A054888 -id:A054888 - cp's OEIS frontend

A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).

1, 3, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.
Equivalently, coordination sequence for (3,3,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
Equivalently, spherical growth series for modular group.
Also, number of sequences of length n with terms 1, 2, and 3, with no adjacent terms equal, and no three consecutive terms (1, 2, 3) or (3, 2, 1). - Pontus von Brömssen, Jan 03 2022

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

G. C. Greubel, Table of n, a(n) for n = 0..999 [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (1,1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Essentially the same as A006355.
Cf. A000045, A054888.

Join[{1,3},2Fibonacci[Range[4,40]]] (* Harvey P. Dale, Jan 06 2012 *)

my(x='x+O('x^50)); Vec((1+2*x+2*x^2+x^3)/(1-x-x^2)) \\ G. C. Greubel, Aug 06 2017

G.f.: (1+2*x+2*x^2+x^3)/(1-x-x^2) = (x^2+x+1)*(1+x)/(1-x-x^2).
a(n) = 2*F(n+2) for n >= 2, with F(n) the n-th Fibonacci number (cf. A000045).
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1 - x. - Stefano Spezia, Apr 18 2022

Offset changed to 0 by N. J. A. Sloane, Jan 03 2022 at the suggestion of Pontus von Brömssen

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1

Offset: 0

Philippe Deléham, Apr 22 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

# Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017

from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A203976 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=4.

Original entry on oeis.org

0, 1, 5, 4, 15, 11, 40, 29, 105, 76, 275, 199, 720, 521, 1885, 1364, 4935, 3571, 12920, 9349, 33825, 24476, 88555, 64079, 231840, 167761, 606965, 439204, 1589055, 1149851, 4160200, 3010349, 10891545, 7881196, 28514435, 20633239, 74651760, 54018521, 195440845

Offset: 0

Michael Somos, Jan 08 2012

a(n+1) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = Lucas(k) for k = 1, ..., n+1.
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
a(n) = row sums of triangle A226377(n), based on differences among Lucas Numbers.  - Richard R. Forberg, Aug 01 2013
A strong divisibility sequence, i.e., gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -5, 1, -5, ...] = 1/(1 - 1/(5 - 1/(1 - 1/(5 - ...)))) = 1/2*(5 - sqrt(5)) begins [0/1, 1/1, 5/4, 4/3, 15/11, 11/8, 40/29,...]. The present sequence is the sequence of numerators; the sequence of denominators [1, 1, 4, 3, 11, 8, 29,...] is A005013. - Peter Bala, May 19 2014
It appears that the first homology group of the branched n-th cyclic covering of the group of figure-eight knot is the direct sum of cyclic groups of orders a(n) and A005013(n), so the order of that group is the product of these numbers, i. e. A004146(n); see the table on p. 156 of the paper by Fox. - Andrey Zabolotskiy, Mar 16 2023

			a(3) = 4 since p(x) = (-x^2 + 7*x - 4) / 2 interpolates p(1) = 1, p(2) = 3, p(3) = 4, and p(4) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. H. Fox, A quick trip through knot theory, pages 120-167 in: Topology of 3-manifolds and related topics (Proceedings of The University of Georgia Institute, 1961), Prentice-Hall, 1962.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000032, A000045, A201157 (bisection), A002878 (bisection). A005013.

a203976 n = a203976_list !! n
a203976_list = 0 : 1 : 5 : 4 : zipWith (-)
   (map (* 3) $ drop 2 a203976_list) a203976_list
-- Reinhard Zumkeller, Jan 10 2012

I:=[0,1,5,4]; [n le 4 select I[n] else 3*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 29 2016

LinearRecurrence[{0,3,0,-1},{0,1,5,4},40] (* Harvey P. Dale, Apr 06 2013 *)

{a(n) = if( n%2, fibonacci(n+1) + fibonacci(n-1), 5 * fibonacci(n))}

{a(n) = if( n<0, -a(-n), polcoeff( x * (1 + 5*x + x^2) / (1 - 3*x^2 + x^4) + x * O(x^n), n))}

{a(n) = if( n<0, -a(-n), subst( polinterpolate( vector( n, k, fibonacci(k-1) + fibonacci(k+1) )), x, n + 1))}

a(1) = 1, a(2) = 5, a(3) = 4, a(n) * a(n-3) = a(n-1) * a(n-2) - 5. a(-n) = -a(n).
G.f.: x * (1 + 5*x + x^2) /  ( (x^2+x-1)*(x^2-x-1) ).
a(2*n) = 5 * A000045(2*n) (Fibonacci). a(2*n+1) = A000032(2*n+1) (Lucas).
a(A004277(n)) = A054888(n+1). - Reinhard Zumkeller, Jan 11 2012
a(n) = A000032(n+1) - A061084(n). - R. J. Mathar, Jun 23 2013
a(2n) = a(2n-1) + a(2n+1), for n>0. - Richard R. Forberg, Aug 01 2013
a(n) = (2^(-1-n)*((-5-sqrt(5)+(-1)^n*(-5+sqrt(5)))*((-1+sqrt(5))^n-(1+sqrt(5))^n)))/sqrt(5). - Colin Barker, Mar 28 2016
E.g.f.: exp(-phi*x)*(exp(x) - 1)*(phi*exp(sqrt(5)*x) - 1/phi), where phi = (1 + sqrt(5))/2. - G. C. Greubel, Mar 28 2016

A207969 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^4 x^n/n ).

Original entry on oeis.org

1, 5, 15, 60, 295, 1625, 9430, 56465, 345010, 2139595, 13419500, 84926105, 541398665, 3472389210, 22385362895, 144945232375, 942089445030, 6143582084115, 40181143112035, 263482860974570, 1731780213622125, 11406235045261205, 75268685723935940

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k>=0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + 295*x^4 + 1625*x^5 + 9430*x^6 +...
such that
log(A(x))/5 = x + x^2/2 + 2^4*x^3/3 + 3^4*x^4/4 + 5^4*x^5/5 + 8^4*x^6/6 + 13^4*x^7/7 +...+ Fibonacci(n)^4*x^n/n +...

Cf. A054888, A207970, A207971, A207972, A207834, A207835.

A077916, A203804.

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^4*x^k/k)+x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + ... is an algebraic function: A(x)^5 = (1 + 3*x + x^2)^4/( (1 - 7*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A203804. - Peter Bala, Apr 03 2014

a(n) ~ 2^(4/5) * 5^(1/10) * phi^(4*n) / (Gamma(1/5) * 3^(1/5) * n^(4/5)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A207970 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^6 x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 1505, 21875, 319620, 4936985, 77358485, 1236083870, 19982821875, 326511608255, 5379199407890, 89249496596015, 1489580814490755, 24988546214618750, 421055477328447620, 7122346563647277860, 120891417096833214485, 2058225554792946621495

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k >= 0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 1505*x^4 + 21875*x^5 + 319620*x^6 + ...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^6*x^4/4 + 5^6*x^5/5 + 8^6*x^6/6 + 13^6*x^7/7 + ... + Fibonacci(n)^6*x^n/n + ...

Cf. A054888, A207969, A207971, A207972, A207834, A207834. A077916, A203806.

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^6*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + ... is an algebraic function: A(x)^25 = ( (1 + 2*x + x^2)^10*(1 + 7*x + x^2)^6 )/( (1 - 3*x + x^2)^15*(1 - 18*x + x^2) ). Cf. A203806. - Peter Bala, Apr 03 2014

a(n) ~ 2^(17/25) * 5^(13/50) * phi^(6*n) / (Gamma(1/25) * 3^(3/5) * n^(24/25)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A207972 Expansion of g.f.: exp( Sum_{n>=1} 5Fibonacci(n^2) x^n/n ).

Original entry on oeis.org

1, 5, 20, 115, 1665, 82650, 12847310, 5620114060, 6659421195205, 21082748688390045, 177217804775828062850, 3941798437750184226876305, 231505293200405380457355524620, 35848160499603817968830380832049915, 14619744406297572472084577939841875791890

Offset: 0

Paul D. Hanna, Feb 22 2012

Moss and Ward prove that this is an integral sequence. - Peter Bala, Nov 28 2022

Let A(x) be the g.f. for this sequence. Note that the expansion of A(x)^(1/5) = exp( Sum_{n>=1} Fibonacci(n^2) * x^n/n ) does not have integer coefficients.

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 115*x^3 + 1665*x^4 + 82650*x^5 + ...
such that
log(A(x))/5 = x + 3*x^2/2 + 34*x^3/3 + 987*x^4/4 + 75025*x^5/5 + 14930352*x^6/6 + 7778742049*x^7/7 + ... + Fibonacci(n^2)*x^n/n + ...

Patrick Moss and Tom Ward, Fibonacci along even powers is (almost) realizable, arXiv:2011.13068 [math.NT], 2020; Fibonacci Quart. 60 (2022), no. 1, 40-47.

Cf. A054888, A207969, A207970, A207971, A054783, A207834, A211892.

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,16,print1(a(n),", "))

A054887 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3, Pi/5, Pi/7).

Original entry on oeis.org

1, 3, 6, 11, 20, 36, 64, 113, 200, 354, 626, 1107, 1958, 3464, 6128, 10839, 19172, 33913, 59988, 106111, 187696, 332009, 587280, 1038820, 1837534, 3250353, 5749442, 10169998, 17989372, 31820803, 56286764, 99563792, 176115092

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

nonn

The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for linear recurrences with constant coefficients, signature (0,0,2,2,4,3,4,2,2,0,0,-1).

Cf. A054888.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16) )); // G. C. Greubel, Feb 07 2023

LinearRecurrence[{0,0,2,2,4,3,4,2,2,0,0,-1}, {1,3,6,11,20,36,64,113, 200,354,626,1107,1958}, 41] (* G. C. Greubel, Feb 07 2023 *)

def A054887_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16) ).list()
a=A054887_list(40); a[1:] # G. C. Greubel, Feb 07 2023

G.f.: x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16).

A207971 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^(2n) * x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 8795, 9808325, 57315191130, 2812698182891585, 894119494320160426760, 2048089587570930007354766745, 32079229816919862900907520464756250, 3500720882833094608324749707338857577696205, 2633228648869966875007549667526201212159637714889015

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

Give g.f. A(x), note that A(x)^(1/5) is not an integer series.

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 8795*x^4 + 9808325*x^5 +...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^8*x^4/4 + 5^10*x^5/5 + 8^12*x^6/6 + 13^14*x^7/7 +...+ Fibonacci(n)^(2*n)*x^n/n +...

Cf. A054888, A207969, A207970, A207972, A207834, A207834.

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^(2*k)*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 18, 57, 195, 684, 2460, 8970, 33102, 123204, 461868, 1741410, 6597750, 25099584, 95822928, 366943881, 1408947675, 5422742910, 20915079258, 80820382425, 312839889219, 1212812010804, 4708415402772, 18302630040504, 71230126892088, 277514015733168

Offset: 0

Paul D. Hanna, Apr 25 2012

nonn

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^2*x^3/3 + 5^2*x^4/4 + 11^2*x^5/5 + 21^2*x^6/6 + 43^2*x^7/7 +...+ Jacobsthal(n)^2*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Cf. A211893, A211895, A211896, A054888, A207969, A001045 (Jacobsthal).

CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 18 2020 *)

{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^2*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3),n)}

G.f.: (1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^2 / 3 * x^n/n ).
a(n) ~ 3^(1/3) * 2^(2*n) / (n^(2/3) * Gamma(1/3)). - Vaclav Kotesovec, Oct 18 2020

A054889 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.

Original entry on oeis.org

1, 5, 20, 70, 245, 860, 3015, 10570, 37060, 129935, 455560, 1597225, 5599980, 19633910, 68837825, 241350100, 846189875, 2966799290, 10401800220, 36469419475, 127864266640, 448300820765, 1571773187140, 5510743762630

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

The layer sequence is the sequence of the cardinalities of the layers accumulating around a (finite-sided) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

Georg Fischer, Table of n, a(n) for n = 1..500
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for linear recurrences with constant coefficients, signature (3,1,3,-1).

Cf. A054886, A054887, A054888, A054890.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) )); // G. C. Greubel, Feb 08 2023

LinearRecurrence[{3,1,3,-1},{1,5,20,70,245},40] (* Georg Fischer, Apr 13 2020 *)

def A054889_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) ).list()
a=A054889_list(40); a[1:] # G. C. Greubel, Feb 08 2023

G.f.: x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4).

a(21) inserted by Georg Fischer, Apr 13 2020

cp's OEIS Frontend

A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A128908 Riordan array (1, x/(1-x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Sage

Formula

A203976 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

PARI

Formula

A207969 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^4 * x^n/n ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A207970 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^6 * x^n/n ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A207972 Expansion of g.f.: exp( Sum_{n>=1} 5*Fibonacci(n^2) * x^n/n ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A054887 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3, Pi/5, Pi/7).

Views

Author

Keywords

Comments

A207969 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^4 x^n/n ).

A207970 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^6 x^n/n ).

A207972 Expansion of g.f.: exp( Sum_{n>=1} 5Fibonacci(n^2) x^n/n ).

A207971 G.f.: exp( Sum_{n>=1} 5Fibonacci(n)^(2n) * x^n/n ).