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 39 results. Next

A027062 a(n) = A027052(n, 2n-6).

Original entry on oeis.org

1, 1, 3, 11, 37, 123, 401, 1281, 4023, 12461, 38175, 115939, 349701, 1049063, 3133493, 9327357, 27687947, 82009215, 242473197, 715889685, 2111215763, 6220468653, 18314669783, 53892395679, 158512474561, 466071547105
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-6), n=3..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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-6], {n,3,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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-6) for n in (3..30)] # G. C. Greubel, Nov 06 2019

A027063 a(n) = A027052(n, 2n-7).

Original entry on oeis.org

0, 2, 6, 20, 68, 228, 754, 2456, 7884, 24982, 78282, 242998, 748364, 2289584, 6966346, 21098366, 63651808, 191406976, 573998990, 1717334182, 5127933348, 15286303526, 45503354154, 135287179508, 401809091392, 1192336418386
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-7), n=4..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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-7], {n,4,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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-7) for n in (4..30)] # G. C. Greubel, Nov 06 2019

A027064 a(n) = A027052(n, 2n-8).

Original entry on oeis.org

1, 1, 3, 11, 37, 125, 421, 1405, 4637, 15125, 48777, 155665, 492157, 1543269, 4804663, 14865495, 45745953, 140118817, 427445507, 1299383403, 3937901525, 11902380845, 35891429675, 108009437323, 324455779889, 973119941425
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-8), n=4..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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-8], {n,4,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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-8) for n in (4..30)] # G. C. Greubel, Nov 06 2019

A027065 a(n) = A027052(n, 2n-9).

Original entry on oeis.org

0, 2, 6, 20, 68, 230, 776, 2604, 8670, 28606, 93494, 302748, 971810, 3094486, 9782092, 30721056, 95919714, 297938906, 921183940, 2836545972, 8702745304, 26614653494, 81159163058, 246855070144, 749123740876, 2268700324802
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-9), n=5..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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-9], {n,5,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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-9) for n in (5..30)] # G. C. Greubel, Nov 06 2019

A027066 a(n) = A027052(n, 2n-10).

Original entry on oeis.org

1, 1, 3, 11, 37, 125, 423, 1429, 4811, 16111, 53589, 176905, 579407, 1882943, 6073469, 19452705, 61900375, 195799527, 615978629, 1928297807, 6009527345, 18653079889, 57686469763, 177812890843, 546456642501, 1674844848629
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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-10), n=5..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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-10], {n,5,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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-10) for n in (5..30)] # G. C. Greubel, Nov 06 2019

A027068 a(n) = Sum_{i=0..n} Sum_{j=i..2*i} A027052(i, j).

Original entry on oeis.org

1, 2, 6, 16, 43, 120, 340, 972, 2793, 8050, 23256, 67324, 195275, 567448, 1651830, 4816328, 14064569, 41128626, 120425604, 353022920, 1035983443, 3043189688, 8947381566, 26328236756, 77531471737, 228475334594, 673725464150
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027052. Partial sums of A027067.

Programs

  • Maple
    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( add(add(T(k,j), j=k..2*k), k=0..n), n=0..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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[Sum[Sum[T[i, j], {j, i, 2*i}], {i, 0, n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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))
[sum(sum(T(k,j) for j in (k..2*k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

Extensions

Title corrected by Sean A. Irvine, Oct 22 2019

A027069 a(n) = diagonal sum of left-justified array T given by A027052.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 11, 14, 22, 32, 43, 67, 97, 134, 206, 298, 419, 637, 923, 1312, 1978, 2872, 4111, 6161, 8961, 12888, 19232, 28010, 40423, 60129, 87665, 126840, 188216, 274634, 398151, 589689, 861001, 1250210, 1848840, 2700900, 3926839, 5799949, 8476579
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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( add(T(n-k,k), k=0..n), n=0..50); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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[Sum[T[n-k,k], {k,0,n}], {n, 0, 50}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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))
[sum(T(n-k,k) for k in (0..n)) for n in (0..50)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = Sum_{k=0..n} A027052(n - k, k). - Sean A. Irvine, Oct 22 2019

Extensions

More terms from Sean A. Irvine, Oct 22 2019

A027070 a(n) = diagonal sum of right-justified array T given by A027052.

Original entry on oeis.org

1, 1, 1, 4, 6, 12, 31, 73, 183, 476, 1248, 3322, 8943, 24271, 66355, 182538, 504824, 1402682, 3913585, 10959499, 30792445, 86775340, 245204312, 694603032, 1972115945, 5610955925, 15994866669, 45677496204, 130661330526, 374339736820, 1074025873959, 3085699969569, 8876601230175
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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( add(T(n-k,2*n-3*k), k=0..n), n=0..35); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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[Sum[T[n-k, 2*n-3*k], {k, 0, n}], {n,0,35}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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))
[sum(T(n-k,2*n-3*k) for k in (0..n)) for n in (0..35)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = Sum_{k=0..n} A027052(n-k, 2*n-3*k). - Sean A. Irvine, Oct 22 2019

Extensions

More terms from Sean A. Irvine, Oct 22 2019

A027072 a(n) = self-convolution of row n of array T given by A027052.

Original entry on oeis.org

1, 2, 3, 12, 53, 222, 899, 3540, 13657, 51882, 194727, 723760, 2668453, 9771870, 35577935, 128887616, 464885073, 1670362418, 5981289455, 21352860808, 76020123293, 269977176422, 956644165503, 3382864303648, 11940005836537
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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( add(T(n,k)*T(n,2*n-k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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[Sum[T[n,k]*T[n,2*n-k], {k,0,2*n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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))
[sum(T(n,k)*T(n,2*n-k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = Sum_{k=0..2*n} T(n,k)*T(n,2*n-k), where T = A027052. - G. C. Greubel, Nov 06 2019

A027073 a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027052.

Original entry on oeis.org

1, 1, 2, 8, 31, 129, 510, 1970, 7513, 28253, 105176, 388330, 1423691, 5188577, 18812848, 67907520, 244160177, 874821817, 3124747792, 11130097846, 39544807851, 140180597013, 495886522916, 1750852227736, 6171019594129
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Maple
    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( add(T(n,k)*T(n,2*n-k), k=0..n), n=0..30); # G. C. Greubel, Nov 06 2019
  • Mathematica
    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[Sum[T[n, k]*T[n, 2*n - k], {k, 0, n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)
  • Sage
    @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))
[sum(T(n,k)*T(n,2*n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019
Previous Showing 11-20 of 39 results. Next

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