A100252 -id:A100252 - cp's OEIS frontend

A100251 The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.

6, 2, 99, 35, 9, 15, 3, 0, 14, 8, 6, 21, 55, 4, 133, 10, 22, 0, 51, 27, 261, 15, 5, 85, 161, 9, 35, 451, 21, 33, 69, 14, 124, 6, 44, 715, 28, 24, 7421, 217, 34, 16, 23001, 54, 1065, 36, 7, 76, 156, 0, 245

Offset: 3

Charlie Marion, Nov 21 2004

nonn

Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=k; 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.

			a(3)=99 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.

Cf. A100252, A100253.

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]; k, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)

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

a(n)^2 = A100252(n).

A188896 Numbers n such that there is no square n-gonal number greater than 1.

Original entry on oeis.org

10, 20, 52, 164, 340, 580, 884, 1252, 1684, 2180, 2740, 4052, 4804, 5620, 6500, 7444, 8452, 9524, 10660, 11860, 13124, 14452, 15844, 17300, 18820, 20404, 22052, 25540, 27380, 29284, 31252, 33284, 35380, 37540, 39764, 42052, 44404, 46820, 49300, 51844

Offset: 1

T. D. Noe, Apr 13 2011

nonn

It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (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 2k^2+2 (A005893), 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.

Muniru A Asiru, Table of n, a(n) for n = 1..244
Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

Cf. A001107 (10-gonal numbers), A051872 (20-gonal numbers), A188892, A100252, A188950, A005893.

Subsequence of A271624. - Muniru A Asiru, Oct 16 2016

P[n_,k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2,161]; data2=Head[Reduce[m^2==P[n,#] && 1Ant King, Mar 01 2012 *)

A101157 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square, say k^2; then a(n)=k.

Original entry on oeis.org

1, 3, 5, 2, 9, 11, 13, 15, 3, 19, 6, 5, 25, 27, 29, 4, 33, 10, 37, 39, 14, 43, 45, 7, 5, 9, 53, 55, 57, 59, 61, 18, 65, 67, 15, 6, 18, 75, 22, 9, 81, 83, 15, 87, 21, 26, 12, 95, 7, 99, 101, 33, 30, 107, 109, 111, 22, 25, 117, 11, 121, 42, 125, 8, 129

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

a(n) is the least k>0 such that triangular(n-1) + k^2 is a triangular number. - Alex Ratushnyak, May 17 2013

			a(11)=6 since 11+12+13 = 6^2.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101158, A101159, A101160.

a(n) = {j = 0; while(! issquare(v=sum(k=0, j, n+k)), j++); sqrtint(v);} \\ Michel Marcus, Sep 01 2013

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+A101159(n) = a(n)^2 = A101158(n).

a(n^2) = n. - Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A100253 Let j be the smallest integer for which 1+(1+1n)+(1+2n)+...+(1+jn)=k^2=s. Then a(n)=1+jn; if no such j exists, then a(n)=0.

Original entry on oeis.org

8, 3, 241, 97, 26, 49, 8, 0, 55, 31, 23, 97, 274, 15, 721, 49, 120, 0, 305, 161, 1681, 89, 24, 577, 1126, 53, 244, 3361, 146, 241, 528, 97, 991, 35, 351, 6049, 223, 191, 65521, 1921, 288, 127, 213281, 485, 10081, 323, 48, 721, 1520, 0, 2449

Offset: 1

Charlie Marion, Nov 21 2004

nonn

Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

			a(3)=241 since 1 + 4 + 7 +...+ 241 = 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.

1+(1+1*n)+(1+2*n)+...+(1+A100254(n)*n)= 1+(1+1*n)+(1+2*n)+...+a(n)=A100251(n)^2=A100252(n)

A100254 Let j be the smallest integer for which 1 + (1+1n) + (1+2n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.

Original entry on oeis.org

7, 1, 80, 24, 5, 8, 1, 0, 6, 3, 2, 8, 21, 1, 48, 3, 7, 0, 16, 8, 80, 4, 1, 24, 45, 2, 9, 120, 5, 8, 17, 3, 30, 1, 10, 168, 6, 5, 1680, 48, 7, 3, 4960, 11, 224, 7, 1, 15, 31, 0, 48, 24, 16, 12, 288, 8, 48, 6, 26, 80, 117, 1, 136160, 195, 13, 3, 9, 840, 1520, 24, 49, 8, 70, 2, 1680

Offset: 1

Charlie Marion, Nov 21 2004

nonn

Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

			a(3)=80 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.

a[n_] := Block[{k = 1}, While[ !IntegerQ[ Sqrt[(k + 1)(1 + k*n/2)]], k++ ]; k]; a[18] = a[50] = 0; Table[ a[n], {n, 75}] (* Robert G. Wilson v, Nov 27 2004 *)

1 + (1+1*n) + (1+2*n) + ... + (1+a(n)*n) = 1 + (1+1*n) + (1+2*n) + ... + A100253(n) = A100251(n)^2 = A100252(n).

More terms from Robert G. Wilson v, Nov 27 2004

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

T. D. Noe, Apr 18 2011

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.

			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

Eric W. Weisstein, MathWorld: Polygonal Number

Cf. A188891, A100252.

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}]]

A101160 a(n) is the smallest integer j for which n+(n+1)+...+(n+j) is a square.

Original entry on oeis.org

0, 2, 4, 0, 8, 10, 12, 14, 0, 18, 2, 1, 24, 26, 28, 0, 32, 4, 36, 38, 7, 42, 44, 1, 0, 2, 52, 54, 56, 58, 60, 8, 64, 66, 5, 0, 7, 74, 10, 1, 80, 82, 4, 86, 8, 12, 2, 94, 0, 98, 100, 17, 14, 106, 108, 110, 7, 9, 116, 1, 120, 23, 124, 0, 128

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

0 <= a(n) <= 2*(n - 1). - Ctibor O. Zizka, Oct 05 2023

			a(11)=2 since j=2 is the smallest integer for which 11+...+11+j = 6^2 = 36 is a perfect square.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101158, A101159.

Cf. A100251, A100252, A100253, A100254.

a(n) = {j = 0; while(! issquare(sum(k=0, j, n+k)), j++); j;} \\ Michel Marcus, Sep 01 2013

n+(n+1)+...+(n+a(n)) = n+(n+1)+...+A101159(n) = A101157(n)^2 = A101158(n).

a(n^2) = 0. - Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A188891 Least triangular n-gonal number greater than 1, or 0 if none exists.

Original entry on oeis.org

3, 36, 210, 6, 55, 21, 325, 10, 0, 105, 36, 1275, 15, 45, 231, 0, 946, 276, 21, 11935, 66, 136, 351, 1596, 78, 28, 1225, 595, 820, 58653, 190, 325, 1335795, 36, 6670, 0, 561, 4005, 120, 1128, 1485, 203841, 45, 666, 6903, 465, 4950, 20910, 741, 153, 10731, 8911, 55, 1953

Offset: 3

T. D. Noe, Apr 13 2011

nonn

See A188893 and A188894 for the corresponding indices of these terms. Note that a(n) is zero for n = 11, 18, 38 (numbers in A188892). Although the Mathematica program searches only the first 20000 triangular numbers for n-gonal numbers, the Reduce function can show that there are no triangular n-gonal numbers (other than 0 and 1) for these n.

Eric W. Weisstein, MathWorld: Polygonal Number
Eric W. Weisstein, MathWorld: Triangular Number

Cf. A000217 (triangular numbers), A100252 (similar sequence for squares).

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, tr=0]; tr, {n, 3, 50}]
Table[SelectFirst[PolygonalNumber[n,Range[2,1000]],OddQ[Sqrt[8#+1]]&],{n,3,100}]/.Missing["NotFound"]->0 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 10 2019 *)

A101158 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; sequence gives the squares.

Original entry on oeis.org

1, 9, 25, 4, 81, 121, 169, 225, 9, 361, 36, 25, 625, 729, 841, 16, 1089, 100, 1369, 1521, 196, 1849, 2025, 49, 25, 81, 2809, 3025, 3249, 3481, 3721, 324, 4225, 4489, 225, 36, 324, 5625, 484, 81, 6561, 6889, 225, 7569, 441, 676, 144, 9025, 49, 9801

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

			a(11)=36 since 11+12+13 = 36.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101159, A101160.

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+A101159(n) = A101157(n)^2 = a(n).

a(n^2) = n^2. - Michel Marcus, Jun 28 2013

a(21) corrected by Michel Marcus, Jun 29 2013

A101159 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; then a(n) = n+j.

Original entry on oeis.org

1, 4, 7, 4, 13, 16, 19, 22, 9, 28, 13, 13, 37, 40, 43, 16, 49, 22, 55, 58, 28, 64, 67, 25, 25, 28, 79, 82, 85, 88, 91, 40, 97, 100, 40, 36, 44, 112, 49, 41, 121, 124, 47, 130, 53, 58, 49, 142, 49, 148, 151, 69, 67, 160, 163, 166, 64, 67, 175, 61

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

			a(11)=13 since j=13 is the smallest integer such that 11+...+j=6^2=36 is a perfect square.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101158, A101160, A108269.

a(n) = {my(j = 0); while(! issquare(sum(k=0, j, n+k)), j++); n+j;} \\ Michel Marcus, May 02 2018

a(n) = my(s = n, t = 0); while(!issquare(s), s += n + t++); n + t \\ David A. Corneth, May 05 2018

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+a(n) = A101157(n)^2 = A101158(n).
a(n^2) = n^2. - Michel Marcus, Jun 28 2013
a(n) <= 3*n - 2. - David A. Corneth, May 03 2018

More terms from Michel Marcus, Jun 28 2013

cp's OEIS Frontend

A100251 The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.

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A188896 Numbers n such that there is no square n-gonal number greater than 1.

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A101157 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square, say k^2; then a(n)=k.

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A100253 Let j be the smallest integer for which 1+(1+1*n)+(1+2*n)+...+(1+j*n)=k^2=s. Then a(n)=1+j*n; if no such j exists, then a(n)=0.

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A100254 Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.

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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.

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A101160 a(n) is the smallest integer j for which n+(n+1)+...+(n+j) is a square.

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A188891 Least triangular n-gonal number greater than 1, or 0 if none exists.

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A100253 Let j be the smallest integer for which 1+(1+1n)+(1+2n)+...+(1+jn)=k^2=s. Then a(n)=1+jn; if no such j exists, then a(n)=0.

A100254 Let j be the smallest integer for which 1 + (1+1n) + (1+2n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.