A069190 -id:A069190 - cp's OEIS frontend

A195314 Centered 28-gonal numbers.

1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849, 2185, 2549, 2941, 3361, 3809, 4285, 4789, 5321, 5881, 6469, 7085, 7729, 8401, 9101, 9829, 10585, 11369, 12181, 13021, 13889, 14785, 15709, 16661, 17641, 18649, 19685, 20749, 21841, 22961, 24109, 25285, 26489

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Semi-axis opposite to A144555 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195145.
Cf. A003154, A069129, A069133, A069190, A195315, A195316, A195317, A195318.
Cf. A069127, A118277, A144555.

[(14*n^2-14*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[14n^2-14n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,29,85},50]

a(n)=14*n^2-14*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 14*n + 1.
G.f.: -x*(1 + 26*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 01 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5/7)*Pi/2)/(2*sqrt(35)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(14*x^2 + 1) - 1.
a(n) = 2*A069127(n) - 1. (End)

A195315 Centered 32-gonal numbers.

Original entry on oeis.org

1, 33, 97, 193, 321, 481, 673, 897, 1153, 1441, 1761, 2113, 2497, 2913, 3361, 3841, 4353, 4897, 5473, 6081, 6721, 7393, 8097, 8833, 9601, 10401, 11233, 12097, 12993, 13921, 14881, 15873, 16897, 17953, 19041, 20161, 21313, 22497, 23713, 24961, 26241, 27553, 28897, 30273

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195146.
Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
Cf. A016802, A074377.

[(16*n^2-16*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{3,-3,1},{1,33,97},50] (* Harvey P. Dale, Feb 11 2024 *)

a(n)=16*n^2-16*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*n^2 - 16*n + 1.
G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(16*x^2 + 1) - 1.
a(n) = 2*A069129(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195316 Centered 36-gonal numbers.

Original entry on oeis.org

1, 37, 109, 217, 361, 541, 757, 1009, 1297, 1621, 1981, 2377, 2809, 3277, 3781, 4321, 4897, 5509, 6157, 6841, 7561, 8317, 9109, 9937, 10801, 11701, 12637, 13609, 14617, 15661, 16741, 17857, 19009, 20197, 21421, 22681, 23977, 25309, 26677, 28081, 29521, 30997, 32509

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195321 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195147.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195317, A195318.
Cf. A069131, A195160, A195321.

[(18*n^2-18*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[18*n^2 - 18*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

a(n)=18*n^2-18*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 18*n^2 - 18*n + 1.
G.f.: -x*(1 + 34*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7)*Pi/6)/(6*sqrt(7)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(18*x^2 + 1) - 1.
a(n) = 2*A069131(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195318 Centered 44-gonal numbers.

Original entry on oeis.org

1, 45, 133, 265, 441, 661, 925, 1233, 1585, 1981, 2421, 2905, 3433, 4005, 4621, 5281, 5985, 6733, 7525, 8361, 9241, 10165, 11133, 12145, 13201, 14301, 15445, 16633, 17865, 19141, 20461, 21825, 23233, 24685, 26181, 27721, 29305, 30933, 32605, 34321, 36081, 37885, 39733

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 45, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195323 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195149.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195317.
Cf. A069173, A195313, A195323.

[22*n^2 - 22*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Table[22n^2-22n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,45,133},50] (* Harvey P. Dale, Mar 16 2019 *)

a(n)=22*n^2-22*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 22*n^2 - 22*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(3*Pi/(2*sqrt(11)))/(6*sqrt(11)). - Amiram Eldar, Feb 11 2022
G.f.: -x*(1+42*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(22*x^2 + 1) - 1.
a(n) = 2*A069173(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A069132 Centered 19-gonal numbers.

Original entry on oeis.org

1, 20, 58, 115, 191, 286, 400, 533, 685, 856, 1046, 1255, 1483, 1730, 1996, 2281, 2585, 2908, 3250, 3611, 3991, 4390, 4808, 5245, 5701, 6176, 6670, 7183, 7715, 8266, 8836, 9425, 10033, 10660, 11306, 11971, 12655, 13358, 14080, 14821, 15581, 16360, 17158

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Binomial transform of [1, 19, 19, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 19, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011

			a(5)= 191 because (19*5^2 - 19*5 + 2)/2 = (475 - 95 + 2)/2 = 382/2 = 191.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.

FoldList[#1 + #2 &, 1, 19 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(19n^2-19n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,20,58},50] (* Harvey P. Dale, Aug 21 2011 *)

a(n)=19*binomial(n,2)+1 \\ Charles R Greathouse IV, Jul 29 2011

a(n) = (19*n^2 - 19*n + 2)/2.
a(n) = 19*n + a(n-1) - 19 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1 + 17*x + x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=20, a(2)=58. - Harvey P. Dale, Aug 21 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(11/19)*Pi/2)/sqrt(209).
Sum_{n>=1} a(n)/n! = 21*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 21/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 19*x^2/2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A069174 Centered 23-gonal numbers.

Original entry on oeis.org

1, 24, 70, 139, 231, 346, 484, 645, 829, 1036, 1266, 1519, 1795, 2094, 2416, 2761, 3129, 3520, 3934, 4371, 4831, 5314, 5820, 6349, 6901, 7476, 8074, 8695, 9339, 10006

Offset: 1

Terrel Trotter, Jr., Apr 09 2002

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.

FoldList[#1 + #2 &, 1, 23 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)

a(n)=(23*n^2-23*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (23*n^2 - 23*n + 2)/2.
a(n) = 23*n+a(n-1)-23 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(15/23)*Pi/2)/sqrt(345).
Sum_{n>=1} a(n)/n! = 25*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 25/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 23*x^2/2)-1. - Nikolaos Pantelidis, Feb 06 2023

A069178 Centered 21-gonal numbers.

Original entry on oeis.org

1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 1387, 1639, 1912, 2206, 2521, 2857, 3214, 3592, 3991, 4411, 4852, 5314, 5797, 6301, 6826, 7372, 7939, 8527, 9136, 9766, 10417, 11089, 11782, 12496, 13231, 13987, 14764, 15562, 16381, 17221, 18082, 18964

Offset: 1

Terrel Trotter, Jr., Apr 09 2002

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.

FoldList[#1 + #2 &, 1, 21 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,22,64},60] (* Harvey P. Dale, Jun 13 2022 *)

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

a(n) = (21n^2 - 21n + 2)/2
a(n) = 21*n + a(n-1) - 21 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f. -x*(1+19*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
Binomial transform of [1, 21, 21, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 21, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
a(n) = 1 + Sum_{i=1..n} 21*(i-1). - Wesley Ivan Hurt, May 25 2013
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(13/21)*Pi/2)/sqrt(273).
Sum_{n>=1} a(n)/n! = 23*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 23/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 21*x^2/2)-1. - Nikolaos Pantelidis, Feb 06 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

A193252 Great rhombicuboctahedron with faces of centered polygons.

Original entry on oeis.org

1, 75, 365, 1015, 2169, 3971, 6565, 10095, 14705, 20539, 27741, 36455, 46825, 58995, 73109, 89311, 107745, 128555, 151885, 177879, 206681, 238435, 273285, 311375, 352849, 397851, 446525, 499015, 555465, 616019, 680821, 750015, 823745, 902155, 985389, 1073591

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a great rhombicosidodecahedron. Each iteration requires the addition of (n-2) edge units and (n-1) vertices to complete the centered polygon of each face: centered squares, centered octagons and centered hexagons.

Bruno Berselli, Table of n, a(n) for n = 1..1000
OEIS Wiki, (Centered polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences in 2*A158591.

Cf. A001844 (centered square numbers), A016754 (centered octagonal numbers), A003215 (centered hexagonal numbers).

Excel
```
=24*ROW()^3-36*ROW()^2+14*ROW()-1
```

List([1..40], n-> 24*n^3 -36*n^2 +14*n -1); # G. C. Greubel, Feb 26 2019

A069190:=func; [(2*n-1)*A069190(n): n in [1..40]];  // Bruno Berselli, Jul 21 2011

Table[24n^3-36n^2+14n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,75,365,1015},40] (* Harvey P. Dale, Jul 27 2011 *)

for(n=1,40, print1(24*n^3-36*n^2+14*n-1", "));  \\ Bruno Berselli, Jul 21 2011

[24*n^3 -36*n^2 +14*n -1 for n in (1..40)] # G. C. Greubel, Feb 26 2019

a(n) = 24*n^3 - 36*n^2 + 14*n - 1.
G.f.: x*(1+x)*(1+70*x+x^2)/(1-x)^4; a(n) = (2*n-1)*A069190(n). - Bruno Berselli, Jul 21 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=75, a(2)=365, a(3)=1015. - Harvey P. Dale, Jul 27 2011
a(n) = 72 * A000330(n-1) + A005408(n-1). - Bruce J. Nicholson, Feb 23 2019
E.g.f.: 1 + (-1 + 2*x + 36*x^2 + 24*x^3)*exp(x). - G. C. Greubel, Feb 26 2019

cp's OEIS Frontend

A195314 Centered 28-gonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A195315 Centered 32-gonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A195316 Centered 36-gonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A195318 Centered 44-gonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A069132 Centered 19-gonal numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A069174 Centered 23-gonal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A069178 Centered 21-gonal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI