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.

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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

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

A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   2;
  1, 3,  5,   6,   4;
  1, 4,  9,  14,  15,  10;
  1, 5, 14,  27,  38,  39,   25;
  1, 6, 20,  46,  79, 104,  102,   64;
  1, 7, 27,  72, 145, 229,  285,  270,  166;
  1, 8, 35, 106, 244, 446,  659,  784,  721,  436;
  1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;

Links

Crossrefs

Cf. A026269, A026270, A026288, A026289, A026290, A026291, A026292, A026293.
Cf. A026294, A026295, A026296, A026297, A026299, A026519, A026536, A026552.
Cf. A026584, A027926, A212342.

Programs

  • Magma
    f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n,k) // T = A026268
  if k eq 0 or n lt 3 then return 1;
  elif k eq 1 then return n-1;
  elif k eq 2 then return f(n);
  elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
  else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)
  • SageMath
    def T(n,k): # T = A026268
    if n<3 or k==0: return 1
    elif k==1: return n-1
    elif k==2: return (n^2 -n -2)//2 + int(n==2)
    elif k==n: return T(n-1, n-2) + T(n-1, n-1)
    else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022

Formula

From G. C. Greubel, Sep 24 2022: (Start)
T(n, 1) = A000027(n-1), n >= 1.
T(n, 2) = A212342(n-1), n >= 2.
T(n, n-1) = A026270(n), n >= 2.
T(n, n-2) = A026288(n), n >= 2.
T(n, n-3) = A026289(n), n >= 3.
T(n, n-4) = A026290(n), n >= 4.
T(n, n) = A026269(n), n >= 2.
T(n, floor(n/2)) = A026297(n), n >= 0.
T(2*n, n) = A026292(n).
T(2*n, n-1) = A026295(n), n >= 1.
T(2*n, n+1) = A026296(n), n >= 1.
T(2*n-1, n-1) = A026291(n), n >= 2.
T(3*n, n) = A026293(n), n >= 0.
T(4*n, n) = A026294(n), n >= 0.
Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)

Extensions

Updated by Clark Kimberling, Aug 29 2014
Indices of b-file corrected by Sidney Cadot, Jan 06 2023.

A026572 a(n) = T(n,n-3), T given by A026568. 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, 19, 56, 140, 376, 953, 2474, 6286, 16097, 40880, 104069, 264052, 670414, 1699831, 4310546, 10924970, 27690075, 70168812, 177820791, 450618964, 1142004584, 2894347667, 7336297080, 18597140982, 47148420564
Offset: 3

Views

Author

Clark Kimberling

Keywords

Comments

Also, a(n) = T(n,n-3), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.

Crossrefs

Cf. A026568, A026584.

Formula

Conjectured g.f.: (1/2)*((-3*x^2-x+1)*((x-1)/(4*x^2+x-1))^(1/2)-1+x+x^2)/x^3. - Mark van Hoeij, Oct 30 2011
Conjecture: (n+3)*a(n) +(-3*n-5)*a(n-1) +4*(-n-1)*a(n-2) +(13*n-9)*a(n-3) +(5*n-9)*a(n-4) +6*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026570 a(n) = A026568(n,n-1), also a(n) = number of integer strings s(0),...,s(n) counted by A026568 such that s(n)=1.

Original entry on oeis.org

1, 1, 4, 7, 20, 43, 111, 259, 648, 1565, 3885, 9533, 23662, 58547, 145630, 362151, 903110, 2253615, 5633359, 14094035, 35304658, 88511733, 222115782, 557819793, 1401987930, 3526066273, 8874034647, 22346581133, 56304982154
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Also a(n) = T'(n,n-1), T' given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T' such that s(n)=1.

Crossrefs

Cf. A026568, A026569, A026584, A026585.

Formula

Conjecture: (n+1)*a(n) -2*n*a(n-1) +(-3*n-1)*a(n-2) +2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013
If recurrence is correct then a(n) = (A026569(n+1)-A026569(n))/2 = A026585(n+1)/2. - Mark van Hoeij, Nov 29 2024
Previous Showing 21-25 of 25 results.

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

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