A048790 -id:A048790 - cp's OEIS frontend

A094100 Fit a polynomial of degree k-1 to column k of array in A048790, evaluate it at dimension n = -1.

Original entry on oeis.org

1, -2, 9, -64, 560, -5370, 53788, -555864, 5957685, -66459200, 763983132, -8919566196, 105678848821, -1286858544734

Offset: 1

Dan Hoey

Might be thought of as number of rooted (-1)-dimensional "polycubes" with n cells, with no symmetries removed.

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Sebastian Luther and Stephan Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565; arXiv:1106.1078 [cond-mat.stat-mech], 2011.
R. C. Schroeppel, A few mathematical experiments

Cf. A094101, A048663-A048868.

a(9)-a(14) using Luther & Mertens's formulas added by Andrei Zabolotskii, Jun 27 2025

A094161 Column 5 of A048790.

Original entry on oeis.org

0, 5, 315, 2670, 10810, 30475, 69405, 137340, 246020, 409185, 642575, 963930, 1392990, 1951495, 2663185, 3553800, 4651080, 5984765, 7586595, 9490310, 11731650, 14348355, 17380165, 20868820, 24858060, 29393625, 34523255, 40296690, 46765670, 53983935, 62007225

Offset: 1

N. J. A. Sloane, May 05 2004

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Colin Barker, Table of n, a(n) for n = 1..1000
R. C. Schroeppel, A few mathematical experiments
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

concat(0, Vec(-5*x^2*(112*x^3+229*x^2+58*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Feb 28 2015

a(n) = 5*(n-1) + 305*C(n-1,2) + 1740*C(n-1,3) + 2000*C(n-1,4) where C(n,k) is the binomial coefficient. - Joshua Zucker, Aug 14 2006
a(n) = 5*(672-1715*n+1595*n^2-652*n^3+100*n^4)/6. - Colin Barker, Feb 28 2015
G.f.: -5*x^2*(112*x^3+229*x^2+58*x+1) / (x-1)^5. - Colin Barker, Feb 28 2015

More terms from Joshua Zucker, Aug 14 2006

A094160 Column 4 of A048790.

Original entry on oeis.org

0, 4, 76, 344, 936, 1980, 3604, 5936, 9104, 13236, 18460, 24904, 32696, 41964, 52836, 65440, 79904, 96356, 114924, 135736, 158920, 184604, 212916, 243984, 277936, 314900, 355004, 398376, 445144, 495436, 549380, 607104, 668736, 734404, 804236

Offset: 1

N. J. A. Sloane, May 05 2004

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, California, March 30, 2004.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[64/3*n^3-30*n^2+38/3*n: n in [0..60]]; // Vincenzo Librandi, Aug 28 2016

Table[(64/3 n^3 - 30 n^2 + 38/3 n), {n, 0, 80}] (* Vincenzo Librandi, Aug 28 2016 *)

concat(0, Vec(4*x^2*(1+15*x+16*x^2)/(1-x)^4 + O(x^60))) \\ Colin Barker, Aug 28 2016

A polynomial in n of degree 3.
a(n) = 64/3 n^3 - 30 n^2 + 38/3 n. - Joshua Zucker, Aug 14 2006
From Colin Barker, Aug 28 2016: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: 4*x^2*(1+15*x+16*x^2) / (1-x)^4.
(End)

More terms from Joshua Zucker, Aug 14 2006

A094159 3 times hexagonal numbers: a(n) = 3n(2*n-1).

Original entry on oeis.org

0, 3, 18, 45, 84, 135, 198, 273, 360, 459, 570, 693, 828, 975, 1134, 1305, 1488, 1683, 1890, 2109, 2340, 2583, 2838, 3105, 3384, 3675, 3978, 4293, 4620, 4959, 5310, 5673, 6048, 6435, 6834, 7245, 7668, 8103, 8550, 9009, 9480, 9963, 10458, 10965, 11484

Offset: 0

N. J. A. Sloane, May 05 2004

Column 3 of A048790.
Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
a(n) is the sum of all perimeters of triangles having two sides of length n. For n=4 one has seven triangles with two sides of length 4 and the other of lengths 1..7. - J. M. Bergot, Mar 26 2014
a(n) is the Wiener index of the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 07 2017
Sequence found by reading the line from 0, in the direction 0, 3, ..., in a spiral on an equilateral triangular lattice. - Hans G. Oberlack, Dec 08 2018

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hans G. Oberlack, Triangle spiral.
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, California, March 30, 2004.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A048790.
3 times n-gonal numbers: A045943, A033428, A062741, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Essentially a bisection of A045943. - Omar E. Pol, Sep 17 2011
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=12: see Comments lines of A226492.

List([0..50], n -> 3*n*(2*n-1)); # G. C. Greubel, Dec 07 2018

[3*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Dec 07 2018

A094159:=n->3*n*(2*n-1); seq(A094159(n), n=0..40); # Wesley Ivan Hurt, Mar 28 2014

CoefficientList[Series[3x(1+3x)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 19 2013 *)
Table[3n(2n-1), {n, 0, 50}] (* or *) 3*PolygonalNumber[6, Range[0, 50]] (* or *) LinearRecurrence[{3, -3, 1}, {3, 18, 45}, {0, 50}] (* Eric W. Weisstein, Sep 07 2017 *)

a(n)=3*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015

[3*n*(2*n-1) for n in range(50)] # G. C. Greubel, Dec 07 2018

a(n) = 6*n^2 - 3*n = 3*n*(2*n-1) = 3*A000384(n). - Omar E. Pol, Dec 11 2008
a(n) = 12*n + a(n-1) - 9 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 16 2010
G.f.: 3*x*(1+3*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
Sum_{n>0} 1/a(n) = (2/3)*log(2). - Enrique Pérez Herrero, Jun 04 2015
E.g.f.: 3*x*(1+2*x)*exp(x). - G. C. Greubel, Dec 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/3. - Amiram Eldar, Jan 10 2022

More terms from Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Definition improved, offset corrected and edited by Omar E. Pol, Dec 11 2008

A048663 Number of rooted polycubes with n cells, with no symmetries removed.

Original entry on oeis.org

1, 6, 45, 344, 2670, 20886, 164514, 1303304, 10375830, 82947380, 665440039, 5354470860, 43196001173, 349254823554, 2829388506690, 22961191276080, 186622811691276, 1518914831183982, 12377727000122043

Offset: 1

Richard C. Schroeppel, Dan Hoey

nonn

"Rooted" means some cell of the polycube is designated as the origin. This has the effect of multiplying the count by the volume of the polycube.

			There are six dicubes, each consisting of the origin cube together with one adjacent cube, in each of the six directions.

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

A row of the array in A048790.

Cf. A001931.

A001931 = Cases[Import["https://oeis.org/A001931/b001931.txt", "Table"], {, }][[All, 2]];
a[n_] := n * A001931[[n]];
Array[a, 19] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = n * A001931(n). - Andrew Howroyd, Dec 04 2018

a(12)-a(19) from Andrew Howroyd, Dec 04 2018

A048668 Number of rooted 7-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 14, 273, 5936, 137340, 3307878, 81972296, 2075032808, 53403322203, 1392729138920, 36718293579198, 976872759337356, 26189947759482689, 706808197553825794

Offset: 1

Dan Hoey

nonn

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 7 of A048790.

Cf. A048663, A151833.

a(n) = n * A151833(n). - Andrew Howroyd, Dec 05 2018

a(8)-a(14) from Andrew Howroyd, Dec 05 2018

A094101 Number of rooted 8-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 16, 360, 9104, 246020, 6940128, 201819688, 6003642144, 181770021702, 5581576203840, 173384554507648, 5438172832075920

Offset: 1

Dan Hoey

nonn

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 8 of A048790.

Cf. A094100, A048663-A048868, A151834.

a(n) = n * A151834(n). - Andrew Howroyd, Dec 05 2018

a(9)-a(12) from Andrew Howroyd, Dec 05 2018

A048666 Number of rooted 5-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 10, 135, 1980, 30475, 483702, 7847525, 129419240, 2161766520, 36481155310, 620845213890, 10640356142700, 183453873965570, 3179310190998270, 55345614317169210

Offset: 1

Dan Hoey

nonn

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 5 of A048790.

Cf. A048663, A151831.

a(n) = n * A151831(n). - Andrew Howroyd, Dec 05 2018

a(9)-a(15) from Andrew Howroyd, Dec 05 2018

A048667 Number of rooted 6-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 12, 198, 3604, 69405, 1386048, 28403620, 593399416, 12584663901, 270123960220, 5855607723702, 127986261470436, 2817048848634449, 62378907950601228, 1388516401122627270

Offset: 1

Dan Hoey

nonn

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 6 of A048790.

Cf. A048663, A151832.

a(n) = n * A151832(n). - Andrew Howroyd, Dec 05 2018

a(9)-a(15) from Andrew Howroyd, Dec 05 2018

a(13) corrected by Andrei Zabolotskii, Jun 27 2025

A048665 Number of rooted 4-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 8, 84, 936, 10810, 127632, 1531180, 18589840, 227826873, 2813450960, 34963217388, 436807761192, 5482017092760, 69070750692496, 873243394317660, 11073530439895728

Offset: 1

Dan Hoey

nonn

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 4 of A048790.

Cf. A048663, A151830.

a(n) = n * A151830(n). - Andrew Howroyd, Dec 05 2018

a(10)-a(16) from Andrew Howroyd, Dec 05 2018

cp's OEIS Frontend

A094100 Fit a polynomial of degree k-1 to column k of array in A048790, evaluate it at dimension n = -1.

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A094161 Column 5 of A048790.

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A094160 Column 4 of A048790.

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A094159 3 times hexagonal numbers: a(n) = 3*n*(2*n-1).

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A048663 Number of rooted polycubes with n cells, with no symmetries removed.

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Mathematica

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A048668 Number of rooted 7-dimensional "polycubes" with n cells, with no symmetries removed.

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A094101 Number of rooted 8-dimensional "polycubes" with n cells, with no symmetries removed.

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Author

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Extensions

A048666 Number of rooted 5-dimensional "polycubes" with n cells, with no symmetries removed.

A094159 3 times hexagonal numbers: a(n) = 3n(2*n-1).