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 21 results. Next

A026530 a(n) = T(n, floor(n/2)), T given by A026519.

Original entry on oeis.org

1, 1, 1, 2, 5, 8, 16, 28, 65, 111, 251, 436, 1016, 1763, 4117, 7176, 16913, 29521, 69865, 122182, 290455, 508595, 1212905, 2126312, 5085224, 8923136, 21389824, 37563930, 90226449, 158563368, 381519416, 670893296, 1616684241, 2844444761
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, T[n, Floor[n/2]] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 21 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n, n//2) for n in (0..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = A026519(n, floor(n/2)).

A026519 Irregular triangular array T read by rows: T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 4, 14, 32, 65, 104, 150, 180, 196, 180, 150, 104, 65, 32, 14, 4, 1
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, |s(i) - s(i-1)| <= 1 if i is odd.

Examples

			First 5 rows:
1
1 ... 1 ... 1
1 ... 1 ... 2 ... 1 ... 1
1 ... 2 ... 4 ... 4 ... 4 ... 2 ... 1
1 ... 2 ... 5 ... 6 ... 8 ... 6 ... 5 ... 2 ... 1

Links

Crossrefs

Cf. A026552, A026536, A026568, A026584, A027926.
Cf. A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    z = 12; t[n_, 0]:= 1; t[n_, k_]:= 1/; k==2n; t[n_, 1]:= Floor[(n+1)/2]; t[n_, k_] := Floor[(n+1)/2] /; k==2n-1; t[n_, k_]:= t[n, k]= If[EvenQ[n], t[n-1, k-2] + t[n-1, k], t[n-1, k-2] + t[n-1, k-1] + t[n-1, k]];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2n}];
TableForm[u]  (* A026519 array *)
Flatten[u] (* A026519 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Dec 19 2021

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

Extensions

Updated by Clark Kimberling, Aug 29 2014
Offset changed to 0 by G. C. Greubel, Dec 19 2021

A026520 a(n) = T(n,n), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 0.

Original entry on oeis.org

1, 1, 2, 4, 8, 20, 38, 104, 196, 556, 1052, 3032, 5774, 16778, 32146, 93872, 180772, 529684, 1024256, 3008864, 5837908, 17184188, 33433996, 98577712, 192239854, 567591142, 1109049320, 3278348608, 6416509142, 18986482250
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
Table[T[n, n], {n,0,40}] (* G. C. Greubel, Dec 19 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n,n) for n in (0..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n).
For n>1, a(n) = 2*A026554(n-1).

A026527 a(n) = T(2*n, n-2), where T is given by A026519.

Original entry on oeis.org

1, 3, 14, 55, 231, 952, 3976, 16614, 69750, 293557, 1238952, 5240599, 22212645, 94318875, 401143304, 1708558480, 7286677479, 31113264579, 132994055090, 569048532612, 2437033824302, 10445705817063, 44807461337160, 192342179361800, 826205908069555, 3551172735996756, 15272395383833658
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026536.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n, n-2] ];
Table[a[n], {n,2,40}] (* G. C. Greubel, Dec 20 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(2*n,n-2) for n in (2..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n, n-2).
a(n) = A026536(2*n, n-2).

Extensions

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

A026534 a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).

Original entry on oeis.org

1, 4, 10, 28, 64, 172, 388, 1036, 2332, 6220, 13996, 37324, 83980, 223948, 503884, 1343692, 3023308, 8062156, 18139852, 48372940, 108839116, 290237644, 653034700, 1741425868, 3918208204, 10448555212, 23509249228, 62691331276
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A027262, A027263, A027264, A027265, A027266.
Cf. A026532, A026551.

Programs

  • Magma
    I:=[1,4,10]; [n le 3 select I[n] else Self(n-1) +6*Self(n-2) -6*Self(n-3): n in [1..40]]; // G. C. Greubel, Dec 20 2021
  • Mathematica
    LinearRecurrence[{1,6,-6}, {1,4,10}, 40] (* G. C. Greubel, Dec 20 2021 *)
  • PARI
    Vec((1+3*x)/((1-x)*(1-6*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jan 24 2022
  • Sage
    @CachedFunction
def T(n, k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( sum( T(j,i) for i in (0..2*n) ) for j in (0..n-1) )
[a(n) for n in (1..40)]

Formula

a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).
G.f.: x*(1+3*x)/((1-x)*(1-6*x^2)). - Ralf Stephan, Feb 03 2004
a(n) = (1/60)*( 6^((n+1)/2)*( (4*sqrt(6) - 9)*(-1)^n + (4*sqrt(6) + 9) ) - 48 ). - G. C. Greubel, Dec 20 2021

A026522 a(n) = T(n, n-2), where T is given by A026519. Also number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.

Original entry on oeis.org

1, 2, 5, 13, 27, 76, 150, 434, 845, 2470, 4797, 14085, 27377, 80584, 156900, 462620, 902394, 2664276, 5205950, 15387670, 30114073, 89097932, 174609162, 517058502, 1014555607, 3006637946, 5906040623, 17514547015, 34438443075
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
Table[T[n, n-2], {n,2,40}] (* G. C. Greubel, Dec 19 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n,n-2) for n in (2..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n-2).

A026523 a(n) = T(n, n-3), T given by A026519. 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, 2, 8, 16, 52, 104, 319, 635, 1910, 3786, 11304, 22344, 66514, 131264, 390266, 769578, 2286996, 4508580, 13397075, 26412001, 78489235, 154773696, 460030947, 907432695, 2697786052, 5323519838, 15830906756, 31251588060
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
Table[T[n, n-3], {n,3,40}] (* G. C. Greubel, Dec 19 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n,n-3) for n in (3..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n-3).

A026524 a(n) = T(n, n-4), T given by A026519. 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, 9, 28, 65, 201, 430, 1316, 2721, 8259, 16793, 50680, 102102, 306958, 615024, 1844304, 3682545, 11024331, 21963161, 65675764, 130648089, 390374193, 775797750, 2316881892, 4601346295, 13737041045, 27270124455
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
Table[T[n, n-4], {n,4,40}] (* G. C. Greubel, Dec 19 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+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n,n-4) for n in (4..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n-4).

A026525 a(n) = T(2*n, n), where T is given by A026519.

Original entry on oeis.org

1, 1, 5, 16, 65, 251, 1016, 4117, 16913, 69865, 290455, 1212905, 5085224, 21389824, 90226449, 381519416, 1616684241, 6863544233, 29187402749, 124305180842, 530108333515, 2263423401745, 9674857844129, 41396075156859, 177285394355336, 759895396193376, 3259667597627576, 13992851410449865
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026536.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, T[2 n, n] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 20 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(2*n, n) for n in (0..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n, n).
a(n) = A026536(2*n, n).

Extensions

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

A026526 a(n) = T(2n,n-1), T given by A026519.

Original entry on oeis.org

1, 2, 9, 32, 130, 516, 2107, 8632, 35703, 148390, 619850, 2598960, 10933507, 46124274, 195055005, 826617216, 3509650697, 14926011714, 63572290247, 271125967840, 1157705297385, 4948808238110, 21175676836110, 90692557152240, 388751132082600, 1667665994149376, 7159105163384127, 30753762496639504
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026536.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, T[2*n, n-1] ];
Table[a[n], {n,40}] (* G. C. Greubel, Dec 20 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(2*n,n-1) for n in (1..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n, n-1).
a(n) = A026536(2*n, n-1).

Extensions

Terms a(20) onward added by G. C. Greubel, Dec 20 2021
Showing 1-10 of 21 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.