A026536 -id:A026536 - cp's OEIS frontend

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

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

A026525 a(n) = T(2*n, n), where T is given by A026519.

Original entry on oeis.org

1, 1, 5, 16, 65, 251, 1016, 4117, 16913, 69865, 290455, 1212905, 5085224, 21389824, 90226449, 381519416, 1616684241, 6863544233, 29187402749, 124305180842, 530108333515, 2263423401745, 9674857844129, 41396075156859, 177285394355336, 759895396193376, 3259667597627576, 13992851410449865

Offset: 0

Clark Kimberling

nonn

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

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026526, A026527, 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, 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[2 n, n] ];
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)
[T(2*n, n) for n in (0..40)] # G. C. Greubel, Dec 20 2021

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

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

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

A026526 a(n) = T(2n,n-1), T given by A026519.

Original entry on oeis.org

1, 2, 9, 32, 130, 516, 2107, 8632, 35703, 148390, 619850, 2598960, 10933507, 46124274, 195055005, 826617216, 3509650697, 14926011714, 63572290247, 271125967840, 1157705297385, 4948808238110, 21175676836110, 90692557152240, 388751132082600, 1667665994149376, 7159105163384127, 30753762496639504

Offset: 1

Clark Kimberling

nonn

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

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026527, 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, 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[2*n, n-1] ];
Table[a[n], {n,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)
[T(2*n,n-1) for n in (1..40)] # G. C. Greubel, Dec 20 2021

a(n) = A026519(2*n, n-1).

a(n) = A026536(2*n, n-1).

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

A026549 Ratios of successive terms are 2, 3, 2, 3, 2, 3, 2, 3, ...

Original entry on oeis.org

1, 2, 6, 12, 36, 72, 216, 432, 1296, 2592, 7776, 15552, 46656, 93312, 279936, 559872, 1679616, 3359232, 10077696, 20155392, 60466176, 120932352, 362797056, 725594112, 2176782336, 4353564672, 13060694016, 26121388032, 78364164096, 156728328192, 470184984576, 940369969152

Offset: 0

Clark Kimberling

Appears to be the number of permutations p of {1,2,...,n} such that p(i)+p(i+1)>=n for every i=1,2,...,n-1 (if offset is 1). - Vladeta Jovovic, Dec 15 2003
Equals eigensequence of a triangle with 1's in even columns and (1,3,3,3,...) in odd columns. a(5) = 72 = (1, 3, 1, 3, 1, 1) dot (1, 1, 2, 6, 12, 36) = (1 + 3 + 2 + 18 + 12 + 36), where (1, 3, 1, 3, 1, 1) = row 5 of the generating triangle. - Gary W. Adamson, Aug 02 2010
Partial products of A010693. - Reinhard Zumkeller, Mar 29 2012
Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017
For n >= 2, a(n) is the least k > a(n-1) such that both k and a(n-2) + a(n-1) + k have exactly n prime factors, counted with multiplicity. - Robert Israel, Aug 06 2024

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 36*x^4 + 72*x^5 + 216*x^6 + ... - _Michael Somos_, Apr 09 2022

Arno Berger and Theodore P. Hill, An Introduction to Benford's Law, Princeton University Press, 2015.

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, International Journal of Combinatorics, Vol. 2014 (2014), Article ID 301394, 7 pages; arXiv preprint, arXiv:1312.0583 [math.CO], 2013.
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,6).
Index entries for sequences related to Benford's law.

Cf. A026532, A026536, A026551, A026567.

Cf. A010693, A208131, A109827.

a026549 n = a026549_list !! n
a026549_list = scanl (*) 1 $ a010693_list
-- Reinhard Zumkeller, Mar 29 2012

[(1/2)*(3-(-1)^n)*6^Floor(n/2): n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

seq(seq(2^i*3^j, i=j..j+1),j=0..30); # Robert Israel, Aug 06 2024

LinearRecurrence[{0,6},{1,2},30] (* Harvey P. Dale, May 29 2016 *)

{a(n) = 6^(n\2) * (n%2+1)}; /* Michael Somos, Apr 09 2022 */

[(1+(n%2))*6^(n//2) for n in (0..30)] # G. C. Greubel, Apr 09 2022

Equals T(n, 0) + T(n, 1) + ... + T(n, 2n), T given by A026536.
a(n) = 2*A026532(n), for n > 0.
G.f.: (1+2*x)/(1-6*x^2) - Paul Barry, Aug 25 2003
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (1/2)*(3 - (-1)^n)*6^floor(n/2), or a(n) = 6*a(n-2). - Vincenzo Librandi, Jun 08 2011
a(n) = 1/a(-n) if n is even and (2/3)/a(-n) if n is odd for all n in Z. - Michael Somos, Apr 09 2022
Sum_{n>=0} 1/a(n) = 9/5. - Amiram Eldar, Feb 13 2023

New definition from Ralf Stephan, Dec 01 2004

A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098

Offset: 0

Clark Kimberling

a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.

			Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   2;
  1, 3,  5,   6,   4;
  1, 4,  9,  14,  15,  10;
  1, 5, 14,  27,  38,  39,   25;
  1, 6, 20,  46,  79, 104,  102,   64;
  1, 7, 27,  72, 145, 229,  285,  270,  166;
  1, 8, 35, 106, 244, 446,  659,  784,  721,  436;
  1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;

Clark Kimberling, Rows 0..100, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A026269, A026270, A026288, A026289, A026290, A026291, A026292, A026293.
Cf. A026294, A026295, A026296, A026297, A026299, A026519, A026536, A026552.
Cf. A026584, A027926, A212342.

f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n,k) // T = A026268
  if k eq 0 or n lt 3 then return 1;
  elif k eq 1 then return n-1;
  elif k eq 2 then return f(n);
  elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
  else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022

T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)

def T(n,k): # T = A026268
    if n<3 or k==0: return 1
    elif k==1: return n-1
    elif k==2: return (n^2 -n -2)//2 + int(n==2)
    elif k==n: return T(n-1, n-2) + T(n-1, n-1)
    else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022

From G. C. Greubel, Sep 24 2022: (Start)
T(n, 1) = A000027(n-1), n >= 1.
T(n, 2) = A212342(n-1), n >= 2.
T(n, n-1) = A026270(n), n >= 2.
T(n, n-2) = A026288(n), n >= 2.
T(n, n-3) = A026289(n), n >= 3.
T(n, n-4) = A026290(n), n >= 4.
T(n, n) = A026269(n), n >= 2.
T(n, floor(n/2)) = A026297(n), n >= 0.
T(2*n, n) = A026292(n).
T(2*n, n-1) = A026295(n), n >= 1.
T(2*n, n+1) = A026296(n), n >= 1.
T(2*n-1, n-1) = A026291(n), n >= 2.
T(3*n, n) = A026293(n), n >= 0.
T(4*n, n) = A026294(n), n >= 0.
Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)

Updated by Clark Kimberling, Aug 29 2014

Indices of b-file corrected by Sidney Cadot, Jan 06 2023.

A026551 Expansion of 3(1+2x-2x^2)/((1-x)(1-6*x^2)).

Original entry on oeis.org

3, 9, 21, 57, 129, 345, 777, 2073, 4665, 12441, 27993, 74649, 167961, 447897, 1007769, 2687385, 6046617, 16124313, 36279705, 96745881, 217678233, 580475289, 1306069401, 3482851737, 7836416409, 20897110425, 47018498457

Offset: 0

Clark Kimberling

The even terms are the number of holes of Sierpiński triangle-like fractals. The odd terms are the total number of holes and triangles. - Kival Ngaokrajang, Mar 30 2014
All terms are divisible by 3 (see g.f.). - Joerg Arndt, Dec 20 2014
Former title a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k) was incorrect. - G. C. Greubel, Apr 12 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1, 6, -6).

Cf. A026534, A026565.

[(3/5)*(-1 + 6^(1+Floor(n/2))*((n+1) mod 2) + 16*6^(Floor((n-1)/2))*(n mod 2)): n in [0..40]]; // G. C. Greubel, Apr 12 2022

Table[(3/5)*(-1 +3*6^(n/2)*(1+(-1)^n) +8*6^((n-1)/2)*(1-(-1)^n)), {n, 0, 40}] (* G. C. Greubel, Apr 12 2022 *)

Vec( 3*(1+2*x-2*x^2)/((1-x)*(1-6*x^2))+O(x^33)); \\ Joerg Arndt, Dec 20 2014

[(3/5)*(-1 + 6*6^(n/2)*((n+1)%2)  + 16*6^((n-1)/2)*(n%2)) for n in (0..40)] # G. C. Greubel, Apr 12 2022

G.f.: 3*(1+2*x-2*x^2)/((1-x)*(1-6*x^2)). - Ralf Stephan, Feb 03 2004
From G. C. Greubel, Apr 12 2022: (Start)
a(n) = (3/5)*( -1 + 3*6^(n/2)*(1 + (-1)^n) + 8*6^((n-1)/2)*(1 - (-1)^n) ).
a(2*n) = (3/5)*(6^(n+1) - 1).
a(2*n+1) = (3/5)*(16*6^n -1).
a(n) = a(n-1) + 6*a(n-2) - a(n-3). (End)

Name corrected by G. C. Greubel, Apr 12 2022

cp's OEIS Frontend

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

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A026525 a(n) = T(2*n, n), where T is given by A026519.

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A026526 a(n) = T(2n,n-1), T given by A026519.

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A026549 Ratios of successive terms are 2, 3, 2, 3, 2, 3, 2, 3, ...

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A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

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A026551 Expansion of 3*(1+2*x-2*x^2)/((1-x)*(1-6*x^2)).

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Magma

Mathematica

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SageMath

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Extensions

A026551 Expansion of 3(1+2x-2x^2)/((1-x)(1-6*x^2)).