A027052 -id:A027052 - cp's OEIS frontend

A027053 a(n) = T(n,n+2), T given by A027052.

1, 4, 9, 18, 35, 66, 123, 228, 421, 776, 1429, 2630, 4839, 8902, 16375, 30120, 55401, 101900, 187425, 344730, 634059, 1166218, 2145011, 3945292, 7256525, 13346832, 24548653, 45152014, 83047503, 152748174, 280947695, 516743376

Offset: 2

Clark Kimberling

Second differences of (A027026(n)-1)/2.
Pairwise sums of A089068.
a(n) is also the number of fixed polyominoes with n cells of height (or width) 2. - David Bevan, Sep 09 2009

G. C. Greubel, Table of n, a(n) for n = 2..1001
Doron Zeilberger, The generating functions and series expansions for 2D lattice-animals of globally bounded width. [Local copies of generating functions and series expansions].
Index entries for sequences related to polyominoes
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

2nd column of A308359.

a:=[1,4,9,18];; for n in [5..30] do a[n]:=2*a[n-1]-a[n-4]; od; a; # G. C. Greubel, Nov 05 2019

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!( x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 05 2019

seq(coeff(series(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)), x, n+1), x, n), n = 2 ..30); # G. C. Greubel, Nov 05 2019

LinearRecurrence[{2,0,0,-1}, {1,4,9,18}, 30] (* G. C. Greubel, Nov 05 2019 *)

my(x='x+O('x^32)); Vec(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 05 2019

def A027053_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))).list()
a=A027053_list(32); a[2:] # G. C. Greubel, Nov 05 2019

G.f.: x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)).
a(n) = A089068(n-1) + A089068(n).
a(n) = a(n-1) + a(n-2) + a(n-3) + 4. - David Bevan, Sep 09 2009
a(n) = A001590(n+3) - 2. - David Bevan, Sep 09 2009
a(n+1) - a(n) = A000213(n+1). - R. J. Mathar, Aug 04 2013

A027067 a(n) = Sum_{k=n..2*n} T(n,k), T given by A027052.

Original entry on oeis.org

1, 1, 4, 10, 27, 77, 220, 632, 1821, 5257, 15206, 44068, 127951, 372173, 1084382, 3164498, 9248241, 27064057, 79296978, 232597316, 682960523, 2007206245, 5904191878, 17380855190, 51203234981, 150943862857, 445250129556

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

T:= proc(n, k) option remember;
      if k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n, k), k=n..2*n), n=0..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k], {k,n,2*n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n, k) for k in (n..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

a(n) ~ 3^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Nov 06 2019

A027054 a(n) = T(n, n+3), T given by A027052.

Original entry on oeis.org

1, 8, 23, 52, 107, 210, 401, 754, 1405, 2604, 4811, 8872, 16343, 30086, 55365, 101862, 187385, 344688, 634015, 1166172, 2144963, 3945242, 7256473, 13346778, 24548597, 45151956, 83047443, 152748112, 280947631, 516743310

Offset: 3

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 3..1002
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

a:=[1,8,23,52,107];; for n in [6..33] do a[n]:=3*a[n-1]-2*a[n-2] -a[n-4]+a[n-5]; od; a; # G. C. Greubel, Nov 05 2019

R:=PowerSeriesRing(Integers(), 33); Coefficients(R!( x^3*(1+5*x+x^2-x^3-2*x^4)/((1-x)^2*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 05 2019

seq(coeff(series(x^3*(1+5*x+x^2-x^3-2*x^4)/((1-x)^2*(1-x-x^2-x^3)), x, n+1), x, n), n = 3..33); # G. C. Greubel, Nov 05 2019

LinearRecurrence[{3,-2,0,-1,1}, {1,8,23,52,107}, 30] (* G. C. Greubel, Nov 05 2019 *)

my(x='x+O('x^33)); Vec( x^3*(1+5*x+x^2-x^3-2*x^4)/((1-x)^2*(1-x-x^2-x^3)) ) \\ G. C. Greubel, Nov 05 2019

def A027053_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^3*(1+5*x+x^2-x^3-2*x^4)/((1-x)^2*(1-x-x^2-x^3)) ).list()
a=A027053_list(33); a[3:] # G. C. Greubel, Nov 05 2019

From Colin Barker, Feb 19 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-4) + a(n-5) for n>6.
G.f.: x^3*(1+5*x+x^2-x^3-2*x^4)/((1-x)^2*(1-x-x^2-x^3)). (End)
a(n) = A001590(n+4) -2*n -4, n>=3. - R. J. Mathar, Jun 15 2020

A027055 a(n) = T(n, n+4), T given by A027052.

Original entry on oeis.org

1, 18, 59, 146, 319, 652, 1281, 2456, 4637, 8670, 16111, 29822, 55067, 101528, 187013, 344276, 633561, 1165674, 2144419, 3944650, 7255831, 13346084, 24547849, 45151152, 83046581, 152747190, 280946647, 516742262, 950438067

Offset: 4

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 4..1003
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

a:=[1,18,59,146,319,652];; for n in [7..40] do a[n]:=4*a[n-1] -5*a[n-2]+2*a[n-3]-a[n-4]+2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Nov 06 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^4*(1+14*x-8*x^2-2*x^3-5*x^4+4*x^5)/((1-x)^3*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 06 2019

seq(coeff(series(x^4*(1+14*x-8*x^2-2*x^3-5*x^4+4*x^5)/((1-x)^3*(1-x-x^2-x^3)), x, n+1), x, n), n = 4..40); # G. C. Greubel, Nov 06 2019

LinearRecurrence[{4,-5,2,-1,2,-1}, {1,18,59,146,319,652}, 40] (* G. C. Greubel, Nov 06 2019 *)

my(x='x+O('x^40)); Vec(x^4*(1+14*x-8*x^2-2*x^3-5*x^4+4*x^5)/((1-x)^3*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 06 2019

def A027053_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4*(1+14*x-8*x^2-2*x^3-5*x^4+4*x^5)/((1-x)^3*(1-x-x^2-x^3))).list()
a=A027053_list(40); a[4:] # G. C. Greubel, Nov 06 2019

From Colin Barker, Feb 19 2016: (Start)
a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) -a(n-4) +2*a(n-5) -a(n-6) for n>9.
G.f.: x^4*(1+14*x-8*x^2-2*x^3-5*x^4+4*x^5)/((1-x)^3*(1-x-x^2-x^3)).
(End)
a(n) = A001590(n+5) -n*(5+n), n>=4. - R. J. Mathar, Jun 15 2020

A027056 a(n) = A027052(n, 2n-1).

Original entry on oeis.org

0, 2, 4, 8, 18, 42, 102, 256, 658, 1722, 4570, 12264, 33212, 90626, 248892, 687360, 1907506, 5316266, 14873082, 41751944, 117567784, 331979650, 939807344, 2666718976, 7583071868, 21605822594, 61672362872, 176338826728, 505001067346, 1448365610778, 4159725843526, 11962301199744

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-1), n=1..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-1], {n,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-1) for n in (1..30)] # G. C. Greubel, Nov 06 2019

Conjecture:b D-finite with recurrence (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-7*n+3)*a(n-2) +4*(-2*n+13)*a(n-3) +(5*n-29)*a(n-4) +2*(n-2)*a(n-5) +3*(n-5)*a(n-6)=0. - R. J. Mathar, Jun 15 2020

a(n) ~ 3^(n + 3/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

More terms from Sean A. Irvine, Oct 22 2019

A027057 a(n) = (1/2) * A027052(n, 2n-1).

Original entry on oeis.org

1, 2, 4, 9, 21, 51, 128, 329, 861, 2285, 6132, 16606, 45313, 124446, 343680, 953753, 2658133, 7436541, 20875972, 58783892, 165989825, 469903672, 1333359488, 3791535934, 10802911297, 30836181436, 88169413364, 252500533673, 724182805389, 2079862921763, 5981150599872

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-1)/2, n=2..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-1]/2, {n,2,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-1)/2 for n in (2..30)] # G. C. Greubel, Nov 06 2019

More terms from Sean A. Irvine, Oct 22 2019

A027058 a(n) = A027052(n, 2n-2).

Original entry on oeis.org

1, 1, 3, 9, 23, 59, 153, 401, 1063, 2847, 7693, 20947, 57413, 158265, 438467, 1220145, 3408759, 9556815, 26878861, 75815839, 214411865, 607827693, 1726911631, 4916352891, 14022750725, 40066540277, 114666463855, 328662240617

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-2), n=1..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-2], {n,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-2) for n in (1..30)] # G. C. Greubel, Nov 06 2019

Conjecture: D-finite with recurrence n*a(n) +(-7*n+4)*a(n-1) +(13*n-10)*a(n-2) +(n-34)*a(n-3) +(-13*n+84)*a(n-4) +(3*n-32)*a(n-5) +(-n+6)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Jun 15 2020

a(n) ~ 3^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

Offset changed to 1 and a(1)=1 prepended to sequence by G. C. Greubel, Nov 06 2019

A027059 a(n) = A027052(n, 2n-3).

Original entry on oeis.org

0, 2, 6, 18, 52, 146, 406, 1126, 3124, 8684, 24202, 67640, 189576, 532786, 1501254, 4240550, 12005780, 34063896, 96844082, 275848044, 787104288, 2249633916, 6439678858, 18460717684, 52994100984, 152323413890, 438363476086

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-3), n=2..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-3], {n,2,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-3) for n in (2..30)] # G. C. Greubel, Nov 06 2019

Conjecture D-finite with recurrence (n+1)*a(n) +2*(-4*n-1)*a(n-1) +(19*n-5)*a(n-2) +6*(-n-5)*a(n-3) +3*(-7*n+41)*a(n-4) +2*(4*n-29)*a(n-5) +(n+1)*a(n-6) +6*(n-5)*a(n-7)=0. - R. J. Mathar, Jun 15 2020

a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A027060 a(n) = T(n,2n-4), T given by A027052.

Original entry on oeis.org

1, 1, 3, 11, 35, 107, 319, 935, 2713, 7825, 22491, 64523, 184945, 530001, 1519151, 4356471, 12501301, 35901325, 103188123, 296844379, 854701935, 2463133311, 7104685935, 20510632575, 59262772629, 171373598341, 495968905267

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-4), n=2..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-4], {n,2,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-4) for n in (2..30)] # G. C. Greubel, Nov 06 2019

A027061 a(n) = A027052(n, 2n-5).

Original entry on oeis.org

0, 2, 6, 20, 66, 210, 652, 1988, 5982, 17830, 52782, 155480, 456364, 1336066, 3904280, 11394244, 33222902, 96812174, 282009512, 821327088, 2391918708, 6966267782, 20291422370, 59116724728, 172271893036, 502157965938

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..750

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( T(n,2*n-5), n=3..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-5], {n,3,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n,2*n-5) for n in (3..30)] # G. C. Greubel, Nov 06 2019

cp's OEIS Frontend

A027053 a(n) = T(n,n+2), T given by A027052.

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A027067 a(n) = Sum_{k=n..2*n} T(n,k), T given by A027052.

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A027054 a(n) = T(n, n+3), T given by A027052.

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A027055 a(n) = T(n, n+4), T given by A027052.

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A027056 a(n) = A027052(n, 2n-1).

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A027057 a(n) = (1/2) * A027052(n, 2n-1).

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Maple

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Sage

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A027058 a(n) = A027052(n, 2n-2).

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