A206399 -id:A206399 - cp's OEIS frontend

A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

1, 26, 98, 218, 386, 602, 866, 1178, 1538, 1946, 2402, 2906, 3458, 4058, 4706, 5402, 6146, 6938, 7778, 8666, 9602, 10586, 11618, 12698, 13826, 15002, 16226, 17498, 18818, 20186, 21602, 23066, 24578, 26138, 27746, 29402, 31106, 32858, 34658, 36506, 38402, 40346

Offset: 0

N. J. A. Sloane

Number of points of L_infinity norm n in the simple cubic lattice Z^3. - N. J. A. Sloane, Apr 15 2008
Numbers of cubes needed to completely "cover" another cube. - Xavier Acloque, Oct 20 2003
First bisection of A005897. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Xavier Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Victor Kamel, Hanxueyu Yan, and Sean Chester, CLOVER: A GPU-native, Spatio-graph-based Approach to Exact kNN, ACM Int'l Conf. Supercomp. (ICS 2025). See p. 3.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 24 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

a(n) = if (n==0, 1, 24*n^2 + 2);
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 29 2015

a(n) = (2*n+1)^3 - (2*n-1)^3 for n >= 1. - Xavier Acloque, Oct 20 2003
G.f.: (1+x)*(1+22*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (2*n-1)^2 + (2*n+1)^2 + (4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*24+2)*exp(x)-1. - Gopinath A. R., Feb 14 2012
a(n) = A005899(n) + A195322(n), n > 0. - R. J. Cano, Sep 29 2015
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(3)/24*Pi*coth(Pi*sqrt(3)/6) = 1.065052868574... - R. J. Mathar, May 07 2024
a(n) = 2*A158480(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069190(n)+A069190(n+1). - R. J. Mathar, May 07 2024

More terms from Xavier Acloque, Oct 20 2003

A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

Original entry on oeis.org

1, 34, 130, 290, 514, 802, 1154, 1570, 2050, 2594, 3202, 3874, 4610, 5410, 6274, 7202, 8194, 9250, 10370, 11554, 12802, 14114, 15490, 16930, 18434, 20002, 21634, 23330, 25090, 26914, 28802, 30754, 32770, 34850, 36994, 39202, 41474, 43810, 46210, 48674, 51202

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Apr 21 2021: (Start)
Sequence found by reading the line segment from 1 to 34 together with the line from 34, in the direction 34, 130, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
The spiral begins as follows:
       46_   _   _   _   _   _   _   _   _   _18
        |                                                       |
        |                           0                           |
        |                           |  _   _   _   _  |
        |                           1                           15
        |
       51
(End)

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

Cf. A206399, A158563, A158575, A244082.

Cf. A274979 (generalized 18-gonal numbers).

Join[{1}, 32 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
CoefficientList[Series[(1 + x) (1 + 30 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2014 *)

G.f.: (1+x)*(1+30*x+x^2)/(1-x)^3. [Bruno Berselli, Feb 07 2012]
a(n) = A005893(4n) = A008527(2n); a(n+1) = A108100(2n+2). [Bruno Berselli, Feb 07 2012]
E.g.f.: (x*(x+1)*32+2)*e^x-1. - Gopinath A. R., Feb 14 2012
a(n) = (4n+1)^2+(4n-1)^2 for n>0. [Bruno Berselli, Jun 24 2014]
a(n) = A244082(n) + 2, n >= 1. - Omar E. Pol, Apr 21 2021
Sum_{n>=0} 1/a(n) = 3/4 + Pi/16*coth(Pi/4) = 1.04940725316131.. - R. J. Mathar, May 07 2024
a(n) = 2*A108211(n). - R. J. Mathar, May 07 2024
a(n) = A195315(n)+A195315(n+1). - R. J. Mathar, May 07 2024

A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

Original entry on oeis.org

1, 22, 82, 182, 322, 502, 722, 982, 1282, 1622, 2002, 2422, 2882, 3382, 3922, 4502, 5122, 5782, 6482, 7222, 8002, 8822, 9682, 10582, 11522, 12502, 13522, 14582, 15682, 16822, 18002, 19222, 20482, 21782, 23122, 24502, 25922, 27382, 28882, 30422, 32002, 33622

Offset: 0

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 0..1000
Volodymyr Mazorchuk and Xiaoyu Zhu, Combinatorics of infinite rank module categories over finite dimensional sl_3-modules in Lie-algebraic context, arXiv:2501.00291 [math.RT], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

[1] cat [20*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 20 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {22, 82, 182}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

a(n) = A033571(n)+A158186(n) = A158187(n)*2 for n>0. - Reinhard Zumkeller, Mar 13 2009
G.f.: (1+x)*(1+18*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*20+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/40*Pi*coth(Pi/sqrt(10)) = 1.0772981051444036327... - R. J. Mathar, May 07 2024
a(n) = A069133(n)+A069133(n+1). - R. J. Mathar, May 07 2024

A005919 Number of points on surface of tricapped prism: a(n) = 7*n^2 + 2 for n > 0, a(0)=1.

Original entry on oeis.org

1, 9, 30, 65, 114, 177, 254, 345, 450, 569, 702, 849, 1010, 1185, 1374, 1577, 1794, 2025, 2270, 2529, 2802, 3089, 3390, 3705, 4034, 4377, 4734, 5105, 5490, 5889, 6302, 6729, 7170, 7625, 8094, 8577, 9074, 9585, 10110, 10649, 11202, 11769, 12350, 12945, 13554

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

A005919:=-(z+1)*(z**2+5*z+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

Join[{1},7*Range[50]^2+2] (* or *) CoefficientList[Series[(-x^3-6x^2-6x-1)/(x-1)^3,{x,0,50}],x] (* Harvey P. Dale, Jan 13 2013 *)

From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: (1 + x)*(1 + 5*x + x^2)/(1-x)^3.
E.g.f.: exp(x)*(7*x^2 + 7*x + 2) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

More terms from Erich Friedman, Aug 08 2005

A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

Original entry on oeis.org

1, 20, 74, 164, 290, 452, 650, 884, 1154, 1460, 1802, 2180, 2594, 3044, 3530, 4052, 4610, 5204, 5834, 6500, 7202, 7940, 8714, 9524, 10370, 11252, 12170, 13124, 14114, 15140, 16202, 17300, 18434, 19604, 20810, 22052, 23330, 24644, 25994, 27380, 28802, 30260

Offset: 0

N. J. A. Sloane

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2=4 can be written as a(n+1)^2-A010002(n+1)*A008588(n+1)^2=4. - Vincenzo Librandi, Feb 07 2012

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

After 20, all terms are in A000408.

Cf. A206399.

[1] cat [18*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 18 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {20, 74, 164}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+16*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (3*n-1)^2+(3*n+1)^2 = (n-1)^2+(n+1)^2+(4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*18+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+ (1/12)*Pi*coth(Pi/3) = 1.0853330948... - R. J. Mathar, May 07 2024
a(n) = 2*A247792(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069131(n)+A069131(n+1). - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A005903 Number of points on surface of dodecahedron: 30n^2 + 2 for n > 0.

Original entry on oeis.org

1, 32, 122, 272, 482, 752, 1082, 1472, 1922, 2432, 3002, 3632, 4322, 5072, 5882, 6752, 7682, 8672, 9722, 10832, 12002, 13232, 14522, 15872, 17282, 18752, 20282, 21872, 23522, 25232, 27002, 28832, 30722, 32672, 34682, 36752, 38882, 41072, 43322, 45632, 48002

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bruno Berselli, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Polyhedral Numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

A005903:=-(z+1)*(z**2+28*z+1)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]

Join[{1}, 30 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)

a(n) = if (n==0, 1, 30*n^2+2); \\ Michel Marcus, Mar 04 2014

G.f.: (1+x)*(1+28*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)

Sum_{n>=0} 1/a(n) = 3/4 + Pi*sqrt(15)*coth(Pi/sqrt 15)/60 = 1.052567... - R. J. Mathar, Apr 27 2024

A005905 Number of points on surface of truncated tetrahedron: a(n) = 14*n^2 + 2 for n > 0, a(0)=1.

Original entry on oeis.org

1, 16, 58, 128, 226, 352, 506, 688, 898, 1136, 1402, 1696, 2018, 2368, 2746, 3152, 3586, 4048, 4538, 5056, 5602, 6176, 6778, 7408, 8066, 8752, 9466, 10208, 10978, 11776, 12602, 13456, 14338, 15248, 16186, 17152, 18146, 19168, 20218, 21296, 22402, 23536, 24698

Offset: 0

N. J. A. Sloane

Also sequence found by reading the segment (1,16) together with the line from 16, in the direction 16, 58, ... , in the square spiral whose vertices are the generalized enneagonal numbers A118277. - Omar E. Pol, Nov 05 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. S. M. Coxeter, Polyhedral Numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A118277, A206399.

A005905:=-(z+1)*(z**2+12*z+1)/(z-1)**3; # [Simon Plouffe in his 1992 dissertation.]

a[0] = 1; a[n_] := 14 n^2 + 2; Table[a[n], {n, 0, 50}] (* Wesley Ivan Hurt, Mar 04 2014 *)

a(n) = if (n==0, 1, 14*n^2+2); \\ Michel Marcus, Mar 04 2014

From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: (1 + x)*(1 + 12*x + x^2)/(1-x)^3.
E.g.f.: 2*exp(x)*(7*x^2 + 7*x + 1) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

More terms from Michel Marcus, Mar 04 2014

A010002 a(0) = 1, a(n) = 9*n^2 + 2 for n>0.

Original entry on oeis.org

1, 11, 38, 83, 146, 227, 326, 443, 578, 731, 902, 1091, 1298, 1523, 1766, 2027, 2306, 2603, 2918, 3251, 3602, 3971, 4358, 4763, 5186, 5627, 6086, 6563, 7058, 7571, 8102, 8651, 9218, 9803, 10406, 11027, 11666, 12323, 12998, 13691, 14402, 15131, 15878, 16643

Offset: 0

N. J. A. Sloane

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=1, s=2. After 1, all terms are in A000408. [Bruno Berselli, Feb 06 2012]

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2 = 4 can be written as A010008(n+1)^2-a(n+1)*A008588(n+1)^2 = 4. - Vincenzo Librandi, Feb 07 2012

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

Cf. A206399.

[1] cat [9*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 9 Range[43]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {11, 38, 83}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

A010002(n)=9*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1+7*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*9+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(2)/12 *Pi*coth(Pi/3*sqrt 2) = 1.1606262038.. - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A010005 a(0) = 1, a(n) = 15*n^2 + 2 for n>0.

Original entry on oeis.org

1, 17, 62, 137, 242, 377, 542, 737, 962, 1217, 1502, 1817, 2162, 2537, 2942, 3377, 3842, 4337, 4862, 5417, 6002, 6617, 7262, 7937, 8642, 9377, 10142, 10937, 11762, 12617, 13502, 14417, 15362, 16337, 17342, 18377, 19442, 20537, 21662, 22817, 24002, 25217, 26462

Offset: 0

N. J. A. Sloane

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

Cf. A206399.

[1] cat [15*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 15 Range[42]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {17, 62, 137}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

A010005(n)=15*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1+13*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*15+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(30)/60*Pi*coth(Pi *sqrt(30)/15) = 1.101107302494... - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A010020 a(0) = 1, a(n) = 31*n^2 + 2 for n>0.

Original entry on oeis.org

1, 33, 126, 281, 498, 777, 1118, 1521, 1986, 2513, 3102, 3753, 4466, 5241, 6078, 6977, 7938, 8961, 10046, 11193, 12402, 13673, 15006, 16401, 17858, 19377, 20958, 22601, 24306, 26073, 27902, 29793, 31746, 33761, 35838, 37977, 40178, 42441, 44766, 47153, 49602

Offset: 0

N. J. A. Sloane

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

Cf. A206399.

[1] cat [31*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 31 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {33, 126, 281}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+29*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 07 2012
E.g.f.: (x*(x+1)*31+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(62)/124 *Pi*coth(Pi*sqrt(62)/31) = 1.05093832062... - R. J. Mathar, May 07 2024

cp's OEIS Frontend

A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

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A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

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A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

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A005919 Number of points on surface of tricapped prism: a(n) = 7*n^2 + 2 for n > 0, a(0)=1.

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A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

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Magma

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A005903 Number of points on surface of dodecahedron: 30n^2 + 2 for n > 0.

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A005905 Number of points on surface of truncated tetrahedron: a(n) = 14*n^2 + 2 for n > 0, a(0)=1.

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