A076765 -id:A076765 - cp's OEIS frontend

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

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

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 10, 32, 101, 319, 1006, 3172, 10001, 31531, 99410, 313416, 988125, 3115319, 9821846, 30965900, 97627977, 307797347, 970410426, 3059468848, 9645763669, 30410754735, 95877738174, 302279267892, 953013259777, 3004619799579, 9472837914274, 29865561746840

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-2).

Partial sums of S(n, x), for x=1...10, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784.

Partial sums of A077833.

LinearRecurrence[{3,1,-1,-2},{1,3,10,32},30] (* Harvey P. Dale, May 12 2024 *)

Vec((1-x)^(-1)/(1-2*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

From Wesley Ivan Hurt, Jun 26 2022: (Start)
G.f.: (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).
a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4). (End)

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

Original entry on oeis.org

1, 3, 11, 42, 157, 588, 2204, 8259, 30949, 115977, 434606, 1628619, 6103000, 22870056, 85702025, 321155187, 1203479779, 4509855786, 16899992477, 63330128340, 237319937332, 889320046107, 3332590398005, 12488371098225, 46798254229006, 175369276077483

Offset: 0

N. J. A. Sloane, Nov 17 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,3).

Partial sums of S(n, x), for x=1...15, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829.

CoefficientList[Series[1/(1-3x-2x^2-3x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,3},{1,3,11},40] (* Harvey P. Dale, Nov 05 2021 *)

A145608 Numbers a(n)=k such that (1/3)(5(2k+1)^2-2) is A057080(n)^2.

Original entry on oeis.org

0, 3, 27, 216, 1704, 13419, 105651, 831792, 6548688, 51557715, 405913035, 3195746568, 25160059512, 198084729531, 1559517776739, 12278057484384, 96664942098336, 761041479302307, 5991666892320123, 47172293659258680, 371386682381749320, 2923921165394735883, 23019982640776137747

Offset: 0

Richard Choulet, Oct 14 2008

Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A057080, A131571, A145607.

RecurrenceTable[{a[0]==0,a[1]==3,a[n]==8a[n-1]-a[n-2]+3},a,{n,30}] (* or *) LinearRecurrence[{9,-9,1},{0,3,27},30] (* Harvey P. Dale, May 06 2013 *)

a(n+2) = 8*a(n+1) - a(n) + 3.
a(n) = (A070997(n)-1)/2 = 3*A076765(n-1). - R. J. Mathar, Oct 16 2008
G.f.: -3*x / ( (x-1)*(x^2-8*x+1) ). - R. J. Mathar, Nov 27 2011
a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=27. - Harvey P. Dale, May 06 2013

Made definition and sequence consistent. Changed offset to 0. - R. J. Mathar, Oct 16 2008

A356836 Coordination sequence of the {5,3,4} hyperbolic honeycomb.

Original entry on oeis.org

1, 12, 102, 812, 6402, 50412, 396902, 3124812, 24601602, 193688012, 1524902502, 12005532012, 94519353602, 744149296812, 5858675020902, 46125250870412, 363143331942402, 2859021404668812, 22509027905408102, 177213201838596012, 1395196586803360002, 10984359492588284012

Offset: 0

Eryk Kopczynski, Aug 31 2022

nonn

a(n) is the number of cells n steps from an (arbitrarily chosen) central cell in the {5,3,4} honeycomb.

			Each dodecahedral cell has 12 neighbors, so a(1) = 12.

Dorota Celińska-Kopczyńska and Eryk Kopczyński, Generating Regular Hyperbolic Honeycombs, arXiv:2208.13816 [cs.CG], 2022.

Cf. A076765, A095004, A356835.

It appears thata(n) = 10*A095004(n) + 2. - Hugo Pfoertner, Aug 30 2022

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

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A077826 Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

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A077827 Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).

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A077830 Expansion of 1/(1-3*x-2*x^2-3*x^3).

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A145608 Numbers a(n)=k such that (1/3)*(5*(2k+1)^2-2) is A057080(n)^2.

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A356836 Coordination sequence of the {5,3,4} hyperbolic honeycomb.

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Formula

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

A145608 Numbers a(n)=k such that (1/3)(5(2k+1)^2-2) is A057080(n)^2.