A195040 -id:A195040 - cp's OEIS frontend

A195047 Concentric 17-gonal numbers.

0, 1, 17, 35, 68, 103, 153, 205, 272, 341, 425, 511, 612, 715, 833, 953, 1088, 1225, 1377, 1531, 1700, 1871, 2057, 2245, 2448, 2653, 2873, 3095, 3332, 3571, 3825, 4081, 4352, 4625, 4913, 5203, 5508, 5815, 6137, 6461, 6800, 7141, 7497, 7855, 8228, 8603, 8993

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric heptadecagonal numbers or concentric heptakaidecagonal numbers.

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

Column 17 of A195040.

Cf. A032527, A032528, A061925, A069130, A151978, A195046, A195048, A195146, A195147.

LinearRecurrence[{2,0,-2,1},{0,1,17,35},50] (* Harvey P. Dale, Dec 23 2017 *)

a(n)=17*n^2/4+13*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 17*n^2/4+13*((-1)^n-1)/8. [Typo fixed by Ivan Panchenko, Nov 08 2013]
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+15*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n)+a(n+1) = A069130(n+1). (End)
From Bruno Berselli, Sep 29 2011: (Start)
a(n) = a(-n) = (34*n^2+13*(-1)^n-13)/8.
a(n) = A151978(A061925(n)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/102 + tan(sqrt(13/17)*Pi/2)*Pi/sqrt(221). - Amiram Eldar, Jan 16 2023

A195058 Concentric 23-gonal numbers.

Original entry on oeis.org

0, 1, 23, 47, 92, 139, 207, 277, 368, 461, 575, 691, 828, 967, 1127, 1289, 1472, 1657, 1863, 2071, 2300, 2531, 2783, 3037, 3312, 3589, 3887, 4187, 4508, 4831, 5175, 5521, 5888, 6257, 6647, 7039, 7452, 7867, 8303, 8741, 9200, 9661, 10143, 10627

Offset: 0

Omar E. Pol, Sep 28 2011

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

Column 23 of A195040.

Cf. A032527, A032528, A195049, A195149, A195158.

Table[23n^2/4 + 19((-1)^n - 1)/8, {n, 0, 49}] (* Alonso del Arte, Jan 23 2015 *)
LinearRecurrence[{2,0,-2,1},{0,1,23,47},50] (* Harvey P. Dale, Jul 22 2023 *)

a(n)=23*n^2/4+19*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 23*n^2/4 + 19*((-1)^n-1)/8.
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1 + 21*x + x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/138 + tan(sqrt(19/23)*Pi/2)*Pi/sqrt(437). - Amiram Eldar, Jan 17 2023

A195158 Concentric 24-gonal numbers.

Original entry on oeis.org

0, 1, 24, 49, 96, 145, 216, 289, 384, 481, 600, 721, 864, 1009, 1176, 1345, 1536, 1729, 1944, 2161, 2400, 2641, 2904, 3169, 3456, 3745, 4056, 4369, 4704, 5041, 5400, 5761, 6144, 6529, 6936, 7345, 7776, 8209, 8664, 9121, 9600, 10081, 10584, 11089

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., and the same line from 1, in the direction 1, 49, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Main axis, perpendicular to A049598 in the same spiral.

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

Column 24 of A195040.

Cf. A032527, A032528, A195143, A195149, A195058.

[(12*n^2+5*(-1)^n-5)/2: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,24,49},50] (* Harvey P. Dale, Jan 28 2021 *)

a(n)=6*n^2+5*((-1)^n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*n^2 + 5*((-1)^n-1)/2.
a(n) = -a(n-1) + A069190(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1+22*x+x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/144 + tan(sqrt(5/6)*Pi/2)*Pi/(4*sqrt(30)). - Amiram Eldar, Jan 17 2023

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A195047 Concentric 17-gonal numbers.

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A195058 Concentric 23-gonal numbers.

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A195158 Concentric 24-gonal numbers.

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