A032527 -id:A032527 - cp's OEIS frontend

A195043 Concentric 11-gonal numbers.

0, 1, 11, 23, 44, 67, 99, 133, 176, 221, 275, 331, 396, 463, 539, 617, 704, 793, 891, 991, 1100, 1211, 1331, 1453, 1584, 1717, 1859, 2003, 2156, 2311, 2475, 2641, 2816, 2993, 3179, 3367, 3564, 3763, 3971, 4181, 4400, 4621, 4851, 5083, 5324, 5567

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric hendecagonal numbers. A033584 and A069173 interleaved.

Partial sums of A175885. - Reinhard Zumkeller, Jan 07 2012

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 11 of A195040.

Cf. A032527, A032528, A033584, A069173, A175885, A195042, A195142, A195143, A195045.

a195043 n = a195043_list !! n
a195043_list = scanl (+) 0 a175885_list
-- Reinhard Zumkeller, Jan 07 2012

[11*n^2/4+7*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,11,23},50] (* Harvey P. Dale, May 20 2019 *)

Vec(-x*(x^2+9*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = 11*n^2/4 + 7*((-1)^n - 1)/8.
a(n) = -a(n-1) + A069125(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 15 2013: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(x^2+9*x+1) / ((x-1)^3*(x+1)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/66 + tan(sqrt(7/11)*Pi/2)*Pi/sqrt(77). - Amiram Eldar, Jan 16 2023

A195045 Concentric 13-gonal numbers.

Original entry on oeis.org

0, 1, 13, 27, 52, 79, 117, 157, 208, 261, 325, 391, 468, 547, 637, 729, 832, 937, 1053, 1171, 1300, 1431, 1573, 1717, 1872, 2029, 2197, 2367, 2548, 2731, 2925, 3121, 3328, 3537, 3757, 3979, 4212, 4447, 4693, 4941, 5200, 5461, 5733, 6007, 6292, 6579, 6877, 7177, 7488, 7801, 8125

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric tridecagonal numbers or concentric triskaidecagonal numbers.

Partial sums of A175886. - Reinhard Zumkeller, Jan 07 2012

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 13 of A195040.

Cf. A032527, A032528, A069126, A175886, A195043, A195046, A195143, A195145.

a195045 n = a195045_list !! n
a195045_list = scanl (+) 0 a175886_list
-- Reinhard Zumkeller, Jan 07 2012

[13*n^2/4+9*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

A195045:=n->13*n^2/4+9*((-1)^n-1)/8: seq(A195045(n), n=0..70); # Wesley Ivan Hurt, Nov 22 2015

Table[13 n^2/4 + 9 ((-1)^n - 1)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Nov 22 2015 *)

a(n)=13*n^2/4+9*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(-x*(1+11*x+x^2)/((1+x)*(x-1)^3) + O(x^50))) \\ Altug Alkan, Nov 22 2015

a(n) = 13*n^2/4+9*((-1)^n-1)/8.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+11*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n)+a(n+1) = A069126(n+1). (End)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3. - Wesley Ivan Hurt, Nov 22 2015
Sum_{n>=1} 1/a(n) = Pi^2/78 + tan(3*Pi/(2*sqrt(13)))*Pi/(3*sqrt(13)). - Amiram Eldar, Jan 16 2023

A195048 Concentric 19-gonal numbers.

Original entry on oeis.org

0, 1, 19, 39, 76, 115, 171, 229, 304, 381, 475, 571, 684, 799, 931, 1065, 1216, 1369, 1539, 1711, 1900, 2091, 2299, 2509, 2736, 2965, 3211, 3459, 3724, 3991, 4275, 4561, 4864, 5169, 5491, 5815, 6156, 6499, 6859, 7221, 7600, 7981, 8379, 8779, 9196

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric enneadecagonal 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 19 of A195040.

Cf. A032527, A032528, A195047, A195147, A195148, A195049.

LinearRecurrence[{2,0,-2,1},{0,1,19,39},50] (* Harvey P. Dale, May 17 2016 *)

a(n)=19*n^2/4+15*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (19/4)*n^2 + (15/8)*((-1)^n - 1).
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1 + 17*x + x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/114 + tan(sqrt(15/19)*Pi/2)*Pi/sqrt(285). - Amiram Eldar, Jan 17 2023

A195049 Concentric 21-gonal numbers.

Original entry on oeis.org

0, 1, 21, 43, 84, 127, 189, 253, 336, 421, 525, 631, 756, 883, 1029, 1177, 1344, 1513, 1701, 1891, 2100, 2311, 2541, 2773, 3024, 3277, 3549, 3823, 4116, 4411, 4725, 5041, 5376, 5713, 6069, 6427, 6804, 7183, 7581, 7981, 8400, 8821, 9261, 9703, 10164

Offset: 0

Omar E. Pol, Sep 27 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 21 of A195040.

Cf. A032527, A032528, A195048, A195148, A195149, A195058.

A195049:=n->21*n^2/4+17*((-1)^n-1)/8: seq(A195049(n), n=0..100); # Wesley Ivan Hurt, Jan 17 2017

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 21, 43}, 50] (* Amiram Eldar, Jan 17 2023 *)

a(n)=21*n^2/4+17*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 21*n^2/4 + 17*((-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+19*x+x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/126 + tan(sqrt(17/21)*Pi/2)*Pi/sqrt(357). - Amiram Eldar, Jan 17 2023

A194275 Concentric pentagonal numbers of the second kind: a(n) = floor(5n(n+1)/6).

Original entry on oeis.org

0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, 110, 130, 151, 175, 200, 226, 255, 285, 316, 350, 385, 421, 460, 500, 541, 585, 630, 676, 725, 775, 826, 880, 935, 991, 1050, 1110, 1171, 1235, 1300, 1366, 1435, 1505, 1576, 1650, 1725, 1801, 1880, 1960, 2041, 2125

Offset: 0

Omar E. Pol, Aug 20 2011

Quasipolynomial: trisections are (15*x^2 - 15*x + 2)/2, 5*(15*x^2 - 5*x)/2, and 5*(15*x^2 + 5*x)/2. - Charles R Greathouse IV, Aug 23 2011
Appears to be similar to cellular automaton. The sequence gives the number of elements in the structure after n-th stage. Positive integers of A008854 gives the first differences. For a definition without words see the illustration of initial terms in the example section.
Also partial sums of A008854.
Also row sums of an infinite square array T(n,k) in which column k lists 3*k-1 zeros followed by the numbers A008706 (see example).
For concentric pentagonal numbers see A032527. - Omar E. Pol, Sep 27 2011

			Using the numbers A008706 we can write:
0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
0, 0, 0,  0,  1,  5, 10, 15, 20, 25, 30, ...
0, 0, 0,  0,  0,  0,  0,  1,  5, 10, 15, ...
0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  1, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, ...
...
Illustration of initial terms (in a precise representation the pentagons should appear strictly concentric):
.                                             o
.                                           o   o
.                            o            o       o
.                          o   o        o     o     o
.               o        o       o    o     o   o     o
.             o   o    o     o     o   o     o o     o
.      o    o       o   o         o     o           o
.    o   o   o     o     o       o       o         o
. o   o o     o o o       o o o o         o o o o o
.
. 1    5        10          16                25

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

Cf. A000326, A008706, A008854, A032528, A152734, A193273, A193274.

Cf. similar sequences with the formula floor(k*n*(n+1)/(k+1)) listed in A281026.

[Floor(5*n*(n+1)/6): n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

Table[Floor[5 n (n + 1)/6], {n, 0, 50}] (* Arkadiusz Wesolowski, Oct 03 2011 *)

a(n)=5*n*(n+1)\6 \\ Charles R Greathouse IV, Aug 23 2011

G.f.: (-1 - 3*x - x^2)/((-1 + x)^3*(1 + x + x^2)). - Alexander R. Povolotsky, Aug 22 2011

a(n) = floor(5*n*(n+1)/6). - Arkadiusz Wesolowski, Aug 23 2011

Name improved by Arkadiusz Wesolowski, Aug 23 2011

New name from Omar E. Pol, Sep 28 2011

A195046 Concentric 15-gonal numbers.

Original entry on oeis.org

0, 1, 15, 31, 60, 91, 135, 181, 240, 301, 375, 451, 540, 631, 735, 841, 960, 1081, 1215, 1351, 1500, 1651, 1815, 1981, 2160, 2341, 2535, 2731, 2940, 3151, 3375, 3601, 3840, 4081, 4335, 4591, 4860, 5131, 5415, 5701, 6000, 6301, 6615, 6931, 7260, 7591

Offset: 0

Omar E. Pol, Sep 27 2011

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

Column 15 of A195040.

Cf. A032527, A032528, A195045, A195047, A195145, A195146.

Table[15n^2/4+11((-1)^n-1)/8,{n,0,50}] (* or *) LinearRecurrence[ {2,0,-2,1},{0,1,15,31},50] (* Harvey P. Dale, Feb 23 2012 *)

a(n)=15*n^2/4+11*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2/4+11*((-1)^n-1)/8.
From Harvey P. Dale, Feb 23 2012: (Start)
a(0)=0, a(1)=1, a(2)=15, a(3)=31, a(n)=2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: -((x*(1+x*(13+x)))/((-1+x)^3*(1+x))). (End)
Sum_{n>=1} 1/a(n) = Pi^2/90 + tan(sqrt(11/15)*Pi/2)*Pi/sqrt(165). - Amiram Eldar, Jan 16 2023

a(1)=1 added by Harvey P. Dale, Feb 23 2012

A195047 Concentric 17-gonal numbers.

Original entry on oeis.org

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

A032526 a(n) = floor(5*n^2/2).

Original entry on oeis.org

0, 2, 10, 22, 40, 62, 90, 122, 160, 202, 250, 302, 360, 422, 490, 562, 640, 722, 810, 902, 1000, 1102, 1210, 1322, 1440, 1562, 1690, 1822, 1960, 2102, 2250, 2402, 2560, 2722, 2890, 3062, 3240, 3422, 3610, 3802, 4000, 4202, 4410, 4622, 4840, 5062

Offset: 0

N. J. A. Sloane

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

Cf. A032527.

[Floor(5*n^2/2): n in [0..50]]; // Bruno Berselli, Jun 14 2013

A032526:=n->floor(5*n^2/2): seq(A032526(n), n=0..100); # Wesley Ivan Hurt, Feb 03 2017

Table[Floor[5 n^2/2], {n, 0, 50}] (* Bruno Berselli, Jun 14 2013 *)
LinearRecurrence[{2,0,-2,1},{0,2,10,22},50] (* Harvey P. Dale, Dec 14 2016 *)

a(n) = 2n^2 + floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
From Bruno Berselli, Jun 14 2013: (Start)
G.f.: 2*x*(1+3*x+x^2)/((1+x)*(1-x)^3).
a(n) = 2*A032527(n). (End)
Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)). - Amiram Eldar, Aug 15 2025

cp's OEIS Frontend

A195043 Concentric 11-gonal numbers.

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A195045 Concentric 13-gonal numbers.

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A195048 Concentric 19-gonal numbers.

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A195049 Concentric 21-gonal numbers.

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A194275 Concentric pentagonal numbers of the second kind: a(n) = floor(5*n*(n+1)/6).

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A195046 Concentric 15-gonal numbers.

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A195047 Concentric 17-gonal numbers.

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A194275 Concentric pentagonal numbers of the second kind: a(n) = floor(5n(n+1)/6).