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.

Showing 1-10 of 17 results. Next

A026552 Irregular triangular array T read by rows: T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2 + 1), for even n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), otherwise T(n, k) = T(n-1, k-2) + T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)|<=1 if i is even or i = 1, |s(i)-s(i-1)| = 1 if i is odd and i >= 3.

Examples

			First 5 rows:
  1;
  1, 1, 1;
  1, 2, 3,  2,  1;
  1, 2, 4,  4,  4,  2,  1;
  1, 3, 7, 10, 12, 10,  7,  3,  1;

Links

Crossrefs

Cf. A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
Cf. A026519, A026536, A026568, A026584, A027926.
Cf. A026565.

Programs

  • Mathematica
    z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2 + 1]; t[n_, k_] := Floor[n/2 + 1] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A026552 array *)
v = Flatten[u] (* A026552 sequence *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021

Formula

Sum_{k=0..2*n} T(n,k) = A026565(n). - G. C. Greubel, Dec 17 2021

Extensions

Updated by Clark Kimberling, Aug 28 2014

A026567 a(n) = Sum_{i=0..2*n} Sum_{j=0..i} T(i, j), where T is given by A026552.

Original entry on oeis.org

1, 4, 13, 31, 85, 193, 517, 1165, 3109, 6997, 18661, 41989, 111973, 251941, 671845, 1511653, 4031077, 9069925, 24186469, 54419557, 145118821, 326517349, 870712933, 1959104101, 5224277605, 11754624613, 31345665637, 70527747685
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A027272, A027273, A027274, A027275, A027276.
Cf. A026565, A233325.

Programs

  • Magma
    [Truncate((2*(1+(-1)^n)*6^((n+2)/2) + 27*(1-(-1)^n)*6^((n-1)/2) -14)/10): n in [0..40]]; // G. C. Greubel, Dec 19 2021
  • Mathematica
    CoefficientList[Series[(1 +3x +3x^2)/((1-x)(1-6x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 25 2014 *)
LinearRecurrence[{1,6,-6},{1,4,13},30] (* Harvey P. Dale, Aug 23 2014 *)
  • Sage
    [(1/10)*(2*(1+(-1)^n)*6^((n+2)/2) +27*(1-(-1)^n)*6^((n-1)/2) -14) for n in (0..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = Sum_{i=0..2*n} Sum_{j=0..i} A026552(i, j).
G.f.: (1+3*x+3*x^2)/((1-x)*(1-6*x^2)). - Ralf Stephan, Feb 03 2004
a(n) = 6*a(n-2) + 7. - Philippe Deléham, Feb 24 2014
a(2*k) = A233325(k). - Philippe Deléham, Feb 24 2014
From Colin Barker, Nov 25 2016: (Start)
a(n) = (2^(n/2+2) * 3^(n/2+1) - 7)/5 for n even.
a(n) = (2^((n-1)/2) * 3^((n+5)/2) - 7)/5 for n odd. (End)
a(n) = (1/10)*(2*(1+(-1)^n)*6^((n+2)/2) + 27*(1-(-1)^n)*6^((n-1)/2) - 14). - G. C. Greubel, Dec 19 2021

A026554 a(n) = T(n,n-1), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=1.

Original entry on oeis.org

1, 2, 4, 10, 19, 52, 98, 278, 526, 1516, 2887, 8389, 16073, 46936, 90386, 264842, 512128, 1504432, 2918954, 8592094, 16716998, 49288856, 96119927, 283795571, 554524660, 1639174304, 3208254571, 9493241125, 18607536319, 55108565584
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026520.
Cf. A026552, A026553, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]];
Table[T[n, n-1], {n, 40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(n,n-1) for n in (1..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026520(n+1)/2.

A026553 a(n) = T(n,n), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.

Original entry on oeis.org

1, 1, 3, 4, 12, 20, 58, 104, 300, 556, 1608, 3032, 8806, 16778, 48924, 93872, 274644, 529684, 1553940, 3008864, 8846772, 17184188, 50618184, 98577712, 290817566, 567591142, 1676640462, 3278348608, 9694857750, 18986482250
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
Table[T[n,n], {n,0,40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(n,n) for n in (0..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026552(n,n).

A026556 a(n) = T(n, n-3), T given by A026552. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 3.

Original entry on oeis.org

1, 3, 8, 24, 52, 156, 319, 954, 1910, 5696, 11304, 33648, 66514, 197778, 390266, 1159844, 2286996, 6795576, 13397075, 39809076, 78489235, 233262931, 460030947, 1367463642, 2697786052, 8021305890, 15830906756, 47082494816
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
Table[T[n,n-3], {n,3,40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(n,n-3) for n in (3..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026552(n, n-3).

A026557 a(n) = T(n, n-4), T given by A026552. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

Original entry on oeis.org

1, 3, 12, 28, 93, 201, 631, 1316, 4037, 8259, 25052, 50680, 152782, 306958, 921982, 1844304, 5526849, 11024331, 32987492, 65675764, 196323853, 390374193, 1166171943, 2316881892, 6918228187, 13737041045, 41007165500
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026558, A026559, A026560, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
Table[T[n,n-4], {n,4,40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(n,n-4) for n in (4..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026552(n, n-4).

A026558 a(n) = T(2*n, n), where T is given by A026552.

Original entry on oeis.org

1, 2, 7, 24, 93, 362, 1452, 5880, 24089, 99386, 412637, 1721500, 7211536, 30312960, 127790379, 540082784, 2287577537, 9707988994, 41269156159, 175705272784, 749099069183, 3197651758190, 13665035075871, 58456775063400, 250302852165368, 1072680809038112, 4600656305265352, 19746390910296372
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026559, A026560, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n, n]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(2*n,n) for n in (0..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026552(2*n, n).

Extensions

Terms a(20) onward from G. C. Greubel, Dec 17 2021

A026559 a(n) = T(2*n, n-1), where T is given by A026552.

Original entry on oeis.org

1, 3, 12, 45, 180, 721, 2940, 12069, 49935, 207691, 867900, 3640429, 15319395, 64643580, 273431408, 1158988141, 4921651521, 20934115963, 89173404140, 380355072153, 1624282578215, 6943928981859, 29715239620368, 127276313406125, 545605497876400, 2340694589348376, 10048952593607088, 43170264470594302
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026560, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n, n-1]];
Table[a[n], {n,40}] (* G. C. Greubel, Dec 17 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(2*n,n-1) for n in (1..40)] # G. C. Greubel, Dec 17 2021

Formula

a(n) = A026552(2*n, n-1)

Extensions

Terms a(20) onward added by G. C. Greubel, Dec 17 2021

A026560 a(n) = T(2*n, n-2), where T is given by A026552.

Original entry on oeis.org

1, 4, 18, 74, 311, 1296, 5432, 22796, 95958, 404812, 1711600, 7250970, 30772989, 130810512, 556867224, 2373764416, 10130935783, 43285462884, 185129287262, 792525473552, 3395664830670, 14560682746632, 62482560679368, 268307898599664, 1152883194581155, 4956738399534376, 21323028570642414, 91775945084805898
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n, n-2]];
Table[a[n], {n,2,40}] (* G. C. Greubel, Dec 18 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(2*n,n-2) for n in (2..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = A026552(2*n, n-2).

Extensions

Terms a(20) onward from G. C. Greubel, Dec 18 2021

A026563 a(n) = T(n, floor(n/2)), where T is given by A026552.

Original entry on oeis.org

1, 1, 2, 2, 7, 8, 24, 28, 93, 111, 362, 436, 1452, 1763, 5880, 7176, 24089, 29521, 99386, 122182, 412637, 508595, 1721500, 2126312, 7211536, 8923136, 30312960, 37563930, 127790379, 158563368, 540082784, 670893296, 2287577537
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[n, Floor[n/2]]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Dec 18 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
[T(n,n//2) for n in (0..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = A026552(n, floor(n/2)).
Showing 1-10 of 17 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.