A026520 -id:A026520 - cp's OEIS frontend

A291088 a(n) = A026520(2n).

1, 2, 8, 38, 196, 1052, 5774, 32146, 180772, 1024256, 5837908, 33433996, 192239854, 1109049320, 6416509142, 37215072638, 216309089956, 1259663964184, 7347943049432, 42926944354480, 251119894730596, 1470830479288432, 8624336421678788, 50620679934081988

Offset: 0

N. J. A. Sloane, Aug 19 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..500

Cf. A026520, A291089.

a(15)-a(23) from Lars Blomberg, Sep 01 2017

A291089 a(n) = A026520(2n+1).

Original entry on oeis.org

1, 4, 20, 104, 556, 3032, 16778, 93872, 529684, 3008864, 17184188, 98577712, 567591142, 3278348608, 18986482250, 110217131168, 641125473092, 3736134109936, 21807240851480, 127469052615104, 746057665449076, 4371699398312704, 25644387465768860

Offset: 0

N. J. A. Sloane, Aug 19 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..500

Cf. A026520, A291088.

a(15)-a(22) from Lars Blomberg, Sep 01 2017

A026530 a(n) = T(n, floor(n/2)), T given by A026519.

Original entry on oeis.org

1, 1, 1, 2, 5, 8, 16, 28, 65, 111, 251, 436, 1016, 1763, 4117, 7176, 16913, 29521, 69865, 122182, 290455, 508595, 1212905, 2126312, 5085224, 8923136, 21389824, 37563930, 90226449, 158563368, 381519416, 670893296, 1616684241, 2844444761

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, A026531, A026532, A026533, 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}, T[n, Floor[n/2]] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 21 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)
[T(n, n//2) for n in (0..40)] # G. C. Greubel, Dec 21 2021

a(n) = A026519(n, floor(n/2)).

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, 2n) = 1, T(n, 1) = T(n, 2n-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

Clark Kimberling

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.

			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

Clark Kimberling, Table of n, a(n) for n = 0..10200 (Rows 0..100, flattened) [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.
Index entries for triangles and arrays related to Pascal's triangle

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.

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 *)

@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

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).

Updated by Clark Kimberling, Aug 29 2014

Offset changed to 0 by G. C. Greubel, Dec 19 2021

A026527 a(n) = T(2*n, n-2), where T is given by A026519.

Original entry on oeis.org

1, 3, 14, 55, 231, 952, 3976, 16614, 69750, 293557, 1238952, 5240599, 22212645, 94318875, 401143304, 1708558480, 7286677479, 31113264579, 132994055090, 569048532612, 2437033824302, 10445705817063, 44807461337160, 192342179361800, 826205908069555, 3551172735996756, 15272395383833658

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Cf. A026536.

T[n_, k_]:= T[n, k]= 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}, T[2*n, n-2] ];
Table[a[n], {n,2,40}] (* G. C. Greubel, Dec 20 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+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)
[T(2*n,n-2) for n in (2..40)] # G. C. Greubel, Dec 20 2021

a(n) = A026519(2*n, n-2).

a(n) = A026536(2*n, n-2).

Terms a(20) onward added by G. C. Greubel, Dec 20 2021

A026534 a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).

Original entry on oeis.org

1, 4, 10, 28, 64, 172, 388, 1036, 2332, 6220, 13996, 37324, 83980, 223948, 503884, 1343692, 3023308, 8062156, 18139852, 48372940, 108839116, 290237644, 653034700, 1741425868, 3918208204, 10448555212, 23509249228, 62691331276

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A027262, A027263, A027264, A027265, A027266.

Cf. A026532, A026551.

I:=[1,4,10]; [n le 3 select I[n] else Self(n-1) +6*Self(n-2) -6*Self(n-3): n in [1..40]]; // G. C. Greubel, Dec 20 2021

LinearRecurrence[{1,6,-6}, {1,4,10}, 40] (* G. C. Greubel, Dec 20 2021 *)

Vec((1+3*x)/((1-x)*(1-6*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jan 24 2022

@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(j,i) for i in (0..2*n) ) for j in (0..n-1) )
[a(n) for n in (1..40)]

a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).
G.f.: x*(1+3*x)/((1-x)*(1-6*x^2)). - Ralf Stephan, Feb 03 2004
a(n) = (1/60)*( 6^((n+1)/2)*( (4*sqrt(6) - 9)*(-1)^n + (4*sqrt(6) + 9) ) - 48 ). - G. C. Greubel, Dec 20 2021

A026521 a(n) = T(n, n-1), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 1.

Original entry on oeis.org

1, 1, 4, 6, 19, 33, 98, 180, 526, 990, 2887, 5502, 16073, 30863, 90386, 174456, 512128, 992304, 2918954, 5673140, 16716998, 32571858, 96119927, 187675644, 554524660, 1084649644, 3208254571, 6284986554, 18607536319, 36501029265

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Cf. A026519, A026520, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Cf. A026537.

T[n_, k_]:= T[n, k]= 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 *)
Table[T[n, n-1], {n,40}] (* G. C. Greubel, Dec 19 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+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)
[T(n,n-1) for n in (1..40)] # G. C. Greubel, Dec 19 2021

a(n) = A026519(n, n-1).

a(n) = A026537(n+1)/2.

A026522 a(n) = T(n, n-2), where T is given by A026519. Also number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.

Original entry on oeis.org

1, 2, 5, 13, 27, 76, 150, 434, 845, 2470, 4797, 14085, 27377, 80584, 156900, 462620, 902394, 2664276, 5205950, 15387670, 30114073, 89097932, 174609162, 517058502, 1014555607, 3006637946, 5906040623, 17514547015, 34438443075

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Cf. A026519, A026520, A026521, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

T[n_, k_]:= T[n, k]= 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 *)
Table[T[n, n-2], {n,2,40}] (* G. C. Greubel, Dec 19 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+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)
[T(n,n-2) for n in (2..40)] # G. C. Greubel, Dec 19 2021

a(n) = A026519(n, n-2).

A026523 a(n) = T(n, n-3), T given by A026519. 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, 16, 52, 104, 319, 635, 1910, 3786, 11304, 22344, 66514, 131264, 390266, 769578, 2286996, 4508580, 13397075, 26412001, 78489235, 154773696, 460030947, 907432695, 2697786052, 5323519838, 15830906756, 31251588060

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000
Veronika Irvine, Stephen Melczer and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Cf. A026519, A026520, A026521, A026522, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

T[n_, k_]:= T[n, k]= 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 *)
Table[T[n, n-3], {n,3,40}] (* G. C. Greubel, Dec 19 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+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)
[T(n,n-3) for n in (3..40)] # G. C. Greubel, Dec 19 2021

a(n) = A026519(n, n-3).

A026524 a(n) = T(n, n-4), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 4.

Original entry on oeis.org

1, 3, 9, 28, 65, 201, 430, 1316, 2721, 8259, 16793, 50680, 102102, 306958, 615024, 1844304, 3682545, 11024331, 21963161, 65675764, 130648089, 390374193, 775797750, 2316881892, 4601346295, 13737041045, 27270124455

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000
Veronika Irvine, Stephen Melczer and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Cf. A026519, A026520, A026521, A026522, A026523, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

T[n_, k_]:= T[n, k]= 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 *)
Table[T[n, n-4], {n,4,40}] (* G. C. Greubel, Dec 19 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+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)
[T(n,n-4) for n in (4..40)] # G. C. Greubel, Dec 19 2021

a(n) = A026519(n, n-4).

cp's OEIS Frontend

A291088 a(n) = A026520(2n).

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A291089 a(n) = A026520(2n+1).

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A026530 a(n) = T(n, floor(n/2)), T given by A026519.

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Formula

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).

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A026527 a(n) = T(2*n, n-2), where T is given by A026519.

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Mathematica

Sage

Formula

Extensions

A026534 a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).

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A026521 a(n) = T(n, n-1), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 1.

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Mathematica

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Formula

A026522 a(n) = T(n, n-2), where T is given by A026519. Also number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.

Views

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Mathematica

Sage

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, 2n) = 1, T(n, 1) = T(n, 2n-1) = floor((n+1)/2).