(n) for n in range(5,41)] # G. C. Greubel, Aug 09 2023

O.g.f.: 3*x^5*(1 - 5*x + 9*x^2 - 6*x^3)/((1 - 3*x^2)^2*(1 - 3*x)^4).
G.f.: (1/4)*( (x/(1-3*x))^4 - x^4/(1-3*x^2)^2 ).
a(n) = (1/4)*Sum_{d|4,d|n} mu(d)*C(n/d - 1, (n-4)/d)*3^((n-4)/d).
a(n) = (1/4)*C(n-1, 3)*3^(n-4) if n is odd, a(n) = (1/4)*( C(n-1, 3)*3^(n-4) - (n/2-1)*3^((n-4)/2) ) if n is even.
a(n) = (3/4)*( 3^(n-5)*binomial(n-1, 3) - ((n-2)/2)*A254006(n-6) ). - G. C. Greubel, Aug 09 2023

A084156 Binomial transform of sinh(x)cosh(sqrt(3)x).

Original entry on oeis.org

0, 1, 2, 13, 44, 181, 662, 2521, 9368, 35113, 130922, 489061, 1824836, 6811741, 25420670, 94875313, 354076208, 1321442641, 4931681234, 18405321661, 68689566044, 256353060613, 956722558310, 3570537526921, 13325427195080, 49731172316281, 185599261007162

Offset: 0

Paul Barry, May 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,2,-12,3).

Cf. A001075, A084154, A084157, A254006.

I:=[0,1,2,13]; [n le 4 select I[n] else 4*Self(n-1)+2*Self(n-2)-12*Self(n-3)+3*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 13 2014

seq((2*simplify(ChebyshevT(n,2)) - (1+(-1)^n)*3^(n/2))/4, n = 0..30); # G. C. Greubel, Oct 10 2022

LinearRecurrence[{4,2,-12,3},{0,1,2,13},30] (* Harvey P. Dale, Feb 01 2014 *)

def A084156(n): return (chebyshev_T(n, 2) - ((n+1)%2)*3^(n/2))/2
[A084156(n) for n in range(31)] # G. C. Greubel, Oct 10 2022

a(n) = 4*a(n-1) + 2*a(n-2) - 12*a(n-3) + 3*a(n-4).
a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n - (sqrt(3))^n - (-sqrt(3))^n)/4.
G.f.: x*(1-2*x+3*x^2)/((1-3*x^2)(1-4*x+x^2)).
E.g.f. : exp(x)*sinh(x)*cosh(sqrt(3)*x).
a(n) = (2*ChebyshevT(n, 2) - (1+(-1)^n)*3^(n/2))/4 = (A001075(n) - A254006(n))/2. - G. C. Greubel, Oct 10 2022

A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55

Offset: 0

Philippe Deléham, Dec 18 2006

Rising and falling diagonals are A008999, A124400.
Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012
Jointly generated with A209130 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle begins:
  1;
  2, 1;
  3, 4, 2;
  4, 10, 10, 3;
  5, 20, 31, 20, 5;
  6, 35, 76, 78, 40, 8;
  7, 56, 161, 232, 184, 76, 13;
  8, 84, 308, 582, 636, 406, 142, 21;
  9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
  10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 4, 2, 0
  4, 10, 10, 3, 0
  5, 20, 31, 20, 5, 0
  6, 35, 76, 78, 40, 8, 0
  7, 56, 161, 232, 184, 76, 13, 0

Cf. A209130.
Cf. A008999, A124400, A084938, A209130.
Cf. A254006, A000012, A000027, A000244, A015530, A015544.
Cf. A001629, A000045, A000292, A051747.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)
(* Clark Kimberling, Mar 05 2012 *)

Sum_{k=0..n} x^k*T(n,k) = A254006(n), A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for x = -2, -1, 0, 1, 2, 3 respectively.
T(n,n-1) = 2*A001629(n+1) for n>=1.
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n,0) = n+1.
T(n,1) = A000292(n) for n>=1.
T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.
G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627

Offset: 0

G. C. Greubel, Oct 31 2022

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

Cf. A052993, A062318, A164123, A254006.

I:=[3,6,11]; [1,2] cat [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..60]];

LinearRecurrence[{1,3,-3}, {1,2,3,6,11}, 61]

def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def A358027(n): return (1/3)*( 4*A254006(n) + 7*A254006(n-1) +2*int(n==0) + 2*int(n==1) - 3 )
[A358027(n) for n in (0..60)]

a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1)).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.
E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).
G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022

A319234 T(n, k) is the coefficient of x^k of the polynomial p(n) which is defined as the scalar part of P(n) = Q(x, 1, 1, 1) * P(n-1) for n > 0 and P(0) = Q(1, 0, 0, 0) where Q(a, b, c, d) is a quaternion, triangle read by rows.

Original entry on oeis.org

1, 0, 1, -3, 0, 1, 0, -9, 0, 1, 9, 0, -18, 0, 1, 0, 45, 0, -30, 0, 1, -27, 0, 135, 0, -45, 0, 1, 0, -189, 0, 315, 0, -63, 0, 1, 81, 0, -756, 0, 630, 0, -84, 0, 1, 0, 729, 0, -2268, 0, 1134, 0, -108, 0, 1, -243, 0, 3645, 0, -5670, 0, 1890, 0, -135, 0, 1

Offset: 0

Peter Luschny, Sep 14 2018

The symbol '*' in the name refers to the noncommutative multiplication in Hamilton's division algebra. Traditionally Q(a, b, c, d) is written a + b*i + c*j + d*k.

			The list of polynomials starts 1, x, x^2 - 3, x^3 - 9*x, x^4 - 18*x^2 + 9, ... and the list of coefficients of the polynomials starts:
[0] [  1]
[1] [  0,    1]
[2] [ -3,    0,    1]
[3] [  0,   -9,    0,     1]
[4] [  9,    0,  -18,     0,   1]
[5] [  0,   45,    0,   -30,   0,    1]
[6] [-27,    0,  135,     0, -45,    0,   1]
[7] [  0, -189,    0,   315,   0,  -63,   0,    1]
[8] [ 81,    0, -756,     0, 630,    0, -84,    0, 1]
[9] [  0,  729,    0, -2268,   0, 1134,   0, -108, 0, 1]

Wikipedia, Quaternion

Inspired by the sister sequence A181738 of Roger L. Bagula.

Cf. A254006 (T(n,0) up to sign), A138230 (row sums).

Needs["Quaternions`"]
P[x_, 0 ] := Quaternion[1, 0, 0, 0];
P[x_, n_] := P[x, n] = Quaternion[x, 1, 1, 1] ** P[x, n - 1];
Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten

R. = QQ[]
K = R.fraction_field()
H. = QuaternionAlgebra(K, -1, -1)
def Q(a, b, c, d): return H(a + b*i + c*j + d*k)
@cached_function
def P(n):
    return Q(x, 1, 1, 1)*P(n-1) if n > 0 else Q(1, 0, 0, 0)
def p(n): return P(n)[0].numerator().list()
flatten([p(n) for n in (0..10)]) # Kudos to William Stein

cp's OEIS Frontend

A047081 a(n) = Sum_{k=0..n} T(n, k), array T as in A047080.

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A163325 Pick digits at the even distance from the least significant end of the ternary expansion of n, then convert back to decimal.

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A052993 a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3), with a(0)=a(1)=1, a(2)=4.

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A124812 Number of 4-ary Lyndon words of length n with exactly four 1s.

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Magma

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A084156 Binomial transform of sinh(x)*cosh(sqrt(3)*x).

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A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

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A358027 Expansion of g.f.: (1 + x - 2*x^2 + 2*x^4)/((1-x)*(1-3*x^2)).

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A052993 a(n) = a(n-1) + 3a(n-2) - 3a(n-3), with a(0)=a(1)=1, a(2)=4.

A084156 Binomial transform of sinh(x)cosh(sqrt(3)x).

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).