A001542 -id:A001542 - cp's OEIS frontend

A098890 a(n) = A001652(n)*A046090(n)+1 = A098602(n)+1.

1, 13, 421, 14281, 485113, 16479541, 559819261, 19017375313, 646030941361, 21946034630941, 745519146510613, 25325704946729881, 860328449042305321, 29225841562491651013, 992818284675673829101, 33726595837410418538401, 1145711440187278556476513

Offset: 0

Charlie Marion, Nov 03 2004

			a(3) = 14281 = 99^2+70^2-420.

Colin Barker, Table of n, a(n) for n = 0..653
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001541, A001542, A001652, A046090, A098602.

Vec(-(x^2-22*x+1)/((x-1)*(x^2-34*x+1)) + O(x^30)) \\ Colin Barker, Jul 09 2015

For n>0, a(n) = A001541(n)^2 + A001542(n)^2 - A098602(n-1).
G.f.: -(x^2-22*x+1) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Jul 09 2015
a(n) = (5/8+1/16*(17+12*sqrt(2))^(-n)*(3-2*sqrt(2)+(3+2*sqrt(2))*(17+12*sqrt(2))^(2*n))). - Colin Barker, Mar 02 2016

Name and a(0) corrected by Charlie Marion, Jul 09 2015

A123057 Values x of solutions (x, y) to the Diophantine equation (x-y)^4 - 8xy = 0 with x >= y.

Original entry on oeis.org

0, 8, 216, 7000, 235824, 7999592, 271683720, 9228858808, 313507253856, 10650004589000, 361786571934264, 12290092993331992, 417501372591127440, 14182756559891488808, 481796221575048645096, 16366888776474950875000, 555992422175561082535104

Offset: 0

Mohamed Bouhamida, Sep 26 2006

Corresponding y-values (A123116) are y(n) = c(n)*(-1 + d(n)), with c(n) and d(n) defined in formula section.

The pair (x,y) = (A001542(n), a(n)) satisfies the equation 2*x^4 + 2*x*y - y^2 = 0. - Alexander Samokrutov, Sep 04 2015

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Cf. A000129, A001542, A002203, A123056, A123116.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)) )); // G. C. Greubel, Jul 20 2021

CoefficientList[Series[8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)), {x,0,30}], x] (* Vincenzo Librandi, Sep 04 2015 *)
Table[(Fibonacci[4*n, 2] + 2*Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jul 20 2021 *)

concat(0, Vec(8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)) + O(x^20))) \\ Michel Marcus, Sep 05 2015

[(1/2)*(lucas_number1(4*n,2,-1) + 2*lucas_number1(2*n,2,-1)) for n in (0..30)] # G. C. Greubel, Jul 20 2021

a(n) = c(n)*(1 + d(n)) with: c(0) = 0, c(1) = 2 and c(n) = 6*c(n-1) - c(n-2), d(0) = 1, d(1) = 3 and d(n) = 6*d(n-1) - d(n-2).
For n>=4, a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4). - Max Alekseyev, Nov 13 2009
G.f.: 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)). - Colin Barker, Oct 24 2012
a(n) = A123116(n) + 2*A001542(n). - Alexander Samokrutov, Sep 05 2015
a(n) = (1/2)*(A000129(4*n) + 2*A000129(2*n)) = (1/2)*A000129(2*n)*(A002203(2*n) + 2) = 2*A123056(n).  - G. C. Greubel, Jul 20 2021

More terms from Max Alekseyev, Nov 13 2009
a(16) from Vincenzo Librandi, Sep 04 2015
Edited by Michel Marcus, Sep 05 2015

A123116 Values y of solutions (x, y) to the Diophantine equation (x-y)^4 - 8xy = 0 with x >= y.

Original entry on oeis.org

0, 4, 192, 6860, 235008, 7994836, 271656000, 9228697244, 313506312192, 10649999100580, 361786539945408, 12290092806887276, 417501371504448000, 14182756553557856884, 481796221538133532992, 16366888776259793834300, 555992422174307055403008

Offset: 0

Mohamed Bouhamida, Sep 28 2006

Corresponding x-values (A123057) are x(n) = c(n)*(1 + d(n)) with c(n) and d(n) defined in formula section.

The pair (x,y) = (A001542(n), a(n)) satisfies the equation 2*x^4 - 2*x*y - y^2 = 0. - Alexander Samokrutov, Sep 04 2015

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Cf. A000129, A002203, A123057, A123115.

I:=[0,4,192,6860]; [n le 4 select I[n] else 40*Self(n-1) -206*Self(n-2) +40*Self(n-3) -Self(n-4): n in [1..20]]; // Vincenzo Librandi, Sep 22 2015

LinearRecurrence[{40, -206, 40, -1}, {0, 4, 192, 6860}, 40] (* Vincenzo Librandi, Sep 22 2015 *)
Table[(Fibonacci[4*n, 2] - 2*Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jul 21 2021 *)

concat(0, Vec(4*x*(1+8*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)) + O(x^20))) \\ Michel Marcus, Sep 05 2015

[(1/2)*(lucas_number1(4*n, 2, -1) - 2*lucas_number1(2*n, 2, -1)) for n in (0..30)] # G. C. Greubel, Jul 21 2021

a(n) = c(n)*(d(n) - 1) with c(0)=0, c(1)=2 and c(n) = 6*c(n-1) - c(n-2) d(0)=1, d(1)=3 and d(n) = 6*d(n-1) - d(n-2).
For n>=4, a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4). - Max Alekseyev, Nov 13 2009
G.f.: 4*x*(1 +8*x +x^2)/((1 -34*x +x^2)*(1 -6*x +x^2)). - Colin Barker, Oct 25 2012
a(n) = A123057(n) - 2*A001542(n). - Alexander Samokrutov, Sep 05 2015
a(n) = (1/2)*(A000129(4*n) - 2*A000129(2*n)) = (1/2)*A000129(2*n)*(A002203(2*n) - 2) = 2*A123115(n). - G. C. Greubel, Jul 21 2021

More terms from Max Alekseyev, Nov 13 2009

Edited by Michel Marcus, Sep 05 2015

A129346 a(2n) = A100525(n), a(2n+1) = A001653(n+1); a Pellian-related sequence.

Original entry on oeis.org

4, 5, 22, 29, 128, 169, 746, 985, 4348, 5741, 25342, 33461, 147704, 195025, 860882, 1136689, 5017588, 6625109, 29244646, 38613965, 170450288, 225058681, 993457082, 1311738121, 5790292204, 7645370045, 33748296142, 44560482149, 196699484648, 259717522849

Offset: 0

Creighton Dement, Apr 10 2007

Summation of -a(n) and A129345 returns twice Pell numbers A000129 (apart from signs; starting from 2nd term of A000129).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A129345, A000129, A001542, A038761.

LinearRecurrence[{0,6,0,-1},{4,5,22,29},30] (* Harvey P. Dale, Apr 08 2018 *)

Vec((4+5*x-2*x^2-x^3)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^40)) \\ Colin Barker, May 26 2016

O.g.f.: (4 + 5*x - 2*x^2 - x^3) / ((x^2 - 2*x - 1)*(x^2 + 2*x - 1)).
From Colin Barker, May 26 2016: (Start)
a(n) = (-(-1-sqrt(2))^(1+n)+(-1+sqrt(2))^(1+n)+(1-sqrt(2))^n*(-4+3*sqrt(2))+(1+sqrt(2))^n*(4+3*sqrt(2)))/(2*sqrt(2)).
a(n) = 6*a(n-2)-a(n-4) for n>3. (End)
E.g.f.: 2*cosh(sqrt(2)*x)*(sinh(x) + 2*cosh(x)) + (sinh(sqrt(2)*x)*(5*sinh(x) + 3*cosh(x)))/sqrt(2). - Ilya Gutkovskiy, May 26 2016

A166259 Positive integers n such that a centered polygonal number nk(k+1)/2+1 is not a square for any k > 0.

Original entry on oeis.org

2, 18, 32, 50, 72, 98, 128, 162, 200, 242, 338, 392, 450, 512, 578, 648, 722, 882, 968, 1058, 1152, 1250, 1352, 1458, 1682, 1800, 1922, 2048, 2178, 2312, 2401, 2450, 2662, 2738, 2809, 2888, 3042, 3174, 3200, 3362, 3528, 3698, 3750, 4050, 4225, 4232, 4418, 4489, 4608, 4802

Offset: 1

Alexander Adamchuk, Oct 10 2009

nonn

Positive integers n such that A120744(n) = -1.

Eric Weisstein's World of Mathematics, Centered Polygonal Number.

Cf. A120744, A001542, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

Edited and extended by Max Alekseyev, Jan 20 2010

A174001 Number of spanning trees in C_9 X P_n.

Original entry on oeis.org

9, 632025, 23057815104, 763341471963225, 24743382596536452489, 797880028172050676793600, 25694231385152383926116001849, 827147402338052897443922764419225, 26625078176206788678765153788526329856

Offset: 1

Alois P. Heinz, Nov 26 2010

Alois P. Heinz, Table of n, a(n) for n = 1..70
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Spanning Tree

9th column of A173958.

Cf. A000012, A001542, A003690, A003753, A003733, A158880, A158898.

a:= n-> 9* (Matrix([[0, 1, 265, 50616, 9209545, 1658090689, 297747101520, 53431400864569, 9586723471888105][1+abs(i)]$i=-7..8]). Matrix(16, (i, j)-> if i=j-1 then 1 elif j=1 then [[-427427424, 327381265, -146975161, 38357160, -5699687, 457655, -17736, 265, -1][1+abs(k)]$k=-7..8][i] else 0 fi)^n)[1, 8]^2: seq(a(n), n=1..20);

See program.

A174089 Number of spanning trees in C_11 X P_n.

Original entry on oeis.org

11, 10759331, 4435600730891, 1584603178322856659, 545701094921321191290251, 185861400461684004931359802019, 63080339061067311398935095930531419, 21384626538080492686675351682716886393459

Offset: 1

Alois P. Heinz, Nov 26 2010

Alois P. Heinz, Table of n, a(n) for n = 1..180
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Spanning Tree

11th column of A173958.

Cf. A000012, A001542, A003690, A003753, A003733, A158880, A158898.

a:= n-> 11* (Matrix([[0, 1, 989, 635009, 379545563, 222731206721, 129986502957277, 75726985139241127, 44091461282285910613, 25667108238650778993721, 14940759758135641310394029, 8696803311384043382138568704, 5062251640287899331740697744283, 2946638531103878161891572927216367, 1715179927870529863091149494541065923, 998372029710787510889689081784904921409, 581132402632124482558541496059410958698763][1+abs(i)]*
signum(-i)$i=-15..16]). Matrix(32, (i, j)-> if i=j-1 then 1 elif j=1 then [[-9866686348925002518, 8584218556222705486, -5646220475933195574, 2797526034931937278, -1038052511465703094, 286230180847745070, -58096997326051905, 8585065341436957, -911803001143321, 68534901051869, -3574487862001, 125866549709, -2870938929, 39687581, -297177, 989, -1] [1+abs(k)]$k=-15..16][i] else 0 fi)^n)[1, 16]^2: seq(a(n), n=1..20);

See program.

A227972 Two column recursive array A(n,k), relating expressions based on half-squares (A007590) to each other and several other sequences, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 4, 5, 7, 7, 10, 17, 24, 29, 41, 41, 58, 99, 140, 169, 239, 239, 338, 577, 816, 985, 1393, 1393, 1970, 3363, 4756, 5741, 8119, 8119, 11482, 19601, 27720, 33461, 47321, 47321, 66922, 114243, 161564, 195025, 275807, 275807, 390050, 665857, 941664, 1136689, 1607521

Offset: 1

Richard R. Forberg, Aug 01 2013

nonn

The first column (k=1) holds the interleaved integer square roots of these two "Half-Square" expressions in ascending order:  floor(m^2/2 + 1) for m=>0 and floor(m^2/2 - 1) for m=>1.  The second column (k=2) holds the value of m that yields the corresponding integer square root.
The value of m for row n (at n mod 3 = 2) is the value of the square root for the next row (at n mod 3 = 0) which uses the other expression.
There are twice as many results for the expression floor(m^2/2 + 1) as for floor(m^2/2 - 1), interleaved consistently as two of every three results (as shown in the example below).
The first column, for n mod 3 = 1, produces A001541.
The first column, for n mod 3 = 2, produces A001653.
  NOTE: Interleaving of the two sequences above is A079496.
The first column, for n mod 3 = 0, produces A002315 (NSW Numbers).
  NOTE: Interleaving of A001541 and A002315 is A001333.
The second column, for n mod 3 = 1, produces A005319.
The second column, for n mod 3 = 2, produces A002315 (again).
  NOTE: Interleaving of the two sequences above is A143608.
The second column, for n mod 3 = 0, produces A075870.
  NOTE: Interleaving of A005319 and A075870 is A052542 = 2*A000129 (Pell)
The row sums at n mod 3 = 1 and n mod 3 = 0 are used in the recursion to produce values in subsequent rows of the array for both columns.
For rows at n mod 3 = 2,  the ascending interleaved combination of A(n,1) and the row sum (of the same row) produces A000129 (Pell Numbers).
Row sums also hold all the integer square roots (as given in A001542) of the Half-Squares, (A007590), at n mod 3 = 2, and the corresponding values of m in the next row at n mod 3 = 0, corresponding to A001541.
The value of the floor of half the row sum, for n mod 3 =0 and n mod 3 = 1, produces A048739, giving the partial sums of A000129 (Pell Numbers), for the Pell Numbers produced through the prior row at  n mod 3 = 2.
The value of half the row sum, for n mod 3 = 2, produces A001109 (without its initial 0).  This subsequence is also produced from finding the integer square roots of A083374. The value of the indices of that sequence where these roots occur is given by A002315 (NSW Numbers).
The differences of two entries in row n equals the row sum for row n-3, consistently for all rows n > 3.
The ratio of the two entries in the same row converges to sqrt(2).
The ratio of two entries in the same column (either k=1 or k=2) converge as follows:
      A(k,n)/A(k,n-1)--> sqrt(2)       for n mod 3 = 0,
                     --> sqrt(2) + 1   for n mod 3 = 1,
                     --> sqrt(2)/2 + 1 for n mod 3 = 2.
      A(k,n)/A(k,n-3)--> sqrt(8) + 3   for n mod 3 = 0, 1, or 2,
  That last line means: A001541, A001653, A002315, A005319 and A075870 all have the convergence ratio of sqrt(8) + 3 for adjacent terms. In addition alternating Pell Numbers also converge to that ratio.

			The two column array with row number n and the row sum. An extra column on the right shows which expression is applicable to get that row's values: either floor(m^2/2 + 1) indicated as "+1",  or floor(m^2/2 - 1) indicated as "-1". (NOTE: The value of n is immaterial, except as a row number).
The array begins:
Row         k=1         k=2                   Applicable "Half-Square"
n          (sqrt)       (m)         Row Sum        Expression
1            1           0               1             +1
2            1           1               2             +1
3            1           2               3             -1
4            3           4               7             +1
5            5           7              12             +1
6            7          10              17             -1
7           17          24              41             +1
8           29          41              70             +1
9           41          58              99             -1
10          99         140             239             +1
11         169         239             408             +1
12         239         338             577             -1
13         577         816            1393             +1
14         985        1393            2378             +1
15        1393        1970            3363             -1
16        3363        4756            8119             +1
17        5741        8119           13860             +1
18        8119       11482           19601             -1
19       19601       27720           47321             +1
20       33461       47321           80782             +1

Cf. A007590, A001541, A001653, A079496, A002315, A005319, A143608, A075870, A000129, A001109, A083374.

Initialize row 1 as A(1,1) = 1 and A(1,2) = 0, then:
  For rows at n mod 3 = 0:  A(n,1) = A(n-1, 2)
                            A(n,2) = A(n, 1) + A(n-2, 1)
  For rows at n mod 3 = 1:  A(n,1) = A(n-1, 1) + A(n-1, 2)
                            A(n,2) = A(n, 1) + A(n-1, 1)
  For rows at n mod 3 = 2:  A(n,1) = A(n-1,1) + A(n-3, 1)
                            A(n,2) = A(n-1,1) + A(n-1, 2)
Empirical g.f.: -x*(2*x^11-x^10-x^9+x^8-4*x^7+3*x^6-2*x^5-x^4-x^3-x^2-1) / ((x^6-2*x^3-1)*(x^6+2*x^3-1)). - Colin Barker, Aug 08 2013

Some additional comments by Richard R. Forberg, Aug 12 2013

A251867 Numbers n such that n^2 + (n+1)^2 is equal to the sum of the hexagonal numbers H(m) and H(m+1) for some m.

Original entry on oeis.org

0, 14, 492, 16730, 568344, 19306982, 655869060, 22280241074, 756872327472, 25711378892990, 873430010034204, 29670908962269962, 1007937474707144520, 34240203231080643734, 1163158972382034742452, 39513164857758100599650, 1342284446191393385645664

Offset: 1

Colin Barker, Dec 10 2014

Also nonnegative integers y in the solutions to 4*x^2-2*y^2+2*x-2*y = 0, the corresponding values of x being A220185.

			14 is in the sequence because 14^2+15^2 = 196+225 = 421 = 190+231 = H(10)+H(11).

Colin Barker, Table of n, a(n) for n = 1..654
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000290, A000384, A220185, A105635.

I:=[0,14,492]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 06 2015

LinearRecurrence[{35, -35, 1}, {0, 14, 492}, 20] (* Vincenzo Librandi, Sep 06 2015 *)

concat(0, Vec(-2*x^2*(x+7)/((x-1)*(x^2-34*x+1)) + O(x^100)))

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).
G.f.: -2*x^2*(x+7) / ((x-1)*(x^2-34*x+1)).
a(n) = A220185(n) + A001542(n-1)^2. - Alexander Samokrutov, Sep 05 2015
a(n) = (-4+(10+7*sqrt(2))*(17+12*sqrt(2))^(-n)+(10-7*sqrt(2))*(17+12*sqrt(2))^n)/8. - Colin Barker, Mar 02 2016
a(n) = A105635(4*n-4). - Greg Dresden, Aug 30 2021

A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

Original entry on oeis.org

12, 30, 70, 40, 56, 176, 126, 208, 408, 198, 154, 234, 84, 90, 330, 260, 546, 1026, 476, 456, 736, 286, 418, 1218, 828, 1178, 2378, 1188, 800, 1160, 390, 340, 900, 570, 644, 1364, 714, 374, 494, 144, 132, 532, 442, 1044, 1924, 874, 918, 1518, 608, 1116, 3196

Offset: 1

Rémy Sigrist, Nov 30 2018

This sequence gives the perimeters of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.

If we order the terms in this sequence and keep duplicates then we obtain A024364.

			The first rows are:
   12
   30, 70, 40
   56, 176, 126, 208, 408, 198, 154, 234, 84
T(1,1) corresponds to the perimeter of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 + 4 + 5 = 12.

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

Cf. A001542, A002939, A024364, A033586, A321768, A321769, A321770.

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] + t[2, 1] + t[3, 1])

Empirically:
- T(n, 1) = A002939(n+1),
- T(n, (3^(n-1) + 1)/2) = A001542(n+1),
- T(n, 3^(n-1)) = A033586(n).

cp's OEIS Frontend

A098890 a(n) = A001652(n)*A046090(n)+1 = A098602(n)+1.

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A123057 Values x of solutions (x, y) to the Diophantine equation (x-y)^4 - 8*x*y = 0 with x >= y.

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Magma

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Sage

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A123116 Values y of solutions (x, y) to the Diophantine equation (x-y)^4 - 8*x*y = 0 with x >= y.

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Magma

Mathematica

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Sage

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Extensions

A129346 a(2n) = A100525(n), a(2n+1) = A001653(n+1); a Pellian-related sequence.

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Mathematica

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A166259 Positive integers n such that a centered polygonal number n*k*(k+1)/2+1 is not a square for any k > 0.

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A174001 Number of spanning trees in C_9 X P_n.

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Maple

Formula

A174089 Number of spanning trees in C_11 X P_n.

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Keywords

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Crossrefs

Programs

Maple

Formula

A227972 Two column recursive array A(n,k), relating expressions based on half-squares (A007590) to each other and several other sequences, read by rows.

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Author

A123057 Values x of solutions (x, y) to the Diophantine equation (x-y)^4 - 8xy = 0 with x >= y.

A123116 Values y of solutions (x, y) to the Diophantine equation (x-y)^4 - 8xy = 0 with x >= y.

A166259 Positive integers n such that a centered polygonal number nk(k+1)/2+1 is not a square for any k > 0.