cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 29 results. Next

A027033 a(n) = T(n,2n-9), T given by A027023.

Original entry on oeis.org

1, 3, 9, 31, 105, 355, 1197, 4011, 13329, 43883, 143105, 462391, 1481229, 4707743, 14856441, 46585671, 145253757, 450624055, 1391743825, 4281348119, 13124142489, 40105164499, 122213161617, 371496978671, 1126750503081
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-9), n=5..30); # G. C. Greubel, Nov 05 2019
  • Mathematica
    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-9], {n,5,30}] (* G. C. Greubel, Nov 05 2019 *)
  • Sage
    @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-9) for n in (5..30)] # G. C. Greubel, Nov 05 2019

A027034 a(n) = T(n,2n-10), T given by A027023.

Original entry on oeis.org

1, 1, 5, 17, 57, 193, 653, 2205, 7417, 24805, 82373, 271437, 887377, 2878509, 9268429, 29636981, 94163769, 297435285, 934521973, 2922073641, 9096981049, 28209178729, 87163760797, 268462020889, 824451264113, 2525238433145
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-10), n=5..30); # G. C. Greubel, Nov 05 2019
  • Mathematica
    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-10], {n,5,30}] (* G. C. Greubel, Nov 05 2019 *)
  • Sage
    @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-10) for n in (5..30)] # G. C. Greubel, Nov 05 2019

A027037 Diagonal sum of left-justified array T given by A027023.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 7, 11, 16, 21, 33, 48, 65, 101, 146, 203, 311, 450, 635, 963, 1396, 1989, 2993, 4348, 6233, 9329, 13574, 19543, 29135, 42446, 61303, 91123, 132884, 192377, 285309, 416384, 603925, 894069, 1305618, 1896495, 2803611, 4096182, 5957183, 8796287
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    T:= proc(n, k) option remember;
      if n<0 or k>2*n then 0
    elif 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), k=0..n), n=0..30); # G. C. Greubel, Nov 05 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0 || k>2*n, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n-k, k], {k, 0, n}], {n,0,30}] (* G. C. Greubel, Nov 05 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0 or k>2*n): return 0
    elif (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, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..n} A027023(n-k, k). - Sean A. Irvine, Oct 22 2019

Extensions

More terms from Sean A. Irvine, Oct 21 2019

A027038 Diagonal sum of right-justified array T given by A027023.

Original entry on oeis.org

1, 1, 2, 5, 7, 18, 43, 103, 264, 687, 1809, 4836, 13049, 35493, 97218, 267857, 741791, 2063574, 5763595, 16155403, 45429488, 128121191, 362287433, 1026918632, 2917313257, 8304598593, 23685134746, 67669857661, 193652803391
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027023.

Programs

  • Maple
    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,2*n-3*k), k=0..n), n=0..30); # G. C. Greubel, Nov 05 2019
  • Mathematica
    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[Sum[T[n-k, 2*n-3*k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Nov 05 2019 *)
  • Sage
    @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, 2*n-3*k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..n} T(n-k, 2*n-3*k), where T = A027023. - G. C. Greubel, Nov 05 2019
a(n) ~ 3^(n + 7/2) / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 09 2025

A027040 a(n) = self-convolution of row n of array T given by A027023.

Original entry on oeis.org

1, 3, 9, 31, 129, 531, 2129, 8351, 32177, 122211, 458801, 1706015, 6293169, 23057651, 83992313, 304424639, 1098525761, 3948727555, 14145206209, 50515602111, 179904080257, 639103899411, 2265253438745, 8012421964063
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    T:= proc(n, k) option remember;
      if (n<0 or k>2*n) then 0
    elif 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)*T(n,2*n-k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 05 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0 || k>2*n, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k]*T[n, 2*n-k], {k, 0, 2*n}], {n, 0, 30}] (* G. C. Greubel, Nov 05 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0 or k>2*n): return 0
    elif (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)*T(n,2*n-k) for k in (0..2*n)) for n in (4..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..2*n} T(n,k)*T(n,2*n-k), where T = A027023. - G. C. Greubel, Nov 05 2019

A027041 a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027023.

Original entry on oeis.org

1, 2, 5, 20, 77, 306, 1209, 4656, 17713, 66618, 248025, 916020, 3359789, 12250026, 44435997, 160466304, 577185745, 2068826290, 7392167585, 26338879556, 93609302941, 331924381218, 1174482354493, 4147807582672, 14622567051025, 51466158436298, 180869949252245, 634753692067716
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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)*T(n,2*n-k), k=0..n), n=0..30); # G. C. Greubel, Nov 04 2019
  • Mathematica
    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]*T[n,2*n-k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 04 2019 *)
  • Sage
    @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)*T(n,2*n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 04 2019

Extensions

More terms from Sean A. Irvine, Oct 22 2019

A027042 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027023.

Original entry on oeis.org

1, 4, 11, 52, 225, 920, 3695, 14464, 55593, 210776, 789995, 2933380, 10807625, 39556316, 143958335, 521340016, 1879901265, 6753038624, 24176722555, 86294777316, 307179518193, 1090771084252, 3864614381391, 13664531314176, 48225146757337, 169905685271956, 597661852713467
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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)*T(n,2*n-k), k=0..n-1), n=1..30); # G. C. Greubel, Nov 04 2019
  • Mathematica
    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]*T[n,2*n-k], {k,0,n-1}], {n,30}] (* G. C. Greubel, Nov 04 2019 *)
  • Sage
    @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)*T(n,2*n-k) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Nov 04 2019

Extensions

More terms from Sean A. Irvine, Oct 22 2019

A027043 a(n) = Sum_{k=0..2n} (k+1) * A027023(n,k).

Original entry on oeis.org

1, 6, 23, 80, 285, 990, 3367, 11256, 37097, 120862, 390123, 1249728, 3978365, 12598350, 39718403, 124743104, 390491505, 1218875302, 3794984883, 11789335464, 36551285573, 113120066678, 349523991123, 1078402178776
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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((k+1)*T(n,k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 04 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[(k+1)*T[n,k], {k,0,2*n}], {n,0,30}] (* G. C. Greubel, Nov 04 2019 *)
  • Sage
    @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((k+1)*T(n, k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 04 2019

A027044 a(n) = Sum_{k=0..2n} (k+1) * A027023(n,2n-k).

Original entry on oeis.org

1, 6, 19, 56, 165, 486, 1435, 4248, 12601, 37438, 111367, 331608, 988181, 2946662, 8791447, 26241632, 78359825, 234069830, 699404127, 2090385216, 6249236653, 18686125070, 55884824535, 167164064984, 500102988889
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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((k+1)*T(n,2*n-k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 04 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[(k+1)*T[n,2*n-k], {k,0,2*n}], {n,0,30}] (* G. C. Greubel, Nov 04 2019 *)
  • Sage
    @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((k+1)*T(n, 2*n-k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 04 2019

A027045 a(n) = Sum_{k=n+1..2*n} T(n, k), T given by A027023.

Original entry on oeis.org

1, 4, 11, 34, 103, 306, 901, 2636, 7685, 22372, 65111, 189590, 552547, 1612154, 4709369, 13773368, 40329465, 118217992, 346891115, 1018872626, 2995250535, 8812601062, 25948130525, 76456539156, 225427875325, 665066293480
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027023.

Programs

  • Magma
    function T(n,k)
  if k lt 3 or k eq 2*n then return 1;
  else return (&+[T(n-1,k-j): j in [1..3]]);
  end if; return T; end function;
[(&+[T(n,k): k in [n+1..2*n]]): n in [1..15]]; // G. C. Greubel, Nov 20 2019
  • Maple
    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+1..2*n), n=1..30); # G. C. Greubel, Nov 04 2019
  • Mathematica
    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+1,2*n}], {n,30}] (* G. C. Greubel, Nov 04 2019 *)
  • Sage
    @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+1..2*n)) for n in (1..30)] # G. C. Greubel, Nov 04 2019

Extensions

Offset changed by G. C. Greubel, Nov 04 2019
Previous Showing 11-20 of 29 results. Next

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