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 1-9 of 9 results.

A026765 a(n) = Sum_{k=0..n} T(n,k), T given by A026758.

Original entry on oeis.org

1, 2, 4, 9, 18, 41, 82, 188, 376, 867, 1734, 4020, 8040, 18735, 37470, 87735, 175470, 412715, 825430, 1949624, 3899248, 9245721, 18491442, 44003717, 88007434, 210121733, 420243466, 1006390014, 2012780028, 4833517551
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(add(T(n,k), k=0..n), n=0..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[n,k],{k,0,n}], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 31 2019

Formula

Conjecture: G.f.: -(1-2*x-5*x^2+10*x^3 - sqrt(1-10*x^2+29*x^4-20*x^6) )/(2*x*(1-2*x-5*x^2+10*x^3)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
Conjecture: (n+1)*a(n) -2*a(n-1) +2*(-5*n+3)*a(n-2) +12*a(n-3) +(29*n-71)*a(n-4) -10*a(n-5) +20*(-n+5)*a(n-6)=0. - R. J. Mathar, Jun 30 2013
Conjecture: a(n) ~ (2+sqrt(5) + (-1)^n*(2-sqrt(5))) * 5^(n/2) / sqrt(2*Pi*n). - Vaclav Kotesovec, Feb 12 2014

A026759 a(n) = T(2n, n), T given by A026758.

Original entry on oeis.org

1, 2, 7, 27, 109, 453, 1922, 8284, 36155, 159435, 709246, 3178992, 14343567, 65099245, 297015765, 1361584755, 6268757195, 28975155915, 134410918700, 625578384150, 2920488902795, 13672762887465, 64179220019365, 301987822527627
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026760, A026761, A026762, A026763, A026764, A026765, A026766, A026767, A026768.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( ((1-x)*Sqrt(1 - 4*x) - Sqrt(1 - 6*x + 5*x^2))/(2*x^2) )); // G. C. Greubel, Oct 31 2019
  • Maple
    seq(coeff(series(((1-x)*sqrt(1-4*x) - sqrt(1 -6*x +5*x^2))/(2*x^2), x, n+2), x, n), n = 0..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    CoefficientList[Normal[Series[((1-x)Sqrt[1-4x] -Sqrt[1-6x+5x^2])/(2x^2), {x, 0, 30}]], x] (* David Callan, Feb 01 2014 *)
  • PARI
    my(x='x+O('x^30)); Vec(((1-x)*sqrt(1 - 4*x) - sqrt(1 - 6*x + 5*x^2))/(2*x^2)) \\ G. C. Greubel, Oct 31 2019
  • Sage
    def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(((1-x)*sqrt(1-4*x) - sqrt(1-6*x+5*x^2))/(2*x^2)).list()
A077952_list(30) # G. C. Greubel, Oct 31 2019

Formula

a(n) = A002212(n+1) - A000245(n). - David Callan, Feb 01 2014
G.f.: ((1-x)*sqrt(1 - 4*x) - sqrt(1 - 6*x + 5*x^2))/(2*x^2). - G. C. Greubel, Oct 31 2019

A026760 a(n) = T(2n, n-1), T given by A026758.

Original entry on oeis.org

1, 5, 23, 104, 469, 2119, 9607, 43727, 199819, 916631, 4220267, 19497608, 90370622, 420136173, 1958787580, 9156770130, 42912496696, 201579245739, 949002525067, 4477049676288, 21162505063028, 100217666089863, 475421115762173
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026761, A026762, A026763, A026764, A026765, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n,n-1), n=1..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[T[2 n, n-1], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-1) for n in (1..30)] # G. C. Greubel, Oct 31 2019

A026761 a(n) = T(2n, n-2), T given by A026758.

Original entry on oeis.org

1, 8, 48, 259, 1328, 6622, 32483, 157739, 761128, 3657815, 17534231, 83925062, 401363296, 1918822635, 9173429111, 43866599736, 209853869150, 1004463716937, 4810867131369, 23057388013314, 110588897473219, 530808778620583
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026760, A026762, A026763, A026764, A026765, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n,n-2), n=2..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[T[2 n, n-2], {n, 2, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-2) for n in (2..30)] # G. C. Greubel, Oct 31 2019

A026762 a(n) = T(2n-1,n-1), T given by A026758. Also T(2n+1,n+1), T given by A026747.

Original entry on oeis.org

1, 4, 16, 66, 279, 1201, 5242, 23133, 103015, 462269, 2088146, 9487405, 43328580, 198798447, 915950385, 4236322720, 19661850045, 91549502656, 427539667095, 2002120576312, 9399659155395, 44234927105888, 208631813215116
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026747, A026758, A026759, A026760, A026761, A026763, A026764, A026765, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n-1,n-1), n=1..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[T[2n-1, n-1], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-1) for n in (1..30)] # G. C. Greubel, Oct 31 2019

A026763 a(n) = T(2n-1,n-2), T given by A026758.

Original entry on oeis.org

1, 7, 38, 190, 918, 4365, 20594, 96804, 454362, 2132121, 10010203, 47042042, 221337726, 1042837195, 4920447410, 23250646651, 110029743083, 521462857972, 2474929099976, 11762845907633, 55982738983975, 266789302547057
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026764, A026765, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n-1,n-2), n=2..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[T[2n-1, n-2], {n, 2, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-2) for n in (2..30)] # G. C. Greubel, Oct 31 2019

A026764 a(n) = T(n, floor(n/2)), T given by A026758.

Original entry on oeis.org

1, 1, 2, 4, 7, 16, 27, 66, 109, 279, 453, 1201, 1922, 5242, 8284, 23133, 36155, 103015, 159435, 462269, 709246, 2088146, 3178992, 9487405, 14343567, 43328580, 65099245, 198798447, 297015765, 915950385, 1361584755, 4236322720
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026765, A026766, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(n, floor(n/2)), n=0..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]] ]; Table[T[n, Floor[n/2]], {n,0,30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(n, floor(n/2)) for n in (0..30)] # G. C. Greubel, Oct 31 2019

A026766 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026758.

Original entry on oeis.org

1, 1, 3, 5, 13, 24, 59, 115, 273, 552, 1278, 2655, 6031, 12795, 28632, 61775, 136572, 298764, 653948, 1447225, 3141427, 7020833, 15132512, 34106865, 73069892, 165903082, 353576829, 807957495, 1714132308, 3939206346
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026765, A026767, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[n,k], {k,0,Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 31 2019

A026767 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026758.

Original entry on oeis.org

1, 3, 7, 16, 34, 75, 157, 345, 721, 1588, 3322, 7342, 15382, 34117, 71587, 159322, 334792, 747507, 1572937, 3522561, 7421809, 16667530, 35158972, 79162689, 167170123, 377291856, 797535322, 1803925336, 3816705364
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Partial sums of A026765.

Links

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026765, A026766, A026768.

Programs

  • Maple
    T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq( add(add(T(j,k), k=0..n), j=0..n), n=0..30); # G. C. Greubel, Oct 31 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[j,k], {k,0,n}, {j,0,n}], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(sum(T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 31 2019

Formula

Conjecture: (n+1)*a(n) +(-n-3)*a(n-1) +2*(-5*n+4)*a(n-2) +2*(5*n+3)*a(n-3) +(29*n-83)*a(n-4) +(-29*n+61)*a(n-5) +10*(-2*n+11)*a(n-6) +20*(n-5)*a(n-7)=0. - R. J. Mathar, Jun 30 2013
Showing 1-9 of 9 results.

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