A026520 -id:A026520 - cp's OEIS frontend

A027266 a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).

1, 6, 18, 72, 180, 648, 1512, 5184, 11664, 38880, 85536, 279936, 606528, 1959552, 4199040, 13436928, 28553472, 90699264, 191476224, 604661760, 1269789696, 3990767616, 8344332288, 26121388032, 54419558400, 169789022208

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027265.

I:=[1,6,18,72]; [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 21 2021

CoefficientList[Series[(1+6x+6x^2)/(1-6x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{0,12,0,-36},{1,6,18,72},30] (* Harvey P. Dale, Jun 19 2015 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;18;72])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

[((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ) for n in (0..40)] # G. C. Greubel, Dec 21 2021

a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).
G.f.: (1+6*x+6*x^2)/(1-6*x^2)^2.
a(n) = 12*a(n-2) - 36*a(n-4), with a(0)=1, a(1)=6, a(2)=18, a(3)=72. - Harvey P. Dale, Jun 19 2015
a(n) = ((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ). - G. C. Greubel, Dec 21 2021

A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026519.

Original entry on oeis.org

1, 3, 7, 18, 40, 104, 231, 607, 1353, 3575, 7989, 21169, 47384, 125757, 281798, 748638, 1678832, 4463098, 10014074, 26635050, 59787092, 159078450, 357193976, 950678416, 2135189511, 5684158586, 12769030254, 33999245582

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026534, A027262, A027263, A027264, A027265, A027266.

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}, Sum[T[i,j], {i,0,n}, {j,0,i}] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 20 2021 *)

@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(i,j) for j in (0..i)) for i in (0..n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 20 2021

a(n) = Sum_{i=0..n} Sum_{j=0..i} A026519(i,j).

A026554 a(n) = T(n,n-1), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=1.

Original entry on oeis.org

1, 2, 4, 10, 19, 52, 98, 278, 526, 1516, 2887, 8389, 16073, 46936, 90386, 264842, 512128, 1504432, 2918954, 8592094, 16716998, 49288856, 96119927, 283795571, 554524660, 1639174304, 3208254571, 9493241125, 18607536319, 55108565584

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A026520.

Cf. A026552, A026553, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

T[n_, k_]:= T[n, k]= If[k>2*n, 0, 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]]]]];
Table[T[n, n-1], {n, 40}] (* G. C. Greubel, Dec 17 2021 *)

@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)
[T(n,n-1) for n in (1..40)] # G. C. Greubel, Dec 17 2021

a(n) = A026520(n+1)/2.

cp's OEIS Frontend

A027266 a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).

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A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026519.

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Author

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Links

Crossrefs

Programs

Mathematica

Sage

Formula

A026554 a(n) = T(n,n-1), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Sage

Formula