A188896 -id:A188896 - cp's OEIS frontend

A001107 10-gonal (or decagonal) numbers: a(n) = n(4n-3).

0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326

Offset: 0

N. J. A. Sloane

Write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers on the negative y-axis (see Example section).
Number of divisors of 48^(n-1) for n > 0. - J. Lowell, Aug 30 2008
a(n) is the Wiener index of the graph obtained by connecting two copies of the complete graph K_n by an edge (for n = 3, approximately: |>-<|). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. - Emeric Deutsch, Sep 20 2010
This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011
For n > 0: right edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012
Sequence found by reading the line from 0, in the direction 0, 10, ... and the parallel line from 1, in the direction 1, 27, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Jul 18 2012
Partial sums give A007585. - Omar E. Pol, Jan 15 2013
This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - Luciano Ancora, Mar 28 2015
Also the number of undirected paths in the n-sunlet graph. - Eric W. Weisstein, Sep 07 2017
After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018
Number of corona of an H0 hexagon with a T(n) triangle. - Craig Knecht, Dec 13 2024

			On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along the negative y-axis, as seen in the example below:
  99  64--65--66--67--68--69--70--71--72
   |   |                               |
  98  63  36--37--38--39--40--41--42  73
   |   |   |                       |   |
  97  62  35  16--17--18--19--20  43  74
   |   |   |   |               |   |   |
  96  61  34  15   4---5---6  21  44  75
   |   |   |   |   |       |   |   |   |
  95  60  33  14   3  *0*  7  22  45  76
   |   |   |   |   |   |   |   |   |   |
  94  59  32  13   2--*1*  8  23  46  77
   |   |   |   |           |   |   |   |
  93  58  31  12--11-*10*--9  24  47  78
   |   |   |                   |   |   |
  92  57  30--29--28-*27*-26--25  48  79
   |   |                           |   |
  91  56--55--54--53-*52*-51--50--49  80
   |                                   |
  90--89--88--87--86-*85*-84--83--82--81
[Edited by _Jon E. Schoenfield_, Jan 02 2017]

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.3 p. 33.
Emilio Apricena, A version of the Ulam spiral.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344.
Craig Knecht, Corona of the H0 hexagon with a T(n) triangle.
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Leo Tavares, Illustration: Conjoined Hexagon/Square Pairs
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Decagonal Number.
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007585, A028994.
Cf. A093565 ((8, 1) Pascal, column m = 2). Partial sums of A017077.
Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215.

[4*n^2-3*n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 05 2014

A001107:=-(1+7*z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* Harvey P. Dale, May 08 2012 *)
Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
Table[4 n^2 - 3 n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)
PolygonalNumber[10, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

PARI
```
a(n)=4*n^2-3*n
```

a=lambda n: 4*n**2-3*n # Indranil Ghosh, Jan 01 2017
def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 8, y + 8
A001107 = aList()
print([next(A001107) for i in range(49)]) # Peter Luschny, Aug 04 2019

a(n) = A033954(-n) = A074377(2*n-1).
a(n) = n + 8*A000217(n-1). - Floor van Lamoen, Oct 14 2005
G.f.: x*(1 + 7*x)/(1 - x)^3.
Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - Jon Perry, Dec 18 2004
1^3 + 3^3*(n-1)/(n+1) + 5^3*((n-1)*(n-2))/((n+1)*(n+2)) + 7^3*((n-1)*(n-2)*(n-3))/((n+1)*(n+2)*(n+3)) + ... = n*(4*n-3) [Ramanujan]. - Neven Juric, Apr 15 2008
Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Jul 10 2010
a(n) = 8 + 2*a(n-1) - a(n-2). - Ant King, Sep 04 2011
a(n) = A118729(8*n). - Philippe Deléham, Mar 26 2013
a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Aug 28 2016: (Start)
E.g.f.: x*(1 + 4*x)*exp(x).
Sum_{n >= 1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 2*log(2) + 2*sqrt(2)*log(1 + sqrt(2)))/6 = 0.92491492293323294695... (End)
a(n) = A000217(3*n-2) - A000217(n-2). In general, if P(k,n) be the n-th k-gonal number and T(n) be the n-th triangular number, A000217(n), then P(T(k),n) = T((k-1)*n - (k-2)) - T(k-3)*T(n-2). - Charlie Marion, Sep 01 2020
Product_{n>=2} (1 - 1/a(n)) = 4/5. - Amiram Eldar, Jan 21 2021
a(n) = A003215(n-1) + A000290(n) - 1. - Leo Tavares, Jul 23 2022

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

Original entry on oeis.org

1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234

Offset: 0

N. J. A. Sloane

Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - Emeric Deutsch, Dec 25 2004
For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk, May 20 2006; updated by Peter Munn, Aug 25 2017 due to changed offset in A000292
Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008
Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). - Rick L. Shepherd, Sep 30 2009
Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - Frank M Jackson, Sep 07 2011
Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014
Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016
For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - Muniru A Asiru, Apr 19 2016
Union of A188896, A277449, {1,4}. - Muniru A Asiru, Nov 25 2016
Interleaving of A008527 and A108099. - Bruce J. Nicholson, Oct 14 2019

			G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
J. M. Grau, C. Miguel, and A. M. Oller-Marcén, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy.]
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8.
M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Reticular Chemistry Structure Resource, sod.
Aditya Sivakumar and Dmitri Tymoczko, Intuitive Musical Homotopy, 2018.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558. DOI: 10.1021/ic00220a025.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A053545, A206399.
Cf. similar sequences listed in A255843.
The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.
For partial sums see A005894.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

[2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A005893:=-(z+1)*(1+z^2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation

Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{4,10,20},50]] (* Harvey P. Dale, Feb 26 2012 *)
a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *)
a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)

a(n)=2*n^2-0^n+2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1 - x^4)/(1-x)^4.
a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - Ralf Stephan, Apr 26 2003
a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008
a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - Rick L. Shepherd, Sep 30 2009
a(n) = 2*n^2 - 0^n + 2. - Vincenzo Librandi, Sep 27 2011
a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 26 2012
a(n) = A228643(n+1,2) for n > 0. - Reinhard Zumkeller, Aug 29 2013
a(n) = a(-n) for all n in Z. - Michael Somos, May 14 2014
For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016
E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - Ilya Gutkovskiy, Apr 19 2016
a(n) = 2*A002522(n), n>0. - R. J. Mathar, May 30 2022
From Amiram Eldar, Sep 16 2022: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)
Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - Carlo Wood, Dec 04 2023
Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - Charlie Marion, May 15 2024

A051872 20-gonal (or icosagonal) numbers: a(n) = n(9n-8).

Original entry on oeis.org

0, 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, 1417, 1652, 1905, 2176, 2465, 2772, 3097, 3440, 3801, 4180, 4577, 4992, 5425, 5876, 6345, 6832, 7337, 7860, 8401, 8960, 9537, 10132, 10745, 11376, 12025, 12692, 13377, 14080

Offset: 0

N. J. A. Sloane, Dec 15 1999

This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011
Sequence found by reading the line from 0, in the direction 0, 20,... and the parallel line from 1, in the direction 1, 57,..., in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 18 2012
This is also a star decagonal number: a(n) = A001107(n) + 10*A000217(n-1). - Luciano Ancora, Mar 30 2015

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A051872 := proc(n) n*(9*n-8) ;end proc: seq(A051872(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Table[9n^2 - 8n, {n, 0, 59}] (* Alonso del Arte, Dec 20 2014 *)
PolygonalNumber[20,Range[0,50]] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, May 14 2017 *)

n*(9*n-8) \\ Charles R Greathouse IV, Jan 24 2014

a(n) = 18*n + a(n-1) - 17, with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010
G.f.: x*(1+17*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(18*a(n) + 154*n + 1) = a(18*a(n) + 154*n) + a(18*n + 1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 9/10. - Amiram Eldar, Jan 22 2021
For n>0, a(n) = A002378(3*n-2) + n - 2. - Charlie Marion, Jul 18 2022
E.g.f.: exp(x)*(x + 9*x^2). - Nikolaos Pantelidis, Feb 05 2023

A100252 Least square n-gonal number greater than 1, or 0 if none exists.

Original entry on oeis.org

36, 4, 9801, 1225, 81, 225, 9, 0, 196, 64, 36, 441, 3025, 16, 17689, 100, 484, 0, 2601, 729, 68121, 225, 25, 7225, 25921, 81, 1225, 203401, 441, 1089, 4761, 196, 15376, 36, 1936, 511225, 784, 576, 55071241, 47089, 1156, 256, 529046001, 2916, 1134225

Offset: 3

Charlie Marion, Nov 21 2004

nonn

Also, let j be the smallest integer for which 1+(1+1*n)+(1+2*n)+... +(1+j*n)=k^2=s. Then a(n)=s; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

See A100251 and A188898 for the corresponding indices of these terms. Note that a(n) is zero for n = 10, 20, 52 (numbers in A188896). Although the Mathematica program searches only the first 25000 square numbers for n-gonal numbers, the Reduce function can show that there are no square n-gonal numbers (other than 0 and 1) for these n. - T. D. Noe, Apr 19 2011

			a(3)=9801 since 1 + 4 + 7 +...+ (1+80*3)= 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 3..100
Eric W. Weisstein, MathWorld: Polygonal Number
Eric W. Weisstein, MathWorld: Square Number

Cf. A000290 (squares), A188891 (similar sequence for triangular numbers).

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; sqr, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)

1+(1+1*n)+(1+2*n)+...+(1+A100254(n)*n) = 1+(1+1*n)+(1+2*n)+...+A100253(n) = A100251(n)^2 = a(n).

A188892 Numbers n such that there is no triangular n-gonal number greater than 1.

Original entry on oeis.org

11, 18, 38, 102, 198, 326, 486, 678, 902, 1158, 1446, 1766, 2118, 2918, 3366, 3846, 4358, 4902, 5478, 6086, 6726, 7398, 8102, 8838, 9606, 10406, 11238, 12102, 12998, 13926, 14886, 15878, 16902, 17958, 19046, 20166, 21318, 22502, 24966, 26246

Offset: 1

T. D. Noe, Apr 13 2011

nonn

It is easy to find triangular numbers that are square, pentagonal, hexagonal, etc.  So it is somewhat surprising that there are no triangular 11-gonal numbers other than 0 and 1. For these n, the equation x^2 + x = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.
Chu shows how to transform the equation into a generalized Pell equation. When n has the form k^2+2 (A059100), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.
The general case is in A188950.

Robert Israel, Table of n, a(n) for n = 1..10000
Wenchang Chu, Regular polygonal numbers and generalized pell equations, Int. Math. Forum 2 (2007), 781-802.

Cf. A051682 (11-gonal numbers), A051870 (18-gonal numbers), A188891, A188896.

filter:= n -> nops(select(t -> min(subs(t,[x,y]))>=2, [isolve(x^2 + x = (n-2)*y^2 - (n-4)*y)])) = 0:
select(filter, [seq(t^2+2,t=3..200)]); # Robert Israel, May 13 2018

A342709 12-gonal (dodecagonal) square numbers.

Original entry on oeis.org

1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161

Offset: 1

Bernard Schott, Mar 19 2021

The 12-gonal square numbers k correspond to the nonnegative integer solutions of the Diophantine equation k = d*(5*d-4) = c^2, equivalent to (5*d-2)^2 - 5*c^2 = 4. Substituting x = 5*d-2 and y = c gives the Pell-Fermat's equation x^2 - 5*y^2 = 4.
The solutions x are in A342710, while corresponding solutions y that are also the indices c of the squares which are 12-gonal are in A033890.
The indices d of the corresponding 12-gonal which are squares are in A081068.

			142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.

Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Intersection of A000290 (squares) and A051624 (12-gonal numbers).

Similar for n-gonal squares: A001110 (triangular), A036353 (pentagonal), A046177 (hexagonal), A036354 (heptagonal), A036428 (octagonal), A036411 (9-gonal), A188896 (there are no 10-gonal squares > 1), A333641 (11-gonal), this sequence (12-gonal).

with(combinat):
seq(fibonacci(4*n-2)^2, n=1..16);

Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* Amiram Eldar, Mar 19 2021 *)

a(n) = fibonacci(4*n-2)^2; \\ Michel Marcus, Mar 21 2021

G.f.: x*(1 + 16*x + x^2)/((1 - x)*(1 - 47*x + x^2)). - Stefano Spezia, Mar 20 2021
a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3). - Kevin Ryde, Mar 20 2021
a(n) = 9*A161582(n) + 1. - Hugo Pfoertner, Mar 19 2021
a(n) = A033890(n-1)^2.

A188950 Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n.

Original entry on oeis.org

3, 11, 4, 10, 6, 11, 5, 14, 3, 18, 4, 20, 6, 18, 7, 22, 11, 18, 10, 20, 6, 27, 5, 29, 8, 26, 11, 27, 9, 30, 3, 38, 14, 29, 6, 38, 10, 34, 18, 27, 11, 38, 7, 47, 12, 42, 20, 34, 5, 50, 4, 52, 18, 38, 6, 51, 13, 46, 11, 51, 8, 56, 14, 50, 27, 38, 15, 54, 22, 47

Offset: 1

T. D. Noe, Apr 20 2011

These are n and k such that the generalized Pell equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y has no solution in integers x>1 and y>1. The paper by Chu shows how to solve these equations. A necessary condition for a pair to be in this sequence is (n-2)(k-2) is a square. These (n,k) pairs indicate where the zeros are in triangle A189216, which gives the least n-gonal k-gonal number greater than 1. For triangular (n=3) and square (n=4) numbers, see A188892 and A188896 for lists of k.

			The pairs begin (3,11), (4,10), (6,11), (5,14), (3,18), (4,20), (6,18).

Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.
Eric W. Weisstein, MathWorld: Polygonal Number

Cf. A188892, A188896.

maxSum=100; Reap[Do[k=s-n; If[k>n && IntegerQ[Sqrt[(n-2)*(k-2)]] && FindInstance[(k-2)*x^2 - (k-4)*x == (n-2)*y^2 - (n-4)*y && x>1 && y>1, {x,y}, Integers] == {}, Sow[{n,k}]], {s,7,maxSum}, {n,3,s-3}]][[2,1]]

A188898 The index of the least n-gonal number greater than 1 that is also square, or 0 if none exists.

Original entry on oeis.org

8, 2, 81, 25, 6, 9, 2, 0, 7, 4, 3, 9, 22, 2, 49, 4, 8, 0, 17, 9, 81, 5, 2, 25, 46, 3, 10, 121, 6, 9, 18, 4, 31, 2, 11, 169, 7, 6, 1681, 49, 8, 4, 4961, 12, 225, 8, 2, 16, 32, 0, 49, 25, 17, 13, 289, 9, 49, 7, 27, 81, 118, 2, 136161, 196, 14, 4, 10, 841, 1521

Offset: 3

T. D. Noe, Apr 13 2011

nonn

Sequence A188896 gives the n that have no solution.

Cf. A100252 (least square n-gonal number greater than 1).

kMax = 25000; NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < kMax && ! IntegerQ[i], k++]; If[k == kMax, k = sqr = i = 0]; i, {n, 3, 64}]

If n is a square, then a(n)=2.

A277449 Numbers n such that there is exactly one nontrivial square n-gonal number.

Original entry on oeis.org

34, 74, 100, 130, 202, 244, 290, 394, 452, 514, 650, 724, 802, 970, 1060, 1154, 1354, 1460, 1570, 1802, 1924, 2050, 2314, 2452, 2594, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4234, 4420, 4610, 5002, 5204, 5410, 5834, 6052, 6274, 6730, 6964, 7202, 7690, 7940, 8194, 8714, 8980, 9250, 9802, 10084, 10370, 10954, 11252, 11554, 12170, 12484, 12802, 13450, 13780

Offset: 1

Muniru A Asiru, Oct 16 2016

nonn

There are infinitely many squares that are triangular, pentagonal, hexagonal, etc. Also there is no square 10-gonal number, 20-gonal number, 52-gonal number, 64-gonal number, etc. greater than 1 (see A188896). Other than the trivial square n-gonal numbers 0 and 1, there is exactly one square 34-gonal number, one square 74-gonal number, one square 100-gonal number, one square 130-gonal number, etc.

			For n = 34, the square 34-gonal numbers are 0, 1, 196.
For n = 74, the square 74-gonal numbers are 0, 1, 2601.
For n = 100, the square 100-gonal numbers are 0, 1, 100.

Muniru A Asiru, Table of n, a(n) for n = 1..238

Cf. A005893, A188896, A271624.

G:=[];; for g in [5..100000]  do for r in [1..5000] do if 2*g-4=r^2 then Add(G,g); fi; od; od; G; Length(G);
F:=List(G,g->[g,DivisorsInt((g-4)^2)]);;
N:=List([1..Length(F)], i->List([1..Length(F[i][2])],j->[F[i][1],((F[i][1]-4)*(F[i][1]-4+2*F[i][2][j])+F[i][2][j]^2)/((4*F[i][1]-8)*F[i][2][j])] ) );;
N1:=Filtered(List(List([1..Length(N)],k->Filtered(N[k], l->IsPosInt(l[2]))),Set),o->Length(o)>=2);
N2:=Set(Flat(List([1..Length(N1)],i->List([1..Length(N1[i])],j->N1[i][j][1]))));

A189804 Triangle read by rows: T(n,k) is the number of compositions of set {1, 2, ..., k} into exactly n blocks, each of size 1, 2 or 3 (n >= 0, 0 <= k <= 3*n).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 2, 6, 14, 20, 20, 0, 0, 0, 6, 36, 150, 450, 1050, 1680, 1680, 0, 0, 0, 0, 24, 240, 1560, 7560, 29400, 90720, 218400, 369600, 369600, 0, 0, 0, 0, 0, 120, 1800, 16800, 117600, 667800, 3137400, 12243000, 38808000, 96096000, 168168000, 168168000

Offset: 0

Adi Dani, Apr 27 2011

Row n has 3*n+1 entries.

Column sums give A188896, row sums give A144422. - Adi Dani, May 14 2011

			Triangle begins:
[1]
[0, 1, 1, 1]
[0, 0, 2, 6, 14, 20, 20]
[0, 0, 0, 6, 36, 150, 450, 1050, 1680, 1680]
[0, 0, 0, 0, 24, 240, 1560, 7560, 29400, 90720, 218400, 369600, 369600]
[0, 0, 0, 0, 0, 120, 1800, 16800, 117600, 667880, 3137400, 12243000, 3880800, 96096000, 168168000, 168168000]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

T := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else =k *T(n - 1, k - 1) +  (1/2)*k*(k - 1)*T(n - 1, k - 2)+ (1/6)*k* (k - 1)*(k - 2)*T(n - 1, k - 3);
end if;
end proc; for n from 0 to 12 do lprint([seq(T(n, k), k=0..3*n)]); od:

Table[Sum[ n!/(2^(n + j - 2m)3^(m - j))Binomial[m, j]Binomial[j, n + 2j - 3m], {j, 0, 3m - n}], {m, 0, 5}, {n, 0, 3m}]//Flatten

for(m=0,7, for(n=0,3*m, print1(sum(j=0,3*m-n, (n!/(2^(n+j-2*m)*3^(m-j)))*binomial(m, j)*binomial(j, n+2*j-3*m)), ", "))) \\ G. C. Greubel, Jan 16 2018

T(n, k) = k*T(n-1, k-1) + (1/2)*k*(k-1)*T(n-1, k-2) + (1/6)*k*(k-1)*(k-2)*T(n-1, k-3).

E.g.f.: sum(n>=0, T(n, k)*x^k/k!) = (x+x^2/2+x^3/6)^k.

Terms a(44) and a(47) corrected by G. C. Greubel, Jan 16 2018

cp's OEIS Frontend

A001107 10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).

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A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

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A051872 20-gonal (or icosagonal) numbers: a(n) = n*(9*n-8).

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A100252 Least square n-gonal number greater than 1, or 0 if none exists.

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A188892 Numbers n such that there is no triangular n-gonal number greater than 1.

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A342709 12-gonal (dodecagonal) square numbers.

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A188950 Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n.

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A001107 10-gonal (or decagonal) numbers: a(n) = n(4n-3).

A051872 20-gonal (or icosagonal) numbers: a(n) = n(9n-8).