A027023 -id:A027023 - cp's OEIS frontend

A027024 a(n) = T(n,n+2), T given by A027023.

1, 5, 13, 27, 53, 101, 189, 351, 649, 1197, 2205, 4059, 7469, 13741, 25277, 46495, 85521, 157301, 289325, 532155, 978789, 1800277, 3311229, 6090303, 11201817, 20603357, 37895485, 69700667, 128199517, 235795677, 433695869

Offset: 2

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 2..1001
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Pairwise sums of A027053.

a:=[1,5,13,27];; for n in [5..35] do a[n]:=2*a[n-1]-a[n-4]; od; a; # G. C. Greubel, Nov 04 2019

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 04 2019

seq(coeff(series(x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3)), x, n+1), x, n), n = 2..35); # G. C. Greubel, Nov 04 2019

Drop[CoefficientList[Series[x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3)), {x, 0, 35}], x], 2] (* or *) LinearRecurrence[{2,0,0,-1}, {1,5,13,27}, 35] (* G. C. Greubel, Nov 04 2019 *)

my(x='x+O('x^35)); Vec(x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 04 2019

def A027024_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3)) ).list()
a=A027024_list(35); a[2:] # G. C. Greubel, Nov 04 2019

G.f.: x^2*(1+x)^3/((1-x)*(1-x-x^2-x^3)).
a(n) = a(n-1) + a(n-2) + a(n-3) + 8, for n>4. - Greg Dresden, Feb 09 2020
a(n) = A000213(n+2)-4. - R. J. Mathar, Jun 24 2020

A027026 a(n) = T(n,n+4), T given by A027023.

Original entry on oeis.org

1, 25, 85, 215, 477, 985, 1949, 3755, 7113, 13329, 24805, 45959, 84917, 156625, 288573, 531323, 977873, 1799273, 3310133, 6089111, 11200525, 20601961, 37893981, 69699051, 128197785, 235793825, 433693893, 797688967, 1467180389

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:=[1,25,85,215,477,985];; 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 04 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^4*(1 +21*x -10*x^2 -2*x^3 -7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 04 2019

seq(coeff(series(x^4*(1 +21*x -10*x^2 -2*x^3 -7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3)), x, n+1), x, n), n = 4..40); # G. C. Greubel, Nov 04 2019

Drop[CoefficientList[Series[x^4*(1+21*x-10*x^2-2*x^3-7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3)), {x,0,40}], x], 4] (* or *) LinearRecurrence[{4, -5, 2,-1,2,-1}, {1,25,85,215,477,985}, 40] (* G. C. Greubel, Nov 04 2019 *)

my(x='x+O('x^40)); Vec(x^4*(1 +21*x -10*x^2 -2*x^3 -7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 04 2019

def A027026_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4*(1 +21*x -10*x^2 -2*x^3 -7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3))).list()
a=A027026_list(50); a[4:] # G. C. Greubel, Nov 04 2019

G.f.: x^4*(1 +21*x -10*x^2 -2*x^3 -7*x^4 +5*x^5)/((1-x)^3*(1-x-x^2-x^3)). - Ralf Stephan, Feb 11 2004

a(n) = A000213(n+4) -2*n*(n+3), n>3. - R. J. Mathar, Jun 24 2020

A027025 a(n) = T(n,n+3), T given by A027023.

Original entry on oeis.org

1, 11, 33, 77, 161, 319, 613, 1157, 2161, 4011, 7417, 13685, 25217, 46431, 85453, 157229, 289249, 532075, 978705, 1800189, 3311137, 6090207, 11201717, 20603253, 37895377, 69700555, 128199401, 235795557, 433695745, 797690943

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:=[1,11,33,77,161];; for n in [6..30] do a[n]:=3*a[n-1]-2*a[n-2]-a[n-4] +a[n-5]; od; a; # G. C. Greubel, Nov 04 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^3*(1+8*x+2*x^2-3*x^4)/((1-x)^2*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 04 2019

seq(coeff(series(x^4/((1+2*x)*(2*x^3+x^2-2*x+1)), x, n+1), x, n), n = 3..40); # G. C. Greubel, Nov 04 2019

Drop[CoefficientList[Series[x^3*(1+8*x+2*x^2-3*x^4)/((1-x)^2*(1-x-x^2-x^3)), {x,0,40}], x], 3] (* or *) LinearRecurrence[{3,-2,0,-1,1}, {1, 11,33,77,161}, 40] (* G. C. Greubel, Nov 04 2019 *)

my(x='x+O('x^40)); Vec(x^3*(1+8*x+2*x^2-3*x^4)/((1-x)^2*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 04 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^3*(1+8*x+2*x^2-3*x^4)/((1-x)^2*(1-x-x^2-x^3))).list()
a=A077952_list(40); a[3:] # G. C. Greubel, Nov 04 2019

G.f.: x^3*(1+8*x+2*x^2-3*x^4)/((1-x)^2*(1-x-x^2-x^3)).

a(n) = A000213(n+3) -4*(n+1). - R. J. Mathar, Jun 24 2020

A027035 a(n) = Sum_{k=0..n} T(n,n+k), T given by A027023.

Original entry on oeis.org

1, 2, 5, 14, 39, 112, 323, 932, 2693, 7790, 22565, 65466, 190243, 553748, 1614363, 4713432, 13780841, 40343210, 118243273, 346937614, 1018958151, 2995407840, 8812890391, 25948662684, 76457517949, 225429675606, 665069604713

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<3 or k=2*n then 1
    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 04 2019

T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[T[n,k], {k,n,2*n}], {n,0,30}] (* G. C. Greubel, Nov 04 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    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 04 2019

A027027 a(n) = T(n, 2n-3), T given by A027023.

Original entry on oeis.org

1, 3, 9, 27, 77, 215, 597, 1655, 4593, 12775, 35629, 99651, 279501, 786071, 2216437, 6264663, 17746897, 50380895, 143307269, 408388819, 1165819757, 3333448075, 9545909641, 27375525727, 78612676241, 226034151539, 650692800633

Offset: 2

Clark Kimberling

nonn

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

Cf. A027023.

T:= proc(n, k) option remember;
      if k<3 or k=2*n then 1
    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 04 2019

T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]; Table[T[n, 2*n-3], {n, 2, 30}] (* G. C. Greubel, Nov 04 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    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 04 2019

Conjecture: D-finite with recurrence (n+1)*a(n) +(-8*n-1)*a(n-1) +(19*n-14)*a(n-2) +2*(-3*n-1)*a(n-3) +(-21*n+89)*a(n-4) +(8*n-45)*a(n-5) +(n-4)*a(n-6) +6*(n-4)*a(n-7)=0. - R. J. Mathar, Jun 24 2020

a(n) ~ 3^(n + 7/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A027028 a(n) = T(n,2n-4), T given by A027023.

Original entry on oeis.org

1, 1, 5, 17, 53, 161, 477, 1393, 4033, 11617, 33365, 95681, 274209, 785793, 2252509, 6460433, 18542169, 53260481, 153115765, 440572993, 1268830877, 3657435745, 10551936125, 30469329025, 88056216233, 254690980449

Offset: 2

Clark Kimberling

nonn

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

Cf. A027023.

T:= proc(n, k) option remember;
      if k<3 or k=2*n then 1
    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 04 2019

T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]; Table[T[n, 2*n-4], {n, 2, 30}] (* G. C. Greubel, Nov 04 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    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 04 2019

Conjecture D-finite with recurrence -(n+2)*(n-6)*a(n) +3*(n+1)*(2*n-11)*a(n-1) -n*(7*n-31)*a(n-2) -2*(n-3)*(4*n-19)*a(n-3) +(5*n^2-38*n+84)*a(n-4) +(2*n-5)*(n-3)*a(n-5) +3*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 24 2020

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

A027029 a(n) = T(n,2n-5), T given by A027023.

Original entry on oeis.org

1, 3, 9, 31, 101, 319, 985, 2991, 8973, 26687, 78877, 232083, 680645, 1991487, 5817073, 16971415, 49474389, 144149959, 419869065, 1222785111, 3561052305, 10371483259, 30211361481, 88022087975, 256521698097, 747794233779

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<3 or k=2*n then 1
    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 04 2019

T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]; Table[T[n, 2*n-5], {n, 3, 30}] (* G. C. Greubel, Nov 04 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    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 04 2019

A027030 a(n) = T(n,2n-6), T given by A027023.

Original entry on oeis.org

1, 1, 5, 17, 57, 189, 613, 1949, 6097, 18825, 57525, 174353, 525049, 1573077, 4693909, 13960805, 41415089, 122603341, 362343053, 1069436317, 3152995209, 9287942097, 27341397469, 80443393889, 236581555233, 695561538977

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<3 or k=2*n then 1
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq(T(n,2*n-6), n=3..30); # G. C. Greubel, Nov 05 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[T[n, 2*n-6], {n,3,30}] (* G. C. Greubel, Nov 05 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n, 2*n-6) for n in (3..30)] # G. C. Greubel, Nov 04 2019

A027031 a(n) = T(n,2n-7), T given by A027023.

Original entry on oeis.org

1, 3, 9, 31, 105, 351, 1157, 3755, 12013, 37951, 118613, 367383, 1129345, 3449823, 10482869, 31713863, 95589753, 287224199, 860773781, 2573894583, 7681972113, 22890634939, 68116073369, 202458285647, 601159897137

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..750

T:= proc(n, k) option remember;
      if k<3 or k=2*n then 1
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq(T(n,2*n-7), n=4..30); # G. C. Greubel, Nov 05 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[T[n, 2*n-7], {n,4,30}] (* G. C. Greubel, Nov 05 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n, 2*n-7) for n in (4..30)] # G. C. Greubel, Nov 05 2019

A027032 a(n) = T(n,2n-8), T given by A027023.

Original entry on oeis.org

1, 1, 5, 17, 57, 193, 649, 2161, 7113, 23137, 74417, 236913, 747401, 2339137, 7270189, 22460801, 69031105, 211206529, 643684485, 1955082321, 5920720729, 17884040961, 53898818389, 162120056257, 486791503521, 1459448379329

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..750

T:= proc(n, k) option remember;
      if k<3 or k=2*n then 1
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq(T(n,2*n-8), n=4..30); # G. C. Greubel, Nov 05 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[T[n, 2*n-8], {n,4,30}] (* G. C. Greubel, Nov 05 2019 *)

@CachedFunction
def T(n, k):
    if (k<3 or k==2*n): return 1
    else: return sum(T(n-1, k-j) for j in (1..3))
[T(n, 2*n-8) for n in (4..30)] # G. C. Greubel, Nov 05 2019

cp's OEIS Frontend

A027024 a(n) = T(n,n+2), T given by A027023.

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A027026 a(n) = T(n,n+4), T given by A027023.

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A027025 a(n) = T(n,n+3), T given by A027023.

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A027035 a(n) = Sum_{k=0..n} T(n,n+k), T given by A027023.

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A027027 a(n) = T(n, 2n-3), T given by A027023.

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A027028 a(n) = T(n,2n-4), T given by A027023.

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Maple

Mathematica

Sage

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A027029 a(n) = T(n,2n-5), T given by A027023.

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