A189216 -id:A189216 - cp's OEIS frontend

A189217 Triangle read by rows of the index of the least k-gonal number A189216(n,k).

2, 8, 2, 20, 99, 2, 3, 35, 165, 2, 10, 9, 54, 247, 2, 6, 15, 11, 77, 345, 2, 25, 3, 21, 13, 104, 459, 2, 4, 0, 91, 28, 15, 135, 589, 2, 0, 14, 22, 0, 36, 17, 170, 735, 2, 14, 8, 3, 17, 86, 45, 19, 209, 897, 2, 8, 6, 7, 46, 16, 58, 55, 21, 252, 1075, 2, 50, 21, 0, 171, 3081, 861, 153, 66, 23, 299, 1269, 2

Offset: 3

T. D. Noe, Apr 18 2011

The n-th term of the n-th row is 2. The first and second columns are A188893 and A188897.

			The triangle begins:
2,
8,   2
20,  99,  2
3,   35,  165,  2
10,  9,   54,   247,  2
6,   15,  11,   77,   345,  2
25,  3,   21,   13,   104,  459,   2
4,   0,   91,   28,   15,   135,   589,   2
0,   14,  22,   0,    36,   17,    170,   735,   2

A189218 Triangle read by rows of the index of the least n-gonal number A189216(n,k).

Original entry on oeis.org

2, 6, 2, 12, 81, 2, 2, 25, 143, 2, 5, 6, 42, 221, 2, 3, 9, 8, 63, 315, 2, 10, 2, 14, 10, 88, 425, 2, 2, 0, 56, 20, 12, 117, 551, 2, 0, 7, 13, 0, 27, 14, 150, 693, 2, 5, 4, 2, 11, 61, 35, 16, 187, 851, 2, 3, 3, 4, 28, 11, 43, 44, 18, 228, 1025, 2, 15, 9, 0, 99, 1989, 609, 117, 54, 20, 273, 1215, 2

Offset: 3

T. D. Noe, Apr 18 2011

The n-th term of the n-th row is 2. The first and second columns are A188894 and A188898.

			The triangle begins:
2
6,   2
12,  81,   2
2,   25,   143,  2
5,   6,    42,   221,  2
3,   9,    8,    63,   315,  2
10,  2,    14,   10,   88,   425,  2
2,   0,    56,   20,   12,   117,  551,  2
0,   7,    13,   0,    27,   14,   150,  693,  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]]

A342300 Least nonnegative number greater than the previous number which is simultaneously an n-gonal and (n+1)-gonal number.

Original entry on oeis.org

0, 1, 3, 36, 9801, 40755, 121771, 297045, 631125, 1212751, 2158695, 3617601, 5773825, 8851275, 13117251, 18886285, 26523981, 36450855, 49146175, 65151801, 85076025, 109597411, 139468635, 175520325, 218664901, 269900415, 330314391, 401087665, 483498225, 578925051, 688851955, 814871421

Offset: 0

Robert G. Wilson v, Jun 04 2021

Also the least nontrivial number simultaneously an n and (n+1)-gonal number for n greater than one.
0 and 1 are always terms of any sequence of polygonal numbers of a particular rank beginning with index 0.
Since the formula for the k-th n-gonal number P(n,k) is k*(4+k*(n-2)-n)/2, one can extrapolate for the non-geometrical terms 0, 1 and 2.
Indices of the n and (n+1)-gonal numbers by pairs: {0, 0} {1, 1}, {3, 2}, {8, 6}, {99, 81}, {165, 143}, {247, 221}, {345, 315}, {459, 425}, {589, 551}, {735, 693}, {897, 851} ..., .
{x, y} of the above are {8n^2 + 10n - 3, 8n^2 - 10n - 7} for n>3 (A303295).
In the first 1000 terms, 1 is congruent to 0 (mod 6), 333 are congruent to 1 (mod 6), and 666 are congruent to 3 (mod 6).

			a(3) is the least triangular and square number > 3, which is 36: A001110(2).
a(4) is the least square and pentagonal number > 36, which is 9801: A036353(2).

Index entries for linear recurrences with constant coefficients, signature (6,-15, 20,-15,6,-1).

Cf. A001110, A036353, A046180, A048903, A048906, A048924, A203627, A189216.

a[n_] := Intersection[ Table[ PolygonalNumber[n, i], {i, 2, 10000}], Table[ PolygonalNumber[n + 1, i], {i, 2, 10000}]][[1]]; a[0] = 0; a[1] = 1; Array[a, 30, 0] (* Or *)
a[n_] := a[n] = 6a[n - 1] -15a[n - 2] +20a[n - 3] -15a[n - 4] +6a[n - 5] -a[n - 6]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 36; a[4] = 9801; a[5] = 40755; a[6] = 121771; a[7] = 297045; a[8] = 631125; a[9] = 1212751; Array[a, 30, 0]

a(n) = 32n^5 - 112n^4 + 70n^3 + 93n^2 - 57n - 35 for n > 3; a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 36.

G.f.: x*(1 - 3*x + 33*x^2 + 9610*x^3 - 17556*x^4 + 23575*x^5 - 17753*x^6 + 7122*x^7 - 1189*x^8)/(1 - x)^6. - Stefano Spezia, Jun 08 2021

A345038 Triangle T(n,k) read by rows of the smallest centered n-gonal number greater than 1 that is also centered k-gonal, or 0 if none exists, for 1 <= k <= n.

Original entry on oeis.org

2, 7, 3, 4, 31, 4, 0, 13, 85, 5, 16, 31, 31, 181, 6, 7, 7, 19, 61, 331, 7, 22, 43, 316, 841, 106, 547, 8, 121, 0, 361, 25, 22801, 169, 841, 9, 0, 91, 10, 0, 1891, 91, 253, 1225, 10, 11, 31, 31, 61, 31, 61, 2101, 361, 1711, 11, 67, 111, 166, 8581, 1156, 397, 6931, 179479609, 496, 2311, 12

Offset: 1

Mohammed Yaseen, Jun 06 2021

The i-th centered j-gonal number is j*i*(i-1)/2 + 1. Thus if the p-th centered n-gonal number is also the q-th centered k-gonal number, then n*p*(p-1) = k*q*(q-1). Therefore T(n,k) = n*p*(p-1)/2 + 1 = k*q*(q-1)/2 + 1 iff n*p*(p-1) = k*q*(q-1) has a nontrivial positive integer solution. Otherwise T(n,k) = 0. It also implies that when T(n,k) = 0, T(r*n,r*k) = 0 for any positive integer r.

			The triangle begins:
   2;
   7,   3;
   4,  31,   4;
   0,  13,  85,   5;
  16,  31,  31, 181,   6;
   7,   7,  19,  61, 331,   7;
  22,  43, 316, 841, 106, 547,   8;
  ...

Index entries for sequences related to centered polygonal numbers

Cf. A101321, A189216.

iszero(n,k)={if(issquare(n) && issquare(k) && n<>k, my(t=n-k); fordiv(t, d, my(p=(d+t/d)/2/sqrtint(n), q=(d-t/d)/2/sqrtint(k)); if(abs(p)!=1 && !frac(p) && !frac(q) && p%2==1 && q%2==1, return(0))); 1, 0)}
T(n, k)={my(g=gcd(n,k)); n/=g; k/=g; if(iszero(n, k), 0, for(p=2, oo, my(t=n*p*(p-1)/2); if(t%k==0 && ispolygonal(t/k, 3), return(t*g+1))))} \\ Andrew Howroyd, Jun 08 2021

T(n,n) = n+1.

A378245 Numbers that are both k-gonal and (k+1)-gonal for some k >= 3.

Original entry on oeis.org

1, 36, 1225, 9801, 40755, 41616, 121771, 297045, 631125, 1212751, 1413721, 2158695, 3617601, 5773825, 8851275, 13117251, 18886285, 26523981, 36450855, 48024900, 49146175, 65151801, 85076025, 94109401, 109597411, 139468635, 175520325, 218664901, 269900415, 330314391

Offset: 1

Kelvin Voskuijl, Nov 20 2024

nonn

			a(2) = 36 is both the 8th triangular and the 6th square number.
a(3) = 1225 is both the 49th triangular and the 35th square number.
a(5) = 40755 is both the 165th pentagonal number and the 143th hexagonal number.

Cf. A057145, A342300.
Cf. A001110, A036353, A046180, A048903, A048906, A048924 and A203627 (subsequences).
The subdiagonal of A189216 is also a subsequence.

upto(limit) = my(terms=List(1)); for(k=3, oo, my(found=0); for(n=2, oo, my(a = (2*n - 1)^2, b = (4*n*(3*n - 5) + 6), c = (8*(n-1)^2 + 1), s = (a*k^2 - b*k + c), v = n * (n*k - k - 2*n + 4) / 2); if(issquare(s), my(t = sqrtint(s) + k - 3); if(t % (2*(k-1)) == 0, listput(terms, v); found += 1)); if(v >= limit, break)); if(found == 0, break)); Vec(vecsort(terms)); \\ Daniel Suteu, Dec 08 2024

a(12)-a(30) from Pontus von Brömssen, Dec 07 2024

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A189217 Triangle read by rows of the index of the least k-gonal number A189216(n,k).

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A189218 Triangle read by rows of the index of the least n-gonal number A189216(n,k).

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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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A342300 Least nonnegative number greater than the previous number which is simultaneously an n-gonal and (n+1)-gonal number.

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A345038 Triangle T(n,k) read by rows of the smallest centered n-gonal number greater than 1 that is also centered k-gonal, or 0 if none exists, for 1 <= k <= n.

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A378245 Numbers that are both k-gonal and (k+1)-gonal for some k >= 3.

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