A195143 -id:A195143 - 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

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

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

cp's OEIS Frontend

A195043 Concentric 11-gonal numbers.

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A195045 Concentric 13-gonal numbers.

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A195158 Concentric 24-gonal numbers.

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A270693 Alternating sum of centered 25-gonal numbers.

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Magma

Maple

Mathematica

PARI

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