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 11-18 of 18 results.

A026594 a(n) = T(2*n-1, n-2), where T is given by A026584.

Original entry on oeis.org

1, 2, 13, 42, 225, 802, 4235, 15478, 82425, 304156, 1634435, 6064389, 32819839, 122244344, 665162897, 2484851486, 13577768505, 50841782786, 278745377821, 1045763359942, 5749240499515, 21603797860416, 119040956286133, 447922312642212, 2472886893122590, 9315646385012666, 51514464212546865, 194255376492836212
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]]]]; (*T=A026584*)
Table[T[2*n-1, n-2], {n, 2, 40}]  (* G. C. Greubel, Dec 13 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
[T(2*n-1, n-2) for n in (2..40)] # G. C. Greubel, Dec 13 2021

Formula

a(n) = A026584(2*n-1, n-2).

Extensions

Terms a(19) onward added by G. C. Greubel, Dec 13 2021

A026596 Row sums of A026584.

Original entry on oeis.org

1, 1, 4, 8, 23, 54, 143, 354, 914, 2306, 5907, 15012, 38368, 97804, 249865, 637834, 1629729, 4163398, 10640753, 27196246, 69526562, 177757762, 454541197, 1162403180, 2972953385, 7604223184, 19451741733, 49761433640, 127308417226
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:=a[n]= Sum[T[n,k], {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 13 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A026596(n): return sum( T(n, j) for j in (0..n) )
[A026596(n) for n in (0..40)] # G. C. Greubel, Dec 13 2021

Formula

a(n) = Sum_{k=0..n} A026584(n, k).
Conjecture: n*a(n) -3*(n-1)*a(n-1) -(5*n-6)*a(n-2) +3*(5*n-13)*a(n-3) +2*(4*n-9)*a(n-4) -8*(2*n-9)*a(n-5) = 0. - R. J. Mathar, Jun 23 2013

A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.

Original entry on oeis.org

1, 2, 6, 14, 37, 91, 234, 588, 1502, 3808, 9715, 24727, 63095, 160899, 410764, 1048598, 2678327, 6841725, 17482478, 44678724, 114205286, 291963048, 746504245, 1908907425, 4881860810, 12486083994, 31937825727, 81699259367
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026599, A027282, A027283, A027284, A027285, A027286.

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/2], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]];
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 15 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A026598(n): return sum(sum(T(i,j) for j in (0..i)) for i in (0..n))
[A026598(n) for n in (0..40)] # G. C. Greubel, Dec 15 2021

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} A026584(i, j).
Conjecture: n*a(n) - (4*n-3)*a(n-1) - (2*n-3)*a(n-2) + 5*(4*n-9)*a(n-3) - 7*(n-3)*a(n-4) - 6*(4*n-15)*a(n-5) + 8*(2*n-9)*a(n-6) = 0. - R. J. Mathar, Jun 23 2013

A027282 a(n) = self-convolution of row n of array T given by A026584.

Original entry on oeis.org

1, 2, 8, 40, 222, 1296, 7770, 47324, 291260, 1806220, 11266718, 70609316, 444231564, 2803975860, 17748069294, 112609964308, 716010467122, 4561107325336, 29103104031990, 185973253609716, 1189979068401564, 7623432519587692, 48891854980251090, 313874287333373820
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Bisection of A026585.

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027283, A027284, A027285, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, 2*n-k], {k,0,2*n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 15 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027282(n): return sum(T(n,j)*T(n, 2*n-j) for j in (0..2*n))
[A027282(n) for n in (0..40)] # G. C. Greubel, Dec 15 2021

Formula

a(n) = Sum_{k=0..2*n} A026584(n, k)*A026584(n, 2*n-k).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027283 a(n) = Sum_{k=0..2*n-1} T(n,k) * T(n,k+1), with T given by A026584.

Original entry on oeis.org

0, 6, 26, 206, 1100, 7314, 42920, 274010, 1677332, 10616070, 66290046, 419754586, 2648500908, 16818685050, 106781976774, 680250643910, 4337083126232, 27709045093274, 177213890858938, 1135003956744310, 7276652578220372, 46702733068082702, 300013046145979184
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027284, A027285, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}];
Table[a[n], {n, 1, 40}] (* G. C. Greubel, Dec 15 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027283(n): return sum(T(n,j)*T(n, j+1) for j in (0..2*n-1))
[A027283(n) for n in (1..40)] # G. C. Greubel, Dec 15 2021

Formula

a(n) = Sum_{k=0..2*n-1} A026584(n,k) * A026584(n,k+1).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027284 a(n) = Sum_{k=0..2*n-2} T(n,k) * T(n,k+2), with T given by A026584.

Original entry on oeis.org

5, 28, 167, 1024, 6359, 39759, 249699, 1573524, 9943905, 62994733, 399936573, 2543992514, 16210331727, 103453402718, 661164765879, 4230874777682, 27105456280491, 173838468040879, 1115987495619427, 7170725839251598, 46113396476943241, 296773029762031990
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027285, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 15 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027284(n): return sum(T(n,j)*T(n, j+2) for j in (0..2*n-2))
[A027284(n) for n in (2..40)] # G. C. Greubel, Dec 15 2021

Formula

a(n) = Sum_{k=0..2*n-2} A026584(n,k) * A026584(n,k+2).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027285 a(n) = Sum_{k=0..2*n-3} T(n,k) * T(n,k+3), with T given by A026584.

Original entry on oeis.org

12, 116, 682, 4908, 30272, 201648, 1273286, 8275894, 52783298, 340392020, 2180905198, 14035736838, 90149817980, 580197442656, 3732734480794, 24041345351898, 154874693823022, 998441294531516, 6439238635990250, 41552345665859196, 268252644944872486
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027286.

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], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}];
Table[a[n], {n, 3, 40}] (* G. C. Greubel, Dec 15 2021 *)
  • Sage
    @CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027285(n): return sum(T(n,j)*T(n, j+3) for j in (0..2*n-3))
[A027285(n) for n in (3..40)] # G. C. Greubel, Dec 15 2021

Formula

a(n) = Sum_{k=0..2*n-3} A026584(n,k) * A026584(n,k+3).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A026581 Expansion of (1 + 2*x) / (1 - x - 4*x^2).

Original entry on oeis.org

1, 3, 7, 19, 47, 123, 311, 803, 2047, 5259, 13447, 34483, 88271, 226203, 579287, 1484099, 3801247, 9737643, 24942631, 63893203, 163663727, 419236539, 1073891447, 2750837603, 7046403391, 18049753803, 46235367367, 118434382579, 303375852047, 777113382363
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

T(n,0) + T(n,1) + ... + T(n,2n), T given by A026568.
Row sums of Riordan array ((1+2x)/(1+x),x(1+2x)/(1+x)). Binomial transform is A055099. - Paul Barry, Jun 26 2008
Equals row sums of triangle A153341. - Gary W. Adamson, Dec 24 2008
Also, the number of walks of length n starting at vertex 0 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - Sean A. Irvine, Jun 02 2025

Links

Crossrefs

Cf. A006131, A026568, A026583, A026597, A026599, A052923, A055099.
Cf. A153341. - Gary W. Adamson, Dec 24 2008

Programs

  • GAP
    a:=[1,3];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Aug 03 2019
  • Magma
    I:=[1,3]; [n le 2 select I[n] else Self(n-1) +4*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2019
  • Mathematica
    CoefficientList[Series[(1+2x)/(1-x-4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{1,4},{1,3},30] (* Harvey P. Dale, Aug 04 2015 *)
  • PARI
    Vec((1+2*x)/(1-x-4*x^2) + O(x^30)) \\ Colin Barker, Dec 22 2016
  • Sage
    ((1+2*x)/(1-x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

Formula

G.f.: (1 + 2*x) / (1 - x - 4*x^2).
a(n) = a(n-1) + 4*a(n-2), n>1.
a(n) = 2*A006131(n-1) + A006131(n), n>0.
a(n) = (2^(-1-n)*((1-sqrt(17))^n*(-5+sqrt(17)) + (1+sqrt(17))^n*(5+sqrt(17))))/sqrt(17). - Colin Barker, Dec 22 2016

Extensions

Edited by Ralf Stephan, Jul 20 2013
Previous Showing 11-18 of 18 results.

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