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 1-7 of 7 results.

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

A026566 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026552.

Original entry on oeis.org

1, 3, 9, 20, 53, 117, 308, 684, 1806, 4028, 10664, 23844, 63239, 141612, 376026, 842866, 2239900, 5024166, 13359408, 29980384, 79753402, 179044760, 476451644, 1069936084, 2847931619, 6396900694, 17030741437, 38260956765
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, 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}, Sum[T[i,j], {i,0,n}, {j,0,i}]];
Table[a[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+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)
@CachedFunction
def a(n): return sum( sum( T(i,j) for j in (0..i) ) for i in (0..n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} A026552(i, j).

A027272 Self-convolution of row n of array T given by A026552.

Original entry on oeis.org

1, 3, 19, 58, 462, 1608, 13446, 48924, 417440, 1553940, 13409576, 50618184, 440013462, 1676640462, 14649846820, 56201554888, 492944907180, 1900789437276, 16721000706580, 64734185205960, 570792185166764, 2216888144737508, 19584623363041704, 76265067399850848
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, 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}, Sum[T[n, k]*T[n, 2*n-k], {k, 0, 2*n}]];
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)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,2*n-k) for k in (0..2*n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2*n} A026552(n, k)*A026552(n, 2*n-k).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027273 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026552.

Original entry on oeis.org

2, 16, 52, 428, 1516, 12792, 46936, 402164, 1504432, 13015480, 49288856, 429204354, 1639174304, 14340670000, 55108565584, 483825847108, 1868067054968, 16445659005424, 63734526307552, 562323306397388, 2185849699156352, 19320211642880176, 75288454939134992
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, 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}, Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}]];
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)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+1) for k in (0..2*n-1) )
[a(n) for n in (1..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2*n-1} A026552(n, k)*A026552(n, k+1).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027274 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026552.

Original entry on oeis.org

10, 40, 342, 1279, 11016, 41462, 359530, 1365014, 11899516, 45501743, 398306769, 1531614109, 13450930624, 51952990090, 457449811458, 1773182087440, 15646091896400, 60825762159338, 537651887201990, 2095280066101886, 18547910336883720, 72432026278468535
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, 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}, Sum[T[n, k]*T[n, k+2], {k, 0, 2*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)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+2) for k in (0..2*n-2) )
[a(n) for n in (2..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2*n-2} A026552(n,k) * A026552(n,k+2).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027275 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026552.

Original entry on oeis.org

24, 232, 954, 8560, 33648, 297940, 1159844, 10242416, 39809076, 351561242, 1367463642, 12086555584, 47082494816, 416589513644, 1625447736120, 14397549291280, 56265306436584, 498879779964188, 1952476424575980, 17327820010494464, 67907006619888744
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, 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}, Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}]];
Table[a[n], {n,3,40}] (* 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)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+3) for k in (0..2*n-3) )
[a(n) for n in (3..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2*n-3} A026552(n, k) * A026552(n, k+3).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027276 a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).

Original entry on oeis.org

1, 6, 27, 72, 270, 648, 2268, 5184, 17496, 38880, 128304, 279936, 909792, 1959552, 6298560, 13436928, 42830208, 90699264, 287214336, 604661760, 1904684544, 3990767616, 12516498432, 26121388032, 81629337600, 169789022208
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027275.

Programs

  • Magma
    I:= [6,27,72,270]; [1] cat [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 18 2021
  • Mathematica
    Table[-(1/2)*Boole[n==0] + (1/4)*6^(n/2)*(n+1)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)), {n, 0, 40}] (* G. C. Greubel, Dec 18 2021 *)
  • PARI
    a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;27;72])[1,1] \\ Charles R Greathouse IV, Oct 21 2022
  • 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)
@CachedFunction
def a(n): return sum( (k+1)*T(n,k) for k in (0..2*n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
G.f.: (1 +6*x +15*x^2 -18*x^3)/(1-6*x^2)^2.
a(n) = -(1/2)*[n=0] + (1/4)*6^(n/2)*(n + 1)*(3*(1 + (-1)^n) + sqrt(6)*(1 - (-1)^n)). - G. C. Greubel, Dec 18 2021
Showing 1-7 of 7 results.

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