A026595 -id:A026595 - cp's OEIS frontend

A026584 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 4, 1, 1, 1, 2, 5, 7, 8, 7, 5, 2, 1, 1, 2, 8, 9, 20, 14, 20, 9, 8, 2, 1, 1, 3, 9, 19, 28, 43, 40, 43, 28, 19, 9, 3, 1, 1, 3, 13, 22, 56, 62, 111, 86, 111, 62, 56, 22, 13, 3, 1, 1, 4, 14, 38, 69, 140, 167, 259, 222, 259, 167, 140, 69, 38, 14, 4, 1

Offset: 1

Clark Kimberling

Row sums are in A026597. - Philippe Deléham, Oct 16 2006

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 s(i-1) odd, |s(i)-s(i-1)| = 1 if s(i-1) is even, for i = 1..n.

			First 5 rows:
  1
  1  0  1
  1  1  2  1  1
  1  1  4  2  4  1  1
  1  2  5  7  8  7  5  2  1

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

Cf. A026519, A026536, A026552, A026568, A027926.

Cf. A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599.

z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2]; t[n_, k_] := Floor[n/2] /; k == 2 n - 1; t[n_, k_] := t[n, k] = 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]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u]   (* A026584 array *)
v = Flatten[u] (* A026584 sequence *)

@CachedFunction
def T(n,k):
    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)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Dec 11 2021

T(n, k) = T(n-1, k-2) + T(n-1, k) if ( (n+k) mod 2 ) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), where T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2).

Updated by Clark Kimberling, Aug 29 2014

A026585 a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.

Original entry on oeis.org

1, 0, 2, 2, 8, 14, 40, 86, 222, 518, 1296, 3130, 7770, 19066, 47324, 117094, 291260, 724302, 1806220, 4507230, 11266718, 28188070, 70609316, 177023466, 444231564, 1115639586, 2803975860, 7052132546, 17748069294, 44693162266

Offset: 0

Clark Kimberling

nonn

The signed sequence 1,0,2,-2,8,-14,... is the inverse binomial transform of A026569. - Paul Barry, Sep 09 2004

Hankel transform of a(n) is 2^n. Hankel transform of a(n+1) is {0, -4, 0, 16, 0, -64, 0, 256, 0, ...} or -2^(n+1)*[x^n](x/(1+x^2)). Hankel transform of a(n+2) is 2^(n+1)*A109613(n+1). - Paul Barry, Mar 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A026584, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

Cf. A000984, A026569, A109613.

[(&+[Binomial(n-j-1, n-2*j)*Binomial(2*j, j): j in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Dec 12 2021

CoefficientList[Series[Sqrt[(1-x)/(1-x-4*x^2)], {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

[sum(binomial(n-j-1, n-2*j)*binomial(2*j, j) for j in (0..(n//2))) for n in [0..40]] # G. C. Greubel, Dec 12 2021

a(n) = A026584(n, n).
G.f.: sqrt((1-x)/(1-x-4*x^2)). - Ralf Stephan, Jan 08 2004
From Paul Barry, Jul 01 2009: (Start)
G.f.: 1/(1 -2*x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 - ... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(n-k,k)*A000984(k). (End)
From Paul Barry, Mar 23 2011: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*A000984(k).
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*C(2*k,k). (End)
D-finite with recurrence n*a(n) +2*(-n+1)*a(n-1) +(-3*n+2)*a(n-2) +2*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (sqrt(17)+1)^(n-1/2) / (17^(1/4) * sqrt(Pi*n) * 2^(n-3/2)). - Vaclav Kotesovec, Feb 12 2014

A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

Original entry on oeis.org

1, 1, 10, 19, 109, 280, 1261, 3781, 15130, 49159, 185329, 627760, 2295721, 7945561, 28607050, 100117099, 357580549, 1258634440, 4476859381, 15804569341, 56096303770, 198337427839, 703204161769, 2488241012320, 8817078468241, 31211247579121, 110564953793290

Offset: 0

Olivier Gérard

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 10*a(n-2) equals the number of 10-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=3.
Index entries for linear recurrences with constant coefficients, signature (1,9).

Cf. A015443, A015442, A026595, A128099.

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+9*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

m:=25; S:=series(1/(1-x-9*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

CoefficientList[Series[1/(1-x-9*x^2), {x,0,25}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 25] (* G. C. Greubel, Apr 30 2017 *)

a(n)=([0,1; 9,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-9) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*9^k. - Paul Barry, Jul 20 2004
a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2. - Paul Barry, Aug 28 2005
a(n) = Sum_{k=0..n} A109466(n,k)*(-9)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)). - Alexander R. Povolotsky, Oct 13 2010
a(n) = Sum_{k=1..n+1, k odd} C(n+1,k)*37^((k-1)/2)/2^n. - Vladimir Shevelev, Feb 05 2014
G.f.: 1/(1-x-9*x^2). - Philippe Deléham, Feb 19 2020
a(n) = J(n, 9/2), where J(n,x) are the Jacobsthal polynomials. - G. C. Greubel, Feb 18 2020
E.g.f.: exp(x/2)*(sqrt(37)*cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2))/sqrt(37). - Stefano Spezia, Feb 19 2020

Edited by N. J. A. Sloane, Oct 11 2010

A026599 a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).

Original entry on oeis.org

1, 3, 9, 23, 61, 155, 401, 1023, 2629, 6723, 17241, 44135, 113101, 289643, 742049, 1900623, 4868821, 12471315, 31946601, 81831863, 209618269, 536945723, 1375418801, 3523201695, 9024876901, 23117683683, 59217191289, 151687926023

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case cyclic++, sequence c(n) (offset 1).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4).

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597 (first differences), A026598, A027282, A027283, A027284, A027285, A027286.

Cf. A006131, A026581.

[n le 3 select 3^(n-1) else 2*Self(n-1) +3*Self(n-2) -4*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 15 2021

LinearRecurrence[{2,3,-4}, {1,3,9}, 40] (* G. C. Greubel, Dec 15 2021 *)

[( (1+x)/((1-x)*(1-x-4*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 15 2021

G.f.: (1+x)/((1-x)*(1-x-4*x^2)). - Ralf Stephan, Feb 04 2004
From Klaus Purath, Feb 02 2021: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3).
a(n) = Sum_{j=0..n} A026597(j). (End)
a(n) = 2^n*(Fibonacci(n+2, 1/2) + Fibonacci(n+1, 1/2)) - 1/2. - G. C. Greubel, Dec 15 2021

A026587 a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.

Original entry on oeis.org

1, 1, 5, 9, 28, 62, 167, 399, 1024, 2518, 6359, 15819, 39759, 99427, 249699, 626203, 1573524, 3953446, 9943905, 25019005, 62994733, 158680545, 399936573, 1008438757, 2543992514, 6420413940, 16210331727, 40943722115, 103453402718

Offset: 2

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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[n, n-2], {n, 2, 40}] (* G. C. Greubel, Dec 12 2021 *)

@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(n, n-2) for n in (2..40)] # G. C. Greubel, Dec 12 2021

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

Conjecture: (n+2)*a(n) = (3*n+2)*a(n-1) +(3*n+2)*a(n-2) -(11*n-16)*a(n-3) -2*(n-3)*a(n-4) +4*(2*n-9)*a(n-5). - R. J. Mathar, Jun 23 2013

A026589 a(n) = T(n,n-4), T given by A026584. 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, 2, 9, 22, 69, 178, 497, 1294, 3452, 8964, 23430, 60556, 156663, 403214, 1037191, 2660978, 6821200, 17459732, 44657246, 114117628, 291449047, 743904326, 1897956899, 4840429962, 12340947855, 31455453822, 80158533099

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000

Cf. A026584, A026585, A026587, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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[n, n-4], {n, 4, 40}] (* G. C. Greubel, Dec 12 2021 *)

@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(n, n-4) for n in (4..40)] # G. C. Greubel, Dec 12 2021

a(n) = A026584(n, n-4).

Conjecture: -(n+4)*(65*n-269)*a(n) +(-65*n^2+140*n+1933)*a(n-1) +(809*n^2-2431*n-4514)*a(n-2) +(-123*n^2+2496*n-205)*a(n-3) +2*(-726*n^2+3737*n-4395)*a(n-4) +8*(56*n-215)*(2*n-9)*a(n-5) = 0. - R. J. Mathar, Jun 23 2013

A026590 a(n) = T(2*n, n), where T is given by A026584.

Original entry on oeis.org

1, 1, 5, 19, 69, 341, 1203, 6336, 22593, 121483, 438533, 2381512, 8677763, 47419503, 173984792, 954961034, 3522101709, 19397198595, 71831252031, 396646918211, 1473610012405, 8154682794333, 30376120747792, 168394714422722, 628648474795879, 3490216221862041, 13053833414221023, 72566287730964469

Offset: 0

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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]= Block[{$RecursionLimit= Infinity}, T[2*n,n]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 13 2021 *)

@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, n) for n in (0..40)] # G. C. Greubel, Dec 13 2021

a(n) = A026584(n, n).

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

A026591 a(n) = T(2*n, n-1), where T is given by A026584.

Original entry on oeis.org

1, 2, 9, 38, 140, 701, 2534, 13294, 48369, 258430, 947694, 5114572, 18872399, 102539204, 380143356, 2075658454, 7723000261, 42330184638, 157951859953, 868376395790, 3247811317907, 17899895038348, 67075896452000, 370442993383238, 1390392820937920, 7692166179956366, 28910883325637649, 160184255555687056

Offset: 1

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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]= Block[{$RecursionLimit= Infinity}, T[2*n,n-1]];
Table[a[n], {n, 1, 40}] (* G. C. Greubel, Dec 13 2021 *)

@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, n-1) for n in (1..40)] # G. C. Greubel, Dec 13 2021

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

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

A026592 a(n) = T(2*n, n-2), where T is given by A026584.

Original entry on oeis.org

1, 3, 14, 65, 251, 1288, 4830, 25518, 95388, 510532, 1910821, 10309234, 38656462, 209766714, 787912030, 4294635438, 16155375825, 88371236851, 332859949946, 1826080683788, 6885797551334, 37867515477338, 142929375411104, 787637258527505, 2975423924172735, 16425495119248041, 62096233990615140, 343318987947145114

Offset: 2

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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]= Block[{$RecursionLimit= Infinity}, T[2*n,n-2]];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 13 2021 *)

@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, n-2) for n in (2..40)] # G. C. Greubel, Dec 13 2021

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

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

A026593 a(n) = T(2*n-1, n-1), where T is given by A026584.

Original entry on oeis.org

1, 1, 8, 22, 121, 406, 2155, 7624, 40717, 147001, 792351, 2892044, 15703156, 57728737, 315180458, 1164727748, 6385672193, 23691834033, 130316812494, 485018155062, 2674846358141, 9980763478121, 55161813337474, 206262229900060, 1142020843590221, 4277853480389546, 23721423518350124, 88991782850212510

Offset: 1

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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]= Block[{$RecursionLimit= Infinity}, T[2*n-1,n-1]];
Table[a[n], {n, 1, 40}] (* G. C. Greubel, Dec 13 2021 *)

@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-1) for n in (1..40)] # G. C. Greubel, Dec 13 2021

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

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

cp's OEIS Frontend

A026584 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.

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A026585 a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.

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A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

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A026599 a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).

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A026587 a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.

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A026589 a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

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A026590 a(n) = T(2*n, n), where T is given by A026584.

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