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 21-30 of 32 results. Next

A026568 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = [ (i+1)/2 ] for i >= 1; and for i >= 2 and 2 <=j <= i - 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i + j is even, T(i,j) = T(i-1,j-2) + T(i-1,j) if i + j is odd.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 4, 5, 4, 2, 1, 1, 2, 7, 7, 13, 7, 7, 2, 1, 1, 3, 8, 16, 20, 27, 20, 16, 8, 3, 1, 1, 3, 12, 19, 44, 43, 67, 43, 44, 19, 12, 3, 1, 1, 4, 13, 34, 56, 106, 111, 153, 111, 106, 56, 34, 13, 4, 1, 1, 4, 18, 38, 103, 140, 273
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

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

Examples

			First 5 rows:
  1
  1  1  1
  1  1  3  1  1
  1  2  4  5  4  2  1
  1  2  7  7 13  7  7  2  1

Links

Crossrefs

Cf. A026519, A026536, A026552, A026584, A027926.
Cf. T(n,n) is A026569.

Programs

  • Mathematica
    z = 12; t[n_, 0] := 1; t[n_, 1] := Floor[(n + 1)/2]; t[n_, k_] := t[n, k] = Which[k == 2 n, 1, k == 2 n - 1, Floor[(n + 1)/2], EvenQ[n + k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], OddQ[n + k], t[n - 1, k - 2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A026568 array *)
Flatten[u]   (* A026568 sequence *)
  • PARI
    T(k,n)=if(n<0||n>2*k,0,if(n==0||n==2*k,1,if(k>0&&(n==1||n==2*k-1),(k+1)\2,T(k-1,n-2)+T(k-1,n)+if((k+n)%2==0,T(k-1,n-1))))) \\ Ralf Stephan

Extensions

Updated by Clark Kimberling, Aug 28 2014

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

A026521 a(n) = T(n, n-1), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 1.

Original entry on oeis.org

1, 1, 4, 6, 19, 33, 98, 180, 526, 990, 2887, 5502, 16073, 30863, 90386, 174456, 512128, 992304, 2918954, 5673140, 16716998, 32571858, 96119927, 187675644, 554524660, 1084649644, 3208254571, 6284986554, 18607536319, 36501029265
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026537.

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-1], {n,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-1) for n in (1..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n-1).
a(n) = A026537(n+1)/2.

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).

A026528 a(n) = T(2*n-1, n-1), T given by A026519.

Original entry on oeis.org

1, 2, 8, 28, 111, 436, 1763, 7176, 29521, 122182, 508595, 2126312, 8923136, 37563930, 158563368, 670893296, 2844444761, 12081753410, 51400091942, 218990735668, 934228356445, 3990177231742, 17060699906541, 73017457810032, 312785412844736, 1340988707637776, 5753539499846507
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026552.

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-1, 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-1,n-1) for n in (1..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n-1, n-1).
a(n) = A026552(2*n-1, n-1).

Extensions

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

A026529 a(n) = T(2*n-1, n-2), where T is given by A026519.

Original entry on oeis.org

1, 3, 13, 50, 205, 833, 3437, 14232, 59301, 248050, 1041469, 4385888, 18519306, 78376403, 332370925, 1412000824, 6008104249, 25601113893, 109229104313, 466577280830, 1995120743749, 8539562784258, 36583756253885, 156854365793800, 673028595199000, 2889847430222961, 12416501973954798, 53381063233213198
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026552.

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-1, n-2] ];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 20 2021 *)
  • Maxima
    a(n):=sum(binomial(n-1,i-1)*sum(binomial(j,n-j+2*i)*binomial(n,j),j,0,n),i,1,n/2); /* Vladimir Kruchinin, Jan 16 2015 */
  • 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-1,n-2) for n in (2..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n-1, n-2).
a(n) = A026552(2*n-1, n-2).
a(n) = Sum_{i=0..floor(n/2)} C(n-1, i-1)*Sum_{j=0..n} C(j, n-j+2*i)*C(n, j). - Vladimir Kruchinin, Jan 16 2015

Extensions

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

A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 3, 2, 3, 1, 1, 1, 2, 5, 6, 8, 6, 5, 2, 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, 5, 17, 40, 76, 116, 150, 162, 150, 116, 76, 40, 17, 5, 1, 1, 6, 23, 62, 133, 232, 342, 428, 462, 428, 342, 232, 133, 62, 23, 6
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

For n > 2, T(n,k) is the number of integer strings s(0), ..., s(n) such that s(n) = n - k, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2 and <= 1 for i >= 3.

Examples

			                  1
               1  0  1
            1  0  2  0  1
         1  1  3  2  3  1  1
      1  2  5  6  8  6  5  2  1
   1  3  8 13 19 20 19 13  8  3  1

Links

Crossrefs

First differences in n, n direction of array A025177.
Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579.
Cf. A027907, A026552, A024072.

Programs

  • Julia
    using Nemo
function A024996Expansion(prec)
    R, t = PolynomialRing(ZZ, "t")
    S, x = PowerSeriesRing(R, prec+1, "x")
    ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1)
    L = zeros(ZZ, prec^2)
    for k ∈ 0:prec-1, n ∈ 0:2*k
        L[k^2+n+1] = coeff(coeff(ser, k), n)
    end
    L
end
A024996Expansion(8) |> println # Peter Luschny, Jun 25 2020
  • Maple
    A024996 := proc(n,k)
    option remember;
    if n < 0 or k < 0 or k > 2*n then
        0 ;
    elif n <= 2 then
        if k = 2*n or k = 0 then
            1;
        elif k = 2*n-1 or k = 1 then
            0;
        elif k =2 then
            2;
        end if;
    else
        procname(n-1,k-1)+procname(n-1,k-2)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
seq(seq(A024996(n,k), k=0..2*n), n=0..11); # added by Georg Fischer, Jun 24 2020
  • Mathematica
    nmax = 10; CoefficientList[CoefficientList[Series[y*x + (1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)
  • PARI
    T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,0,1][k+1],if(n==2,[1,0,2,0,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k))))) \\ Ralf Stephan, Jan 09 2004
nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n,k),","))) \\ added by _Georg Fischer, Jun 24 2020

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1], [1, 0, 2, 0, 1].
G.f.: y*z + (1-y*z)^2 / (1-z*(1+y+y^2)). - Ralf Stephan, Jan 09 2005 [corrected by Peter Luschny, Jun 25 2020]

Extensions

Edited by Ralf Stephan, Jan 09 2004
Offset corrected by R. J. Mathar, Jun 23 2013
Previous Showing 21-30 of 32 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.