A033581 -id:A033581 - cp's OEIS frontend

A071397 Rounded total surface area of a regular dodecahedron with edge length n.

0, 21, 83, 186, 330, 516, 743, 1012, 1321, 1672, 2065, 2498, 2973, 3489, 4047, 4645, 5285, 5967, 6689, 7453, 8258, 9105, 9993, 10922, 11892, 12904, 13957, 15051, 16186, 17363, 18581, 19841, 21141, 22483, 23866, 25291, 26757, 28264, 29812

Offset: 0

Rick L. Shepherd, May 28 2002

			a(4)=330 because round(3*4^2*sqrt(25 + 10*sqrt(5))) = round(48*6.88190...) = round(330.331...) = 330.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dodecahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071396 (octahedron), A071398 (icosahedron), A071401 (volume of dodecahedron).

[Round(3 * n^2 * Sqrt(25+10*Sqrt(5))): n in [0..50]]; // Vincenzo Librandi, May 21 2011

With[{c=3*Sqrt[25+10*Sqrt[5]]},Round[c*Range[0,40]^2]] (* Harvey P. Dale, Jul 06 2018 *)

for(n=0,100,print1(round(3*n^2*sqrt(25+10*sqrt(5))),","))

a(n) = round(3 * n^2 * sqrt(25 + 10*sqrt(5))).

A071398 Rounded total surface area of a regular icosahedron with edge length n.

Original entry on oeis.org

0, 9, 35, 78, 139, 217, 312, 424, 554, 701, 866, 1048, 1247, 1464, 1697, 1949, 2217, 2503, 2806, 3126, 3464, 3819, 4192, 4581, 4988, 5413, 5854, 6313, 6790, 7283, 7794, 8323, 8868, 9431, 10011, 10609, 11224, 11856, 12505, 13172, 13856, 14558, 15277

Offset: 0

Rick L. Shepherd, May 29 2002

			a(4)=139 because round(5*4^2*sqrt(3)) = round(80*1.73205...) = round(138.56...) = 139.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Icosahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071396 (octahedron), A071397 (dodecahedron), A071402 (volume of icosahedron).

[Round(5 * n^2 * Sqrt(3)): n in [0..50]]; // Vincenzo Librandi, May 21 2011

With[{c=5Sqrt[3]},Round[c Range[0,50]^2]] (* Harvey P. Dale, May 20 2011 *)

for(n=0,100,print1(round(5*n^2*sqrt(3)),","))

from math import isqrt
def A071398(n): return (m:=isqrt(k:=75*n**4))+int(k>m*(m+1)) # Chai Wah Wu, Jun 05 2025

a(n) = round(5 * n^2 * sqrt(3)).

A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 48, 49, 50, 54, 64, 72, 75, 81, 96, 98, 100, 108, 121, 128, 144, 147, 150, 162, 169, 192, 196, 200, 216, 225, 242, 243, 256, 288, 289, 294, 300, 324, 338, 361, 363, 384, 392, 400, 432, 441, 450, 484, 486, 507

Offset: 0

Henry Bottomley, May 06 2003

nonn

It is simple to divide equilateral triangles into these numbers of congruent parts: squares by making smaller equilateral triangles; 6*squares by dividing each small equilateral triangle by its medians into small right triangles; and 2*squares or 3*squares by recombining three or two of these small right triangles.

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A000290, A007913, A001105, A028982, A033428, A033581, A083855, A195055.

mx = 23; Sort@Select[Flatten@Table[{1, 2, 3, 6} n^2, {n, mx}], # <= mx^2 &] (* Ivan Neretin, Nov 08 2016 *)

a(n) is bounded below by 0.137918...*n^2 where 0.137918... = 3*(3-2*sqrt(2))*(2-sqrt(3)); the error appears to be O(n).

Sum_{n>=1} 1/a(n) = Pi^2/3 (A195055). - Amiram Eldar, Dec 19 2020

A202804 a(n) = n(6n+4).

Original entry on oeis.org

0, 10, 32, 66, 112, 170, 240, 322, 416, 522, 640, 770, 912, 1066, 1232, 1410, 1600, 1802, 2016, 2242, 2480, 2730, 2992, 3266, 3552, 3850, 4160, 4482, 4816, 5162, 5520, 5890, 6272, 6666, 7072, 7490, 7920, 8362, 8816, 9282, 9760, 10250, 10752, 11266, 11792, 12330

Offset: 0

Jeremy Gardiner, Dec 24 2011

Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033579 in the same spiral. - Omar E. Pol, Jul 17 2012

Partial sums give A163815. - Leo Tavares, Feb 25 2022

Jeremy Gardiner, Table of n, a(n) for n = 0..3000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Leo Tavares, Illustration: Star Halves.
Pavlos Vavolas, Cellular automaton model of cardiac arrhythmias, University of Sheffield Department of Computer Science (2005), (see page 43). [broken link, abstract]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003154, A005408, A033580, A033581, A049453, A163815, A195319.

Cf. A001318, A033579, A045944, A080859.

A202804:=n->n*(6*n+4): seq(A202804(n), n=0..100); # Wesley Ivan Hurt, Apr 09 2017

Table[n(6n+4),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,10,32},50] (* Harvey P. Dale, Dec 28 2015 *)

x='x + O('x^50); concat([0], Vec(-2*x*(5 + x)/(x - 1)^3)) \\ Indranil Ghosh, Apr 10 2017

a(n) = 2*n(3*n+2) = 6*n^2 + 4*n = 2*A045944(n).
a(n) = A080859(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 28 2015
G.f.: 2*x*(5 + x)/(1 - x)^3. - Indranil Ghosh, Apr 10 2017
a(n) = A003154(n+1) - A005408(n). - Leo Tavares, Feb 25 2022
From Amiram Eldar, Mar 01 2022: (Start)
Sum_{n>=1} 1/a(n) = (Pi/sqrt(3) - 3*log(3) + 3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - 3/8. (End)
E.g.f.: 2*exp(x)*x*(5 + 3*x). - Elmo R. Oliveira, Dec 12 2024

A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 16, 18, 12, 5, 25, 32, 27, 20, 6, 36, 50, 48, 45, 24, 7, 49, 72, 75, 80, 54, 28, 10, 64, 98, 108, 125, 96, 63, 40, 11, 81, 128, 147, 180, 150, 112, 90, 44, 13, 100, 162, 192, 245, 216, 175, 160, 99, 52, 14, 121, 200, 243, 320, 294, 252, 250

Offset: 1

Clark Kimberling, Oct 26 2014

Every positive integer occurs exactly once.

			Northwest corner:
1   4    9    16   25    36    49
2   8    18   32   50    72    98
3   12   27   48   75    108   147
5   20   45   80   125   180   245
6   24   54   96   150   216   294

Cf. A005117, A000037 (is partitioned by the rows of A249327, excluding the first).

z = 20; f = Select[Range[10000], SquareFreeQ[#] &];
u[n_, k_] := f[[n]]*k^2; t = Table[u[n, k], {n, 1, 20}, {k, 1, 20}];
TableForm[t] (* A249327 array *)
Table[u[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* A249327 sequence *)

T(1,k) = A000290(k), T(2,k) = A001105(k), T(3,k) = A033428(k), T(4,k) = A033429(k), T(5,.) through T(10,.) are A033581, A033582, A033583, A033584, A152742 and A144555 without initial 0. - M. F. Hasler, Oct 31 2014

A000535 Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

0, 27, 378, 4536, 48600, 489780, 4738104, 44535456, 409752432, 3708359550, 33125746500, 292779558720, 2565087894720, 22307854940280, 192788833482000, 1657111548654720, 14176605442521312, 120779466450505758, 1025230099571720676, 8674221270307971600

Offset: 1

N. J. A. Sloane

nonn

Number of permutations of 3 distinct letters (ABC) each with n copies such that two (2) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then two fixed points n5=48600. - Zerinvary Lajos, Feb 02 2006

The definition uses notations of Riordan (1958), except for use of n instead of p. - M. F. Hasler, Sep 22 2015

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for sequences related to card matching

Cf. A000279, A000489.

Cf. A033581.

a[n_] := 3*Binomial[n, 2]*Sum[Binomial[n, k+2]*Binomial[n, k]*Binomial[n-2, k], {k, 0, n-2}] + 3n^2*Sum[Binomial[n, k+1]*Binomial[n-1, k+1]*Binomial[ n-1, k], {k, 0, n-2}] (* Jean-François Alcover, Feb 09 2016 *)

A000535(n)=3*binomial(n,2)*sum(k=0,n-2,binomial(n,k+2)*binomial(n,k)*binomial(n-2,k))+3*n^2*sum(k=0,n-2,binomial(n,k+1)*binomial(n-1,k+1)*binomial(n-1,k)) \\ M. F. Hasler, Sep 30 2015

a(n) = 3*binomial(n, 2)*Sum_{k=0..n-2} binomial(n, k+2)*binomial(n, k)*binomial(n-2, k) + 3*n^2*Sum_{k=0..n-2} binomial(n, k+1)*binomial(n-1, k+1)*binomial(n-1, k).
a(n) = 3(n-1)*n^3 3F2(1-n, 1-n, 2-n; 2, 2; -1) + (3/4)(n-1)^2 n^2 3F2(2-n, 2-n, -n; 1, 3; -1), where 3F2 is the hypergeometric function 3F2. - Jean-François Alcover, Feb 09 2016
a(n) ~ 3^(3/2) * 2^(3*n - 2) * n / Pi. - Vaclav Kotesovec, Jun 10 2019

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
More explicit definition by M. F. Hasler, Sep 22 2015

A064763 a(n) = 28*n^2.

Original entry on oeis.org

0, 28, 112, 252, 448, 700, 1008, 1372, 1792, 2268, 2800, 3388, 4032, 4732, 5488, 6300, 7168, 8092, 9072, 10108, 11200, 12348, 13552, 14812, 16128, 17500, 18928, 20412, 21952, 23548, 25200, 26908, 28672, 30492, 32368, 34300, 36288, 38332

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 8-partite graph of order 8n, K_n,n,n,n,n,n,n,n.

Sequence found by reading the line from 0, in the direction 0, 28, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Jul 03 2014

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

Similar sequences are listed in A244630.
Cf. A000217, A000290, A001105, A016742, A033428, A033581, A033582, A033583.
Cf. A144555, A135628.

[28*n^2: n in [0..40]]; // Vincenzo Librandi, Mar 30 2015

I:=[0,28,112]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 30 2015

CoefficientList[Series[28 x (1 + x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 30 2015 *)

a(n)=28*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 56*n + a(n-1) - 28 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = 28*A000290(n) = 14*A001105(n) = 7*A016742(n) = 4*A033582(n) = 2*A144555(n). - Omar E. Pol, Jul 03 2014
From Vincenzo Librandi, Mar 30 2015: (Start)
G.f.: 28*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = t(8*n) - 8*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(8*n) - 8*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 28*x*(1 + x)*exp(x).
a(n) = n*A135628(n). (End)

A307253 Number of triangles larger than size=1 in a matchstick-made hexagon with side length n.

Original entry on oeis.org

0, 0, 14, 62, 166, 346, 624, 1020, 1556, 2252, 3130, 4210, 5514, 7062, 8876, 10976, 13384, 16120, 19206, 22662, 26510, 30770, 35464, 40612, 46236, 52356, 58994, 66170, 73906, 82222, 91140, 100680, 110864, 121712, 133246, 145486, 158454, 172170, 186656, 201932

Offset: 0

John King, Mar 31 2019

nonn

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A033581 (number of size=1 triangles), A045949 (total number of triangles).

The hexagon matchstick sequences are as follows: number of matchsticks: A045945; for T1 triangles: A033581; for larger triangles: this sequence and for total triangles: A045949. There are analogs for triangles (see A045943) and stars (see A045946).

LinearRecurrence[{3,-2,-2,3,-1},{0, 0, 14, 62, 166},166] (* Metin Sariyar, Oct 27 2019 *)

concat([0,0], Vec(2*x^2*(7 + 10*x + 4*x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Apr 02 2019

a(n) = floor(n*(14*n^2+9*n+2)/4)-6*n^2.
G.f.: 2*x^2*(4*x^2+10*x+7)/((x+1)*(x-1)^4).
a(n) = A045949(n) - A033581(n).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>4. - Colin Barker, Apr 02 2019

A354594 a(n) = n^2 + 2*floor(n/2)^2.

Original entry on oeis.org

0, 1, 6, 11, 24, 33, 54, 67, 96, 113, 150, 171, 216, 241, 294, 323, 384, 417, 486, 523, 600, 641, 726, 771, 864, 913, 1014, 1067, 1176, 1233, 1350, 1411, 1536, 1601, 1734, 1803, 1944, 2017, 2166, 2243, 2400, 2481, 2646, 2731, 2904

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A033581, A080859, A247375.

Cf. A213037, A322744 (diagonal), A354595, A354596.

a[n_] := n^2 + 2 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354594 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 6, 11, 24}, 60]

PARI
```
a(n) = n^2 + 2*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 2*A008794(n).
G.f.: x*(1 + 5*x + 3*x^2 + 3*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: (x*(1 + 3*x)*cosh(x) + (1 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jun 07 2022

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

cp's OEIS Frontend

A071397 Rounded total surface area of a regular dodecahedron with edge length n.

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A071398 Rounded total surface area of a regular icosahedron with edge length n.

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A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.

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A202804 a(n) = n*(6*n+4).

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A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

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A000535 Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

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A064763 a(n) = 28*n^2.

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A202804 a(n) = n(6n+4).