A026567 -id:A026567 - cp's OEIS frontend

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

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

Clark Kimberling

nonn

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

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026567, A027272, A027273, A027274, A027275, A027276.

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 *)

@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

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

Clark Kimberling

nonn

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

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027273, A027274, A027275, A027276.

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 *)

@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

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

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

Clark Kimberling

nonn

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

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027274, A027275, A027276.

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 *)

@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

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

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

Clark Kimberling

nonn

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

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027275, A027276.

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 *)

@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

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

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

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027276.

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 *)

@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

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

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

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

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

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 *)

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

@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

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

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

A026564 a(n) = Sum_{j=0..n} T(n, j), where T is given by A026552.

Original entry on oeis.org

1, 2, 6, 11, 33, 64, 191, 376, 1122, 2222, 6636, 13180, 39395, 78373, 234414, 466840, 1397034, 2784266, 8335242, 16620976, 49773018, 99291358, 297406884, 593484440, 1777995535, 3548969075, 10633840743, 21230215328, 63620551947

Offset: 0

Clark Kimberling

nonn

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

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

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], {k,0,n}]];
Table[a[n], {n,0,40}] (* 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+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) for k in (0..n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 19 2021

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

A026565 a(n) = 6*a(n-2), starting with 1, 3, 9.

Original entry on oeis.org

1, 3, 9, 18, 54, 108, 324, 648, 1944, 3888, 11664, 23328, 69984, 139968, 419904, 839808, 2519424, 5038848, 15116544, 30233088, 90699264, 181398528, 544195584, 1088391168, 3265173504, 6530347008, 19591041024, 39182082048

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 (0,6).

Cf. A026532, A026534, A026551, A026552, A026567.

[1] cat [n le 2 select 3^n else 6*Self(n-2): n in [1..35]]; // G. C. Greubel, Dec 17 2021

Table[(1/4)*6^(n/2)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)) - (1/2)*Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Dec 17 2021 *)

def A026565(n): return ( (3/2)*6^(n/2) if (n%2==0) else 3*6^((n-1)/2) ) - bool(n==0)/2
[A026565(n) for n in (0..30)] # G. C. Greubel, Dec 17 2021

a(n) = Sum_{j=0..2*n} A026552(n, j).
G.f.: (1+3*x+3*x^2)/(1-6*x^2). - Ralf Stephan, Feb 03 2004
a(0)=1, a(1)=3; a(n) = 3*a(n-1) if n is even, a(n) = 2*a(n-1) if n is odd. - Vincenzo Librandi, Nov 19 2010
a(n) = (1/4)*6^(n/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) - (1/2)*[n=0]. - G. C. Greubel, Dec 17 2021

Better name from Ralf Stephan, Jul 17 2013

A233325 a(n) = (2*6^(n+1) - 7) / 5.

Original entry on oeis.org

1, 13, 85, 517, 3109, 18661, 111973, 671845, 4031077, 24186469, 145118821, 870712933, 5224277605, 31345665637, 188073993829, 1128443962981, 6770663777893, 40623982667365, 243743896004197, 1462463376025189, 8774780256151141, 52648681536906853

Offset: 0

Philippe Deléham, Feb 23 2014

Sum of n-th row of triangle of powers of 6: 1; 6 1 6; 36 6 1 6 36; 216 36 6 1 6 36 216; ...

			a(0) = 1;
a(1) = 6 + 1 + 6 = 13;
a(2) = 36 + 6 + 1 + 6 + 36 = 85;
a(3) = 216 + 36 + 6 + 1 + 6 + 36 + 216 = 517; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A026567.

[(2*6^(n+1) - 7) / 5: n in [0..30]]; // Vincenzo Librandi, Feb 25 2014

Table[(2 6^(n + 1) - 7)/5, {n, 0, 20}] (* Vincenzo Librandi, Feb 25 2014 *)

G.f.: (1+6*x)/((1-x)*(1-6*x)).
a(n) = 7*a(n-1) - 6*a(n-2) for n>1, a(0)=1, a(1)=13.
a(n) = 6*a(n-1) + 7 for n>0, a(0)=1.
a(n) = A026567(2*n). - Philippe Deléham, Feb 24 2014

cp's OEIS Frontend

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

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

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

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

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

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

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

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