A054888 -id:A054888 - cp's OEIS frontend

A054890 Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.

1, 7, 42, 245, 1428, 8323, 48510, 282737, 1647912, 9604735, 55980498, 326278253, 1901689020, 11083855867, 64601446182, 376524821225, 2194547481168, 12790760065783, 74550012913530, 434509317415397

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.

Michael De Vlieger, Table of n, a(n) for n = 1..1307
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
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 (6,-1).

Cf. A001109, A054886, A054887, A054888, A054889.

[n eq 1 select 1 else 7*Evaluate(ChebyshevSecond(n-1), 3): n in [1..40]]; // G. C. Greubel, Feb 08 2023

Rest@CoefficientList[Series[x*(1+x+x^2)/(1-6*x+x^2), {x,0,30}], x] (* Michael De Vlieger, Dec 29 2020 *)
LinearRecurrence[{6,-1},{1,7,42},20] (* Harvey P. Dale, Jun 06 2021 *)

[7*chebyshev_U(n-2, 3) + int(n==1) for n in range(1,41)] # G. C. Greubel, Feb 08 2023

a(n) = 7*A001109(n-1) + [n=1].
G.f.: x*(1+x+x^2)/(1-6*x+x^2).
a(n) = A001109(n) + A001109(n-1) + A001109(n-2), n>1. - Ralf Stephan, Apr 26 2003

A201157 y-values in the solution to 5*x^2 - 20 = y^2.

Original entry on oeis.org

0, 5, 15, 40, 105, 275, 720, 1885, 4935, 12920, 33825, 88555, 231840, 606965, 1589055, 4160200, 10891545, 28514435, 74651760, 195440845, 511670775, 1339571480, 3507043665, 9181559515, 24037634880, 62931345125, 164756400495, 431337856360, 1129257168585

Offset: 1

Sture Sjöstedt, Nov 27 2011

Except a(1), the same as A054888. - R. J. Mathar, Nov 28 2011

			15 is in the sequence because 15^2 = 5*7^2 - 20.

Michael De Vlieger, Table of n, a(n) for n = 1..2392
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A005248.

Mathematica
```
LinearRecurrence[{3, -1}, {0, 5}, 50]
```

a(n) = 3*a(n-1) - a(n-2), n>2.
G.f.: 5*x^2 / (x^2 - 3*x + 1). - Colin Barker, Apr 08 2013
a(n) = 5*Fibonacci(2*n-2) = Lucas(2*n-1) + Lucas(2*n-3) with Lucas(-1) = -1. - Bruno Berselli, Feb 15 2017
a(n) = Lucas(n)^2 - Lucas(n-2)^2. - Greg Dresden, Apr 15 2022

More terms from Colin Barker, Apr 08 2013

A075269 Product of Lucas numbers and inverted Lucas numbers: a(n)=A000032(n)*A075193(n).

Original entry on oeis.org

2, -3, 12, -28, 77, -198, 522, -1363, 3572, -9348, 24477, -64078, 167762, -439203, 1149852, -3010348, 7881197, -20633238, 54018522, -141422323, 370248452, -969323028, 2537720637, -6643838878, 17393796002, -45537549123, 119218851372, -312119004988, 817138163597

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 11 2002

Index entries for linear recurrences with constant coefficients, signature (-2,2,1).

Cf. A000032, A002878, A075193.

CoefficientList[Series[(2 + x + 2x^2)/(1 + 2x - 2x^2 - x^3), {x, 0, 30}], x]
LinearRecurrence[{-2,2,1},{2,-3,12},30] (* Harvey P. Dale, Jun 30 2022 *)

a(n)=1+(-1)^n*(fibonacci(2*n)+fibonacci(2*n+2))

a(n) = 1 + (-1)^n*A002878(n).
From Michael Somos, Apr 07 2003: (Start)
G.f.: (2+x+2x^2)/((1+3x+x^2)(1-x)).
a(n) = -3a(n-1) - a(n-2)+5 = -2a(n-1) + 2a(n-2) + a(n-3) = a(-1-n). (End)
Sum_{n>=0} 1/a(n) = sqrt(5)/10. - Amiram Eldar, Jan 15 2022
a(n) = (-1)^n*A215602(n). - R. J. Mathar, Jul 09 2024
a(n) - a(n-1) = (-1)^n* A054888(n), n>0. - R. J. Mathar, Jul 09 2024

A237655 G.f.: exp( Sum_{n>=1} 5Fibonacci(n-2)Fibonacci(n+2) * x^n/n ).

Original entry on oeis.org

1, 10, 50, 175, 510, 1376, 3625, 9500, 24875, 65125, 170500, 446375, 1168625, 3059500, 8009875, 20970125, 54900500, 143731375, 376293625, 985149500, 2579154875, 6752315125, 17677790500, 46281056375, 121165378625, 317215079500, 830479859875, 2174224500125, 5692193640500, 14902356421375, 39014875623625

Offset: 0

Paul D. Hanna, May 05 2014

nonn

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

			G.f.: A(x) = 1 + 10*x + 50*x^2 + 175*x^3 + 510*x^4 + 1376*x^5 + 3625*x^6 + ...
where the logarithm begins:
log(A(x)) = 5*1*2*x + 5*0*3*x^2/2 + 5*1*5*x^3/3 + 5*1*8*x^4/4 + 5*2*13*x^5/5 + 5*3*21*x^6/6 + 5*5*34*x^7/7 + 5*8*55*x^8/8 + 5*13*89*x^9/9 + ...

Fung Lam, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3, -1).

Cf. A237654, A054888.

{a(n)=polcoeff(exp(sum(m=1, n, 5*fibonacci(m-2)*fibonacci(m+2) *x^m/m) +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

G.f.: (1+x)^7 / (1-3*x+x^2).

a(n) = 3*a(n-1) - a(n-2), n>=8. - Fung Lam, May 19 2014

A247308 Layer counting sequence in the order-5 cubic honeycomb.

Original entry on oeis.org

1, 7, 37, 163, 661, 2643, 10497, 41511, 164073, 648495, 2562749, 10127291, 40020845, 158152811, 624980489, 2469769903, 9759926065, 38568829879, 152414547541, 602304889075, 2380161078405, 9405812345187, 37169461719153, 146884589311479, 580451843386809, 2293803210617951, 9064547264192237, 35820865853787467

Offset: 0

Tim Hutton, Sep 11 2014

nonn

The number of cubes reachable by at most n steps across faces in the {4,3,5} tessellation of hyperbolic space, for n >= 0.

Eryk Kopczynski, Table of n, a(n) for n = 0..999
Tim Hutton, Generating code in C++, using VTK (gives incorrect terms from some point on!)
Eryk Kopczynski, HyperRogue. Run with parameters -geo 435h -csolve; compile with -DCAP_GMP=0 to get the conjectured formula.

For the {5,3,4} tessellation: A076765.

For the {5,4} tessellation: A054888.

a(d+17) = 3*a(d+16) + 2*a(d+15) + 7*a(d+14) + a(d+13) - 5*a(d+12) + 3*a(d+11) - 2*a(d+10) - 18*a(d+9) + 18*a(d+8) + 2*a(d+7) - 3*a(d+6) + 5*a(d+5) - a(d+4) - 7*a(d+3) - 2*a(d+2) - 3*a(d+1) + a(d) (conjectured, found experimentally and tested from 19 to 135). - Eryk Kopczynski, Jul 04 2020

Conjectured G.f.: (1+x) * (1+2*x+8*x^2+9*x^3+8*x^4+17*x^5+10*x^6+10*x^8+10*x^10+17*x^11+8*x^12+9*x^13+8*x^14+2*x^15+x^16) / ((1-x)^2 * (1-2*x-4*x^2-11*x^3-12*x^4-7*x^5-10*x^6-8*x^7+10*x^8-8*x^9-10*x^10-7*x^11-12*x^12-11*x^13-4*x^14-2*x^15+x^16)). - Natalia L. Skirrow, Apr 29 2025

Offset and terms corrected and more terms added by Eryk Kopczynski, Jul 04 2020

A237654 G.f.: exp( Sum_{n>=1} 5Fibonacci(n-1)Fibonacci(n+1) * x^n/n ).

Original entry on oeis.org

1, 0, 5, 5, 25, 49, 150, 365, 990, 2550, 6726, 17550, 46015, 120390, 315275, 825299, 2160775, 5656855, 14809980, 38772875, 101508876, 265753500, 695751900, 1821501900, 4768754125, 12484760124, 32685526625, 85571819345, 224029931845, 586517975725, 1535523995826, 4020054011225, 10524638038410

Offset: 0

Paul D. Hanna, May 05 2014

nonn

Compare to the g.f. of A054888.

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

			G.f.: A(x) = 1 + 5*x^2 + 5*x^3 + 25*x^4 + 49*x^5 + 150*x^6 + 365*x^7 + ...
where the logarithm begins:
log(A(x)) = 5*1*2*x^2/2 + 5*1*3*x^3/3 + 5*2*5*x^4/4 + 5*3*8*x^5/5 + 5*5*13*x^6/6 + 5*8*21*x^7/7 + 5*13*34*x^8/8 + ...

Index entries for linear recurrences with constant coefficients, signature (0,5,5,0,-1).

Cf. A237655, A054888, A000032.

LinearRecurrence[{0,5,5,0,-1},{1,0,5,5,25},40] (* Harvey P. Dale, Apr 17 2025 *)

{a(n)=polcoeff(exp(sum(m=1, n, 5*fibonacci(m-1)*fibonacci(m+1)*x^m/m) + x*O(x^n)), n)}
for(n=0,36,print1(a(n),", "))

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

a(n) = (2*Lucas(2*n+5) + (28+25*n+5*n^2)*(-1)^(n))/50 where Lucas = A000032. - Greg Dresden, Jan 01 2021

A377322 Number of cells that are a distance of n away in an order-5 hyperbolic square tiling.

Original entry on oeis.org

1, 4, 12, 28, 64, 148, 340, 780, 1792, 4116, 9452, 21708, 49856, 114500, 262964, 603932, 1387008, 3185444, 7315788, 16801660, 38587200, 88620532, 203528596, 467429932, 1073513728, 2465464116, 5662259500, 13004116524, 29865647552, 68590349988, 157526673524

Offset: 0

Lewis Chen, Oct 24 2024

Also known as a {4,5} tiling.

The formula given in the MathOverflow answer (4 * A033303) is erroneous after n=3.

M. Lavrov, Description of the order-5 square tiling of the hyperbolic plane as a graph, MathOverflow
EC Puzzle Hunt, Solution: Tower Navigation
Index entries for coordination sequences.
Index entries for linear recurrences with constant coefficients, signature (2,0,2,-1).

Cf. A008574, A054888 (dual).

Vec((1 + 2*x + 4*x^2 + 2*x^3 + x^4)/(1 - 2*x - 2*x^3 + x^4) + O(x^31)) \\ Andrew Howroyd, Feb 12 2025

G.f.: (1 + 2*x + 4*x^2 + 2*x^3 + x^4)/(1 - 2*x - 2*x^3 + x^4). - Andrew Howroyd, Feb 12 2025

a(20) onwards from Andrew Howroyd, Feb 12 2025

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A054890 Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.

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A201157 y-values in the solution to 5*x^2 - 20 = y^2.

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A075269 Product of Lucas numbers and inverted Lucas numbers: a(n)=A000032(n)*A075193(n).

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A237655 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n-2)*Fibonacci(n+2) * x^n/n ).

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A247308 Layer counting sequence in the order-5 cubic honeycomb.

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A237654 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n-1)*Fibonacci(n+1) * x^n/n ).

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A377322 Number of cells that are a distance of n away in an order-5 hyperbolic square tiling.

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PARI

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Extensions

A237655 G.f.: exp( Sum_{n>=1} 5Fibonacci(n-2)Fibonacci(n+2) * x^n/n ).

A237654 G.f.: exp( Sum_{n>=1} 5Fibonacci(n-1)Fibonacci(n+1) * x^n/n ).