A026599 -id:A026599 - cp's OEIS frontend

A006131 a(n) = a(n-1) + 4*a(n-2), a(0) = a(1) = 1.

1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, 7589, 19305, 49661, 126881, 325525, 833049, 2135149, 5467345, 14007941, 35877321, 91909085, 235418369, 603054709, 1544728185, 3956947021, 10135859761, 25963647845, 66507086889, 170361678269

Offset: 0

N. J. A. Sloane

Length-n words with letters {0,1,2,3,4} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Equals INVERTi transform of A063727: (1, 2, 8, 24, 80, 256, 832, ...). - Gary W. Adamson, Aug 12 2010
a(n) is equal to the permanent of the n X n Hessenberg matrix with 1's along the main diagonal, 2's along the superdiagonal and the subdiagonal, and 0's everywhere else. - John M. Campbell, Jun 09 2011
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, 5*a(n-2) equals the number of 5-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 1, 8, 1, 6, 8, 48, 2, 24, 6,120, 8, 12, 48, 24, 4,136, 24, 18, 6, ... - R. J. Mathar, Aug 10 2012
This is one of only two Lucas-type sequences whose 8th term is a square. The other one is A097705. - Michel Marcus, Dec 07 2012
Numerators of stationary probabilities for the M2/M/1 queue. In this queue, customers arrives in groups of 2. Intensity of arrival = 1. Service rate = 4. There is only one server and an infinite queue. - Igor Kleiner, Nov 02 2018
Number of 4-compositions of n+2 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
From M. Eren Kesim, May 13 2021: (Start)
a(n) is equal to the number of n-step walks from a universal vertex to another (itself or the other) on the diamond graph. It is also equal to the number of (n+1)-step walks from vertex A to vertex B on the graph below.
    B--C
    | /|
    |/ |
    A--D
(End)
From Wolfdieter Lang, Jan 03 2024: (Start)
This sequence {a(n-1)}, with a(-1) = 0, appears in the formula for powers of phi17 := (1 + sqrt(17))/2 = A222132, the fundamental (integer) algebraic number of Q(sqrt(17)): phi17^n = A052923(n) + a(n-1)*phi17, for n >= 0.
Limit_{n->oo} a(n+1)/a(n) = phi17. (End)

			G.f. = 1 + x + 5*x^2 + 9*x^3 + 29*x^4 + 65*x^5 + 181*x^6 + 441*x^7 + 1165*x^8 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015.
Joerg Arndt, Matters Computational (The Fxtbook), pp.317-318.
Jolanta Borowska and Lena Łacińska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix", J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for tridiagonal Toeplitz matrices a=1, b=2.
Andrew Bremner and Nikos Tzanakis, Lucas sequences whose 8th term is a square, arXiv:math/0408371 [math.NT], 2004.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 437
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See pp. 387, 389.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1, 1994.
Index entries for linear recurrences with constant coefficients, signature (1,4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006130, A015440, A026581, A026583, A026597, A026599, A052923, A097705, A102446, A063727, A344236, A344261.

a:=[1,1];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+4*Self(n-2): n in [1..40] ]; // Vincenzo Librandi, Aug 19 2011

A006131:=-1/(-1+z+4*z**2); # conjectured by Simon Plouffe in his 1992 dissertation
seq( simplify((2/I)^n*ChebyshevU(n, I/4)), n=0..30); # G. C. Greubel, Dec 26 2019

m = 16; f[n_] = Product[(1 + m*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]; N[%] (* Roger L. Bagula, Nov 21 2008 *)
a[n_]:=(MatrixPower[{{1,4},{1,0}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1, 4}, {1, 1}, 29] (* Jean-François Alcover, Sep 25 2017 *)
Table[2^n*Fibonacci[n+1, 1/2], {n,0,30}] (* G. C. Greubel, Dec 26 2019 *)

a(n)=([0,1; 4,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

vector(31, n, (2/I)^(n-1)*polchebyshev(n-1, 2, I/4) ) \\ G. C. Greubel, Dec 26 2019

def A006131_list(n):
    list = [1, 1] + [0] * (n - 2)
    for i in range(2, n):
        list[i] = list[i - 1] + 4 * list[i - 2]
    return list
print(A006131_list(29)) # M. Eren Kesim, Jul 19 2021

[lucas_number1(n,1,-4) for n in range(1, 30)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - x - 4*x^2).
a(n) = (((1+sqrt(17))/2)^(n+1) - ((1-sqrt(17))/2)^(n+1))/sqrt(17).
a(n+1) = Sum_{k=0..ceiling(n/2)} 4^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*2^(n-k)/2. - Paul Barry, Aug 28 2005
a(n) = A102446(n)/2. - Zerinvary Lajos, Jul 09 2008
a(n) = Sum_{k=0..n} A109466(n,k)*(-4)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = Product_{k=1..floor((n - 1)/2)} (1 + 16*cos(k*Pi/n)^2). - Roger L. Bagula, Nov 21 2008
Limiting ratio a(n+1)/a(n) is (1 + sqrt(17))/2 = 2.561552812... - Roger L. Bagula, Nov 21 2008
The fraction b(n) = a(n)/2^n satisfies b(n) = 1/2 b(n-1) + b(n-2); g.f. 1/(1-x/2-x^2); b(n) = (( (1+sqrt(17))/4 )^(n+1) - ( (1-sqrt(17))/4 )^(n+1))*2/sqrt(17). - Franklin T. Adams-Watters, Nov 30 2009
G.f.: G(0)/(2-x), where G(k) = 1 + 1/(1 - x*(17*k-1)/(x*(17*k+16) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 4*x)/( x*(4*k+3 + 4*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(k+1 + 4*x)/( x*(k+3/2 + 4*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
G.f.: 1 / (1 - x / (1 - 4*x / (1 + 4*x))). - Michael Somos, Sep 15 2013
a(n) = (Sum_{1<=k<=n+1, k odd} C(n+1,k)*17^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014
a(n) = 2^n*Fibonacci(n+1, 1/2) = (2/i)^n*ChebyshevU(n, i/4). - G. C. Greubel, Dec 26 2019
E.g.f.: exp(x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, Dec 27 2019
a(n) = A344236(n) + A344261(n). - M. Eren Kesim, May 13 2021
With an initial 0 prepended, the sequence [0, 1, 1, 5, 9, 29, 65, ...] satisfies the congruences a(n*p^k) == e*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where e = +1 for the primes p listed in A296938, e = 0 when p = 17, otherwise e = -1. - Peter Bala, Dec 28 2022
a(n) = A052923(n+2)/4. - Wolfdieter Lang, Jan 03 2024
From Peter Bala, Jun 27 2025: (Start)
The following products telescope:
Product_{k >= 0} (1 + 4^k/a(2*k+1)) = 1 + sqrt(17).
Product_{k >= 1} (1 - 4^k/a(2*k+1)) = 1/18 * (1 + sqrt(17)).
Product_{k >= 0} (1 + (-4)^k/a(2*k+1)) = (1/17) * (17 + sqrt(17)).
Product_{k >= 1} (1 - (-4)^k/a(2*k+1)) = (1/18) * (17 + sqrt(17)). (End)

More terms from Roger L. Bagula, Sep 26 2006

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.

Original entry on oeis.org

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

A026595 a(n) = T(n, floor(n/2)), where T is given by A026584.

Original entry on oeis.org

1, 1, 1, 1, 5, 8, 19, 22, 69, 121, 341, 406, 1203, 2155, 6336, 7624, 22593, 40717, 121483, 147001, 438533, 792351, 2381512, 2892044, 8677763, 15703156, 47419503, 57728737, 173984792, 315180458, 954961034, 1164727748, 3522101709

Offset: 0

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, 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, Floor[n/2]], {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(n, n//2) for n in (0..40)] # G. C. Greubel, Dec 13 2021

a(n) = A026584(n, floor(n/2))

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

A006131 a(n) = a(n-1) + 4*a(n-2), a(0) = a(1) = 1.

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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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A026595 a(n) = T(n, floor(n/2)), where T is given by A026584.

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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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