A033581 -id:A033581 - cp's OEIS frontend

A008747 Expansion of (1+x^4)/((1-x)(1-x^2)(1-x^3)).

1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 21, 24, 29, 33, 38, 43, 49, 54, 61, 67, 74, 81, 89, 96, 105, 113, 122, 131, 141, 150, 161, 171, 182, 193, 205, 216, 229, 241, 254, 267, 281, 294, 309, 323, 338, 353, 369, 384, 401, 417, 434, 451, 469, 486, 505, 523, 542, 561

Offset: 0

N. J. A. Sloane

For n>=1, the set {A008747(6n+-1)} is the set of numbers of the form a^2 + 5*(a+1)^2 for -inf < a < inf. Furthermore the set A008747(6n) is A033581(n). - Kieren MacMillan, Dec 19 2007

For n>1, a(n-1) is the number of aperiodic necklaces (Lyndon words) with k<=3 black beads and n-k white beads. For n=4 we have for example a(3)=3 aperiodic necklaces: BWWW, BBWW and BBBW. BWBW is periodic and is not counted. - Herbert Kociemba, Oct 23 2016

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + 14*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Robert Morris, Minimal percolating sets in bootstrap percolation, arXiv:math/0702370 [math.CO], 2007-2008.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A008747, A033581, A051274, A061347, A071619, A077043.

a:=[1,1,2,3,5,6];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4] -a[n-5]+a[n-6]; od; a; # G. C. Greubel, Aug 03 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^4)/((1-x)*(1-x^2)*(1-x^3)) )); // G. C. Greubel, Aug 03 2019

A008747:=n->ceil((n+1)^2/6): seq(A008747(n), n=0..100); # Wesley Ivan Hurt, Oct 25 2016

CoefficientList[Series[(1+x^4)/((1-x)(1-x^2)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,5,6},60] (* Harvey P. Dale, Sep 05 2012 *)

Vec((1+x^4)/((1-x)*(1-x^2)*(1-x^3))+O(x^60)) \\ Charles R Greathouse IV, Sep 25 2012

((1+x^4)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

G.f.: (1+x^4)/((1-x)*(1-x^2)*(1-x^3)).
a(n) = ceiling((n+1)^2/6).
a(n) = (12*n + 23 + 6*n^2 + 9*(-1)^n + 4*A061347(n))/36. - R. J. Mathar, Mar 15 2011
a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=6, a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5. - Harvey P. Dale, Sep 05 2012
From Michael Somos, Oct 25 2016: (Start)
Euler transform of length 8 sequence [ 1, 1, 1, 1, 0, 0, 0, -1].
a(n) = a(-2-n) for all n in Z.
a(2*n-1) = A071619(n).
a(3*n-1) = 2*A077043(n).
a(n) - a(n-1) = A051274(n). (End)

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

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

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A045949 Number of equilateral triangles formed out of matches that can be found in a hexagonal chunk of side length n in hexagonal array of matchsticks.

Original entry on oeis.org

0, 6, 38, 116, 262, 496, 840, 1314, 1940, 2738, 3730, 4936, 6378, 8076, 10052, 12326, 14920, 17854, 21150, 24828, 28910, 33416, 38368, 43786, 49692, 56106, 63050, 70544, 78610, 87268, 96540, 106446, 117008, 128246, 140182, 152836, 166230, 180384, 195320, 211058

Offset: 0

R. K. Guy

nonn

N. J. A. Sloane, Illustration of a(1)=6
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). [From _R. J. Mathar_, Sep 03 2010]

See A008893 for a related sequence.

For hexagons, the number of matches required is A045945, the number of size=1 triangles is A033581, the larger triangles is A307253 and the total number is A045949. For the analogs for triangles see A045943 and for stars see A045946. - John King, Apr 05 2019

List([0..40], n-> (28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8); # G. C. Greubel, Apr 05 2019

[(28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8: n in [0..40]]; // G. C. Greubel, Apr 05 2019

LinearRecurrence[{3,-2,-2,3,-1},{0,6,38,116,262},40] (* or *) CoefficientList[Series[(2*x*(x*(x+2)*(x+5)+3))/((x-1)^4*(x+1)),{x,0,40}],x] (* Harvey P. Dale, Jun 11 2011 *)

A045949(n):=floor(n*(14*n^2+9*n+2)/4)$
makelist(A045949(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

{a(n) = (28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8}; \\ G. C. Greubel, Apr 05 2019

floor(1:25*(14*(1:25)^2+9*(1:25)+2)/4) # Christian N. K. Anderson, Apr 27 2013

[(28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8 for n in (0..40)] # G. C. Greubel, Apr 05 2019

a(n) = floor(n*(14*n^2 + 9*n + 2)/4).
From R. J. Mathar, Sep 03 2010: (Start)
a(n) = +3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).
G.f.: 2*x*(3+10*x+7*x^2+x^3) / ( (1+x)*(1-x)^4 ).
a(n) = (28*n^3 + 18*n^2 + 4*n - 1 + (-1)^n)/8. (End)
a(n) = A033581(n) + A307253(n). - John King, Apr 04 2019
E.g.f.: (x*(25 + 51*x + 14*x^2)*exp(x) - sinh(x))/4. - G. C. Greubel, Apr 05 2019

Edited by N. J. A. Sloane, May 29 2012

A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 6, 6, 0, 0, 0, 1, 9, 24, 10, 0, 0, 0, 1, 12, 54, 80, 15, 0, 0, 0, 1, 15, 96, 270, 240, 21, 0, 0, 0, 1, 18, 150, 640, 1215, 672, 28, 0, 0, 0, 1, 21, 216, 1250, 3840, 5103, 1792, 36, 0, 0, 0, 1, 24, 294, 2160, 9375, 21504, 20412, 4608, 45, 0

Offset: 0

Paul Barry, Mar 08 2003

Rows, of the square array, are three-fold convolutions of sequences of powers.

			The array begins as:
  0,  0,  0,   0,   0,    0, ...
  0,  0,  0,   0,   0,    0, ...
  0,  1,  1,   1,   1,    1, ... A000012
  0,  3,  6,   9,  12,   15, ... A008585
  0,  6, 24,  54,  96,  150, ... A033581
  0, 10, 80, 270, 640, 1250, ... A244729
The antidiagonal triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  0, 0, 1, 0;
  0, 0, 1, 3,  0;
  0, 0, 1, 6,  6,  0;
  0, 0, 1, 9, 24, 10, 0;

G. C. Greubel, Antidiadoganal rows n = 0..50, flattened

Main diagonal: A081131.
Rows: A000012 (n=2), A008585 (n=3), A033581 (n=4), A244729 (n=5).
Columns: A000217 (k=1), A001788 (k=2), A027472 (k=3), A038845 (k=4), A081135 (k=5), A081136 (k=6), A027474 (k=7), A081138 (k=8), A081139 (k=9), A081140 (k=10), A081141 (k=11), A081142 (k=12), A027476 (k=15).

[k eq n select 0 else (n-k)^(k-2)*Binomial(k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021

Table[If[k==n, 0, (n-k)^(k-2)*Binomial[k, 2]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 14 2021 *)

T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));
seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );
seq(12) \\ Michel Marcus, May 14 2021

flatten([[0 if (k==n) else (n-k)^(k-2)*binomial(k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

T(n, k) = k^(n-2)*binomial(n, 2), with T(n, 0) = 0 (square array).
T(n, n) = A081131(n).
Rows have g.f. x^3/(1-k*x)^n.
From G. C. Greubel, May 14 2021: (Start)
T(k, n-k) = (n-k)^(k-2)*binomial(k,2) with T(n, n) = 0 (antidiagonal triangle).
Sum_{k=0..n} T(n, n-k) = A081197(n). (End)

Term a(5) corrected by G. C. Greubel, May 14 2021

A244636 a(n) = 30*n^2.

Original entry on oeis.org

0, 30, 120, 270, 480, 750, 1080, 1470, 1920, 2430, 3000, 3630, 4320, 5070, 5880, 6750, 7680, 8670, 9720, 10830, 12000, 13230, 14520, 15870, 17280, 18750, 20280, 21870, 23520, 25230, 27000, 28830, 30720, 32670, 34680, 36750, 38880, 41070, 43320, 45630, 48000, 50430

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 30, ..., in the square spiral whose vertices are the generalized 17-gonal numbers. - Omar E. Pol, Jul 03 2014

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

Cf. similar sequences listed in A244630.

Cf. A000290, A001105, A033428, A033429, A033581, A033583, A064761, A249674, A330451.

Magma
```
[30*n^2: n in [0..40]];
```

A244636:=n->30*n^2: seq(A244636(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[30 n^2, {n, 0, 40}]
CoefficientList[Series[30x (1+x)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[ {3,-3,1},{0,30,120},50] (* Harvey P. Dale, Dec 02 2021 *)

a(n)=30*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 30*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 30*A000290(n) = 15*A001105(n) = 10*A033428(n) = 6*A033429(n) = 5*A033581(n) = 3*A033583(n) = 2*A064761(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 30*x*(1 + x)*exp(x).
a(n) = n*A249674(n) = A330451(3*n). (End)

A001283 Triangle read by rows, in which row n consists of n(n+m) for m = 1 .. n-1.

Original entry on oeis.org

6, 12, 15, 20, 24, 28, 30, 35, 40, 45, 42, 48, 54, 60, 66, 56, 63, 70, 77, 84, 91, 72, 80, 88, 96, 104, 112, 120, 90, 99, 108, 117, 126, 135, 144, 153, 110, 120, 130, 140, 150, 160, 170, 180, 190, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 156, 168, 180

Offset: 2

N. J. A. Sloane

With a different offset: triangle read by rows: t(n, m) = T(n+1, m) = (n+1)(n+m+1) = radius of C-excircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 21 2001

			The triangle T(n, m) begins:
n\m   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...
2:    6
3:   12  15
4:   20  24  28
5:   30  35  40  45
6:   42  48  54  60  66
7:   56  63  70  77  84  91
8:   72  80  88  96 104 112 120
9:   90  99 108 117 126 135 144 153
10: 110 120 130 140 150 160 170 180 190
11: 132 143 154 165 176 187 198 209 220 231
12: 156 168 180 192 204 216 228 240 252 264 276
13: 182 195 208 221 234 247 260 273 286 299 312 325
14: 210 224 238 252 266 280 294 308 322 336 350 364 378
15: 240 255 270 285 300 315 330 345 360 375 390 405 420 435
...
[Reformatted and extended by _Wolfdieter Lang_, Dec 02 2014]
----------------------------------------------------------------

T. D. Noe, Rows n = 2..100, flattened

Cf. A003991, A063929, A063930.

Row sums are in A085788. Central column is A033581.

Flatten[Table[n*(n+m), {n, 2, 10}, {m, n-1}]] (* T. D. Noe, Jun 27 2012 *)

T(n, m) = n*(n+m), n-1 >= m >= 1.

Edited comment by Wolfdieter Lang, Dec 02 2014

A064762 a(n) = 21*n^2.

Original entry on oeis.org

0, 21, 84, 189, 336, 525, 756, 1029, 1344, 1701, 2100, 2541, 3024, 3549, 4116, 4725, 5376, 6069, 6804, 7581, 8400, 9261, 10164, 11109, 12096, 13125, 14196, 15309, 16464, 17661, 18900, 20181, 21504, 22869, 24276, 25725, 27216, 28749

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n.

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

Similar sequences are listed in A244630.
Cf. A000217, A000290, A033428, A033581, A033582, A033583.
Cf. A008603, A195049.

[21*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 04 2014

21 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3,-3,1},{0,21,84},40] (* Harvey P. Dale, Jul 29 2019 *)

a(n)=21*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 42*n + a(n-1) - 21 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = 21*A000290(n) = 7*A033428(n) = 3*A033582(n). - Omar E. Pol, Jul 03 2014
a(n) = t(7*n) - 7*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(7*n) - 7*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 21*x*(1 + x)/(1-x)^3.
E.g.f.: 21*x*(1 + x)*exp(x).
a(n) = n*A008603(n) = A195049(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A195824 a(n) = 24*n^2.

Original entry on oeis.org

0, 24, 96, 216, 384, 600, 864, 1176, 1536, 1944, 2400, 2904, 3456, 4056, 4704, 5400, 6144, 6936, 7776, 8664, 9600, 10584, 11616, 12696, 13824, 15000, 16224, 17496, 18816, 20184, 21600, 23064, 24576, 26136, 27744, 29400, 31104, 32856, 34656, 36504, 38400, 40344

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818.

Surface area of a cube with side 2n. - Wesley Ivan Hurt, Aug 05 2014

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

Bisection of A195158.

Cf. A000290, A001105, A008606, A016742, A033428, A033581, A135453, A139098, A195818.

[24*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 05 2014

I:=[0,24,96]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 06 2014

A195824:=n->24*n^2: seq(A195824(n), n=0..50); # Wesley Ivan Hurt, Aug 05 2014

24 Range[0, 30]^2 (* or *) Table[24 n^2, {n, 0, 30}] (* or *) CoefficientList[Series[24 x (1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 05 2014 *)
LinearRecurrence[{3,-3,1},{0,24,96},40] (* Harvey P. Dale, Nov 11 2017 *)

a(n) = 24*n^2; \\ Michel Marcus, Aug 05 2014

a(n) = 24*A000290(n) = 12*A001105(n) = 8*A033428(n) = 6*A016742(n) = 4*A033581(n) = 3*A139098(n) = 2*A135453(n).
From Wesley Ivan Hurt, Aug 05 2014: (Start)
G.f.: 24*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 24*x*(1 + x)*exp(x).
a(n) = n*A008606(n) = A195158(2*n). (End)

A000279 Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

3, 24, 216, 1824, 15150, 124416, 1014888, 8241792, 66724398, 538990800, 4346692680, 35009591040, 281699380560, 2264868936960, 18198009147600, 146142982814208, 1173123636533454, 9413509300965936, 75513633110271264, 605598295606296000, 4855626127979443908, 38924245740546950784

Offset: 1

N. J. A. Sloane

nonn

Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150. - 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).

Alois P. Heinz, Table of n, a(n) for n = 1..300
Index entries for sequences related to card matching

Cf. A000489, A000535.

Cf. A033581.

f[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_] := n^2*(f[n]+4*f[n-1])/(n+1); Array[a, 20] (* Jean-François Alcover, Mar 11 2014, after Mark van Hoeij *)

A000279(n)=3*n*sum(k=0,n-1,binomial(n,k+1)*binomial(n,k)*binomial(n-1,k)) \\ M. F. Hasler, Sep 21 2015

a(n) = 3n * sum(C(n, k+1)*C(n, k)*C(n-1, k), k=0..n-1).
G.f.: x * (6*hypergeom([4/3, 5/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^3 - 3*hypergeom([2/3, 4/3],[1],27*x^2/(1-2*x)^3)/(1-2*x)^2). - Mark van Hoeij, Oct 23 2011
a(n) = n^2*(A000172(n)+4*A000172(n-1))/(n+1). - Mark van Hoeij, Oct 26 2011
a(n) ~ 8^n*sqrt(3)/Pi = 8^n*0.5513... - M. F. Hasler, Sep 21 2015
a(n) = 3n*A262407(n). - M. F. Hasler, Sep 23 2015

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
Three lines of data completed and more explicit definition by M. F. Hasler, Sep 21 2015

A071396 Rounded total surface area of a regular octahedron with edge length n.

Original entry on oeis.org

0, 3, 14, 31, 55, 87, 125, 170, 222, 281, 346, 419, 499, 585, 679, 779, 887, 1001, 1122, 1251, 1386, 1528, 1677, 1833, 1995, 2165, 2342, 2525, 2716, 2913, 3118, 3329, 3547, 3772, 4005, 4244, 4489, 4742, 5002, 5269, 5543, 5823, 6111, 6405, 6707, 7015, 7330

Offset: 0

Rick L. Shepherd, May 23 2002

			a(3)=31 because round(2*3^2*sqrt(3)) = round(18*1.73205...) = round(31.1769...) = 31.

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, Octahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071397 (dodecahedron), A071398 (icosahedron), A071400 (volume of octahedron).

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

Table[Round[2n^2 Sqrt[3]],{n,0,50}] (* Harvey P. Dale, Feb 19 2024 *)

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

from math import isqrt
def A071396(n): return (m:=isqrt(k:=3*n**4<<2))+int(k>m*(m+1)) # Chai Wah Wu, Jun 05 2025

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

cp's OEIS Frontend

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