A195043 -id:A195043 - cp's OEIS frontend

A195040 Square array read by antidiagonals with T(n,k) = kn^2/4+(k-4)((-1)^n-1)/8, n>=0, k>=0.

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 3, 2, 1, 0, 1, 4, 5, 3, 1, 0, 0, 7, 8, 7, 4, 1, 0, 1, 9, 13, 12, 9, 5, 1, 0, 0, 13, 18, 19, 16, 11, 6, 1, 0, 1, 16, 25, 27, 25, 20, 13, 7, 1, 0, 0, 21, 32, 37, 36, 31, 24, 15, 8, 1, 0, 1, 25, 41, 48, 49, 45, 37, 28, 17, 9, 1, 0

Offset: 0

Omar E. Pol, Sep 27 2011

Also, if k >= 2 and m = 2*k, then column k lists the numbers of the form k*n^2 and the centered m-gonal numbers interleaved.
For k >= 3, this is also a table of concentric polygonal numbers. Column k lists the concentric k-gonal numbers.
It appears that the first differences of column k are the numbers that are congruent to {1, k-1} mod k, if k >= 3.

			Array begins:
  0,   0,   0,   0,   0,   0,   0,   0,   0,   0, ...
  1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5,   6,   7,   8,   9, ...
  1,   3,   5,   7,   9,  11,  13,  15,  17,  19, ...
  0,   4,   8,  12,  16,  20,  24,  28,  32,  36, ...
  1,   7,  13,  19,  25,  31,  37,  43,  49,  55, ...
  0,   9,  18,  27,  36,  45,  54,  63,  72,  81, ...
  1,  13,  25,  37,  49,  61,  73,  85,  97, 109, ...
  0,  16,  32,  48,  64,  80,  96, 112, 128, 144, ...
  1,  21,  41,  61,  81, 101, 121, 141, 161, 181, ...
  0,  25,  50,  75, 100, 125, 150, 175, 200, 225, ...
  ...

Muniru A Asiru, Rows n=0..100, flattened

Rows n: A000004 (n=0), A000012 (n=1), A001477 (n=2), A005408 (n=3), A008586 (n=4), A016921 (n=5), A008591 (n=6), A017533 (n=7), A008598 (n=8), A215145 (n=9), A008607 (n=10).

Columns k: A000035 (k=0), A004652 (k=1), A000982 (k=2), A077043 (k=3), A000290 (k=4), A032527 (k=5), A032528 (k=6), A195041 (k=7), A077221 (k=8), A195042 (k=9), A195142 (k=10), A195043 (k=11), A195143 (k=12), A195045 (k=13), A195145 (k=14), A195046 (k=15), A195146 (k=16), A195047 (k=17), A195147 (k=18), A195048 (k=19), A195148 (k=20), A195049 (k=21), A195149 (k=22), A195058 (k=23), A195158 (k=24).

nmax:=13;; T:=List([0..nmax],n->List([0..nmax],k->k*n^2/4+(k-4)*((-1)^n-1)/8));; b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 19 2018

A195040 := proc(n,k)
        k*n^2/4+((-1)^n-1)*(k-4)/8 ;
end proc:
for d from 0 to 12 do
        for k from 0 to d do
                printf("%d,",A195040(d-k,k)) ;
        end do:
end do; # R. J. Mathar, Sep 28 2011

t[n_, k_] := k*n^2/4+(k-4)*((-1)^n-1)/8; Flatten[ Table[ t[n-k, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011 *)

A175885 Numbers that are congruent to {1, 10} mod 11.

Original entry on oeis.org

1, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 100, 109, 111, 120, 122, 131, 133, 142, 144, 153, 155, 164, 166, 175, 177, 186, 188, 197, 199, 208, 210, 219, 221, 230, 232, 241, 243, 252, 254, 263, 265, 274, 276, 285, 287, 296, 298

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 11).

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

Cf. A090771 (n==1 or 9 mod 10), A091998 (n==1 or 11 mod 12).
Cf. A195043 (partial sums).
Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A113801, A175886, A175887, A195312, A195313.

a175885 n = a175885_list !! (n-1)
a175885_list = 1 : 10 : map (+ 11) a175885_list
-- Reinhard Zumkeller, Jan 07 2012

[(22*n+7*(-1)^n-11)/4: n in [1..60]]; // Vincenzo Librandi, Sep 19 2011

Rest[Flatten[{#-1,#+1}&/@(11 Range[0,50])]] (* Harvey P. Dale, Nov 05 2010 *)

a(n)=n%2*9 + 1 \\ Charles R Greathouse IV, Aug 01 2016

G.f.: x*(1+9*x+x^2)/((1+x)*(1-x)^2).
a(n) = (22*n + 7*(-1)^n - 11)/4.
a(n) = -a(-n+1) = a(n-2) + 11 = a(n-1) + a(n-2) - a(n-3).
a(n) = 11*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
a(n) = A195312(n) + A195312(n-1) = A195313(n) - A195313(n-2). - Bruno Berselli, Sep 18 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/11)*cot(Pi/11). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((22*x - 11)*exp(x) + 7*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/11).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/11)*cosec(Pi/11). (End)

A195143 a(n) = n-th concentric 12-gonal number.

Original entry on oeis.org

0, 1, 12, 25, 48, 73, 108, 145, 192, 241, 300, 361, 432, 505, 588, 673, 768, 865, 972, 1081, 1200, 1321, 1452, 1585, 1728, 1873, 2028, 2185, 2352, 2521, 2700, 2881, 3072, 3265, 3468, 3673, 3888, 4105, 4332, 4561, 4800, 5041, 5292, 5545, 5808, 6073, 6348

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric dodecagonal numbers. [corrected by Ivan Panchenko, Nov 09 2013]
Sequence found by reading the line from 0, in the direction 0, 12,..., and the same line from 1, in the direction 1, 25,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Main axis, perpendicular to A028896 in the same spiral.
Partial sums of A091998. - Reinhard Zumkeller, Jan 07 2012
Column 12 of A195040. - Omar E. Pol, Sep 28 2011

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

A135453 and A069190 interleaved.
Cf. A001082, A032527, A032528, A077221, A195043, A195045, A195040, A195142, A195145, A195146, A195147, A195148, A195149.
Cf. A016921 (6n+1), A016969 (6n+5), A091998 (positive integers of the form 12*k +- 1), A092242 (positive integers of the form 12*k +- 5).

a195143 n = a195143_list !! n
a195143_list = scanl (+) 0 a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[(3*n^2+(-1)^n-1): n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Table[Sum[2*(-1)^(n - k + 1) + 6*k - 3, {k, n}], {n, 0, 47}] (* L. Edson Jeffery, Sep 14 2014 *)

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 3*n^2+(-1)^n-1.
a(n) = -a(n-1) + 6*n^2 - 6*n + 1. (End)
G.f.: -x*(1+10*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = Sum_{k=1..n} (2*(-1)^(n-k+1) + 3*(2*k-1)), n>0, a(0) = 0. - L. Edson Jeffery, Sep 14 2014
Sum_{n>=1} 1/a(n) = Pi^2/72 + tan(Pi/sqrt(6))*Pi/(4*sqrt(6)). - Amiram Eldar, Jan 16 2023

A195042 Concentric 9-gonal numbers.

Original entry on oeis.org

0, 1, 9, 19, 36, 55, 81, 109, 144, 181, 225, 271, 324, 379, 441, 505, 576, 649, 729, 811, 900, 991, 1089, 1189, 1296, 1405, 1521, 1639, 1764, 1891, 2025, 2161, 2304, 2449, 2601, 2755, 2916, 3079, 3249, 3421, 3600, 3781, 3969, 4159, 4356, 4555, 4761, 4969, 5184, 5401, 5625

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric enneagonal numbers or concentric nonagonal numbers.
A016766 and A069131 interleaved.
Partial sums of A056020. - 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 9 of A195040.

Cf. A016766, A069131, A077221, A195041, A195142, A195043.

a195042 n = a195042_list !! n
a195042_list = scanl (+) 0 a056020_list
-- Reinhard Zumkeller, Jan 07 2012

[(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* Harvey P. Dale, Nov 24 2019 *)

a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A060544(n+1). (End)
Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - Amiram Eldar, Jan 16 2023

A033584 a(n) = 11*n^2.

Original entry on oeis.org

0, 11, 44, 99, 176, 275, 396, 539, 704, 891, 1100, 1331, 1584, 1859, 2156, 2475, 2816, 3179, 3564, 3971, 4400, 4851, 5324, 5819, 6336, 6875, 7436, 8019, 8624, 9251, 9900, 10571, 11264, 11979, 12716

Offset: 0

N. J. A. Sloane

From Roberto E. Martinez II, Jan 07 2002: (Start)
Number of edges of the complete tripartite graph of order 7n, K_n,n,5n.
Number of edges of the complete tripartite graph of order 6n, K_n,2n,3n. (End)
11 times the squares. - Omar E. Pol, Dec 13 2008

			a(1)=22*1+0-11=11; a(2)=22*2+11-11=44; a(3)=22*3+44-11=99 - _Vincenzo Librandi_, Aug 05 2010

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008593, A195043.

Table[11*n^2, {n, 0, 35}] (* Amiram Eldar, Feb 03 2021 *)

a(n)=11*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 11*A000290(n). - Omar E. Pol, Dec 13 2008
a(n) = 22*n + a(n-1) - 11 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/66.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/132.
Product_{n>=1} (1 + 1/a(n)) = sqrt(11)*sinh(Pi/sqrt(11))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(11)*sin(Pi/sqrt(11))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 11*x*(1 + x)/(1-x)^3.
E.g.f.: 11*x*(1 + x)*exp(x).
a(n) = n*A008593(n) = A195043(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

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

A195040 Square array read by antidiagonals with T(n,k) = k*n^2/4+(k-4)*((-1)^n-1)/8, n>=0, k>=0.

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A175885 Numbers that are congruent to {1, 10} mod 11.

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A195143 a(n) = n-th concentric 12-gonal number.

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A195042 Concentric 9-gonal numbers.

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A033584 a(n) = 11*n^2.

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A195045 Concentric 13-gonal numbers.

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A270693 Alternating sum of centered 25-gonal numbers.

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A195040 Square array read by antidiagonals with T(n,k) = kn^2/4+(k-4)((-1)^n-1)/8, n>=0, k>=0.