cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Charlie Marion, Nov 21 2004

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

T. D. Noe, Apr 13 2011

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Maple
    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

A189216 Triangle T(n,k) read by rows of the smallest n-gonal number greater than 1 that is also k-gonal, or 0 if none exists, for 3 <= k <= n.

Original entry on oeis.org

3, 36, 4, 210, 9801, 5, 6, 1225, 40755, 6, 55, 81, 4347, 121771, 7, 21, 225, 176, 11781, 297045, 8, 325, 9, 651, 325, 26884, 631125, 9, 10, 0, 12376, 1540, 540, 54405, 1212751, 10, 0, 196, 715, 0, 3186, 833, 100725, 2158695, 11, 105, 64, 12, 561, 18361, 5985, 1216, 174097, 3617601, 12
Offset: 3

Views

Author

T. D. Noe, Apr 18 2011

Keywords

Comments

The first column (k=3, triangular numbers) is A188891. The second column (k=4, squares) is A100252. The n-th term of the n-th row is n. Observe that 0 occurs for (10,4)-gonal, (11,3)-gonal, and (11,6)-gonal numbers. This can be proved by trying to solve the equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y for integers x>1 and y>1. Other pairs that are zero: (14,5), (18,3), (18,6), (18,11), (20,4), and (20,10). See A188950 for a longer list of pairs.
Sequences A189217 and A189218 give the index of T(n,k) as a k-gonal and n-gonal number, respectively.

Examples

			The triangle begins:
3
36,      4
210,     9801,    5
6,       1225,    40755,   6
55,      81,      4347,    121771,  7
21,      225,     176,     11781,   297045,  8
325,     9,       651,     325,     26884,   631125,  9
10,      0,       12376,   1540,    540,     54405,   1212751, 10
0,       196,     715,     0,       3186,    833,     100725,  2158695,  11

Links

Crossrefs

Cf. A188891, A100252.

Programs

  • Mathematica
    nn = 12; Clear[poly]; Do[poly[n] = Table[i*((n - 2)*i - (n - 4))/2, {i, 2, 20000}], {n, 3, nn}]; Flatten[Table[If[k == n, n, int = Intersection[poly[n], poly[k]]; If[int == {}, 0, int[[1]]]], {n, 3, nn}, {k, 3, n}]]

A188893 The index of the least triangular number greater than 1 that is also an n-gonal number, or 0 if none exists.

Original entry on oeis.org

2, 8, 20, 3, 10, 6, 25, 4, 0, 14, 8, 50, 5, 9, 21, 0, 43, 23, 6, 154, 11, 16, 26, 56, 12, 7, 49, 34, 40, 342, 19, 25, 1634, 8, 115, 0, 33, 89, 15, 47, 54, 638, 9, 36, 117, 30, 99, 204, 38, 17, 146, 133, 10, 62, 70, 18, 41, 49, 34, 4806, 105, 161, 329, 11, 439, 223, 28, 5974, 20, 79
Offset: 3

Views

Author

T. D. Noe, Apr 13 2011

Keywords

Crossrefs

Cf. A188891.

Programs

  • Mathematica
    NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[tr = k*(k+1)/2; i = NgonIndex[n, tr]; k < 20000 && ! IntegerQ[i], k++]; If[k==20000, k=0]; k, {n, 3, 50}]

A188894 The index of the least n-gonal number greater than 1 that is also triangular, or 0 if none exists.

Original entry on oeis.org

2, 6, 12, 2, 5, 3, 10, 2, 0, 5, 3, 15, 2, 3, 6, 0, 11, 6, 2, 35, 3, 4, 6, 12, 3, 2, 10, 7, 8, 63, 4, 5, 285, 2, 20, 0, 6, 15, 3, 8, 9, 99, 2, 6, 18, 5, 15, 30, 6, 3, 21, 19, 2, 9, 10, 3, 6, 7, 5, 621, 14, 21, 42, 2, 55, 28, 4, 725, 3, 10, 11, 195, 7, 8, 460, 2, 14, 6, 28, 56, 10, 4, 36, 2107
Offset: 3

Views

Author

T. D. Noe, Apr 13 2011

Keywords

Crossrefs

Cf. A188891.

Programs

  • Mathematica
    NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[tr = k*(k+1)/2; i = NgonIndex[n, tr]; k < 20000 && ! IntegerQ[i], k++]; If[k==20000, i=0]; i, {n, 3, 50}]

Formula

If n is a triangular number, then a(n)=2.
Showing 1-5 of 5 results.

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