A176542 -id:A176542 - cp's OEIS frontend

A116476 Numbers n such that T(n) + T(n+1) + ... + T(n+10) is a square, where T(m) = A000217(m) is the m-th triangular number.

13, 46, 229, 1608, 7335, 20304, 92391, 635710, 2892133, 8001886, 36403981, 250470288, 1139495223, 3152724936, 14343078279, 98684659918, 448958227885, 1242165625054, 5651136440101, 38881505539560, 176888402293623, 489410103548496

Offset: 1

Edward Fedorovich (chipramy(AT)012.net.il), Mar 29 2006

Positive integers n such that 11*n^2 + 121*n + 440 = 2*m^2 for some integer m. - Max Alekseyev, Jan 20 2010

			13 belongs to this sequence since T(13) + T(14) + ... + T(23) = 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 = 1936 = 44^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,394,-394,0,0,-1,1).

Cf. A176541, A176542, A165517, A202391

For[n = 1, n < 100000, n++, If[IntegerQ[Sqrt[Sum[i*(i+1)/2, {i, n, n + 10}]]], Print[n]]] (* Stefan Steinerberger, Mar 30 2006 *)
LinearRecurrence[{1,0,0,394,-394,0,0,-1,1},{13,46,229,1608,7335,20304,92391,635710,2892133},30] (* Harvey P. Dale, Sep 01 2017 *)

For n>8, a(n) = 394*a(n-4) - a(n-8) + 2156. - Max Alekseyev, Jan 20 2010

G.f.: x*(2*x^8+7*x^7+15*x^6+33*x^5-605*x^4-1379*x^3-183*x^2-33*x-13)/((x-1)*(x^8-394*x^4+1)). - Colin Barker, Nov 22 2012

Extended by Max Alekseyev, Jan 20 2010

A176541 Numbers n such that there exist n consecutive triangular numbers which sum to a square.

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 13, 22, 23, 25, 27, 32, 37, 39, 46, 47, 48, 49, 50, 52, 59, 66, 71, 73, 83, 94, 98, 100, 104, 107, 109, 111, 118, 121, 128, 143, 146, 147, 148, 157, 167, 176, 179, 181, 183, 191, 192, 193, 194, 200, 214, 219, 227, 239, 241, 242, 243, 244, 253, 263

Offset: 1

Andrew Weimholt, Apr 20 2010

nonn

Numbers n such that there exists some x >= 0 such that A000292(x+n) - A000292(x) is a square. Terms of this sequence, for which only a finite number of solutions x exist, are given in A176542.
Integer n is in the sequence if there exist non-degenerate solutions to the Diophantine equation: 8x^2 - n*y^2 - A077415(n) = 0. A degenerate solution is one involving triangular numbers with negative indexes.
The sum of n consecutive triangular numbers starting at the j-th is Sum_{k=j..j+n-1} A000217(k) = n*(n^2 + 3*j*n + 3*j^2 - 1)/6, see A143037. - R. J. Mathar, May 06 2015

			0 is in the sequence because the sum of 0 consecutive triangular numbers is 0 (a square).
1 is in the sequence because there exist triangular numbers which are squares (cf. A001110).
2 is in the sequence because ANY 2 consecutive triangular numbers sum to a square.
3 is in the sequence because there are infinitely many solutions (cf. A165517).
4 is in the sequence because there infinitely many solutions (cf. A202391).
5 is NOT in the sequence because no 5 consecutive triangular numbers sum to a square.
For n=8, solutions to the Diophantine equation exist, but start at A000217(-2) and A000217(-6): 1 + 0 + 0 + 1 + 3 + 6 + 10 + 15 = 36 and 15 + 10 + 6 + 3 + 1 + 0 + 0 + 1 = 36. There are no non-degenerate solutions for n=8. Hence, 8 is not included in the sequence.
For n=11, there exist infinitely many solutions (cf. A116476), so 11 is in the sequence.

Cf. A176542, A000217, A000292, A001110, A077415.

More terms from Max Alekseyev, May 10 2010

A257293 Numbers n such that T(n) + T(n+1) + ... + T(n+12) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

3, 29, 75, 432, 998, 3624, 8310, 44717, 102443, 370269, 848195, 4561352, 10448838, 37764464, 86508230, 465213837, 1065679683, 3851605709, 8822991915, 47447250672, 108688879478, 392826018504, 899858667750, 4839154355357, 11085200027723, 40064402282349

Offset: 1

M. F. Hasler, May 04 2015

It is well known that T(n)+T(n+1) is always a square. T(n)+T(n+1)+T(n+2) is a square for n in A165517. T(n)+T(n+1)+T(n+2)+T(n+3) is a square for n in A202391. There is no sequence of 5, 6, 7, 8, 9 or 10 consecutive T(i)'s which sum to a square, cf. A176541. The next possible length is 11, see A116476. Then comes this sequence, corresponding to length 13.

Positive integers y in the solutions to 2*x^2-13*y^2-169*y-728 = 0. - Colin Barker, May 04 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,102,-102,0,0,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11).

I:=[3,29,75,432,998,3624,8310,44717,102443]; [n le 9 select I[n] else Self(n-1)+102*Self(n-4)-102*Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 05 2015

Select[Range[10^5],IntegerQ[Sqrt[(#^2+13*#+56)*13/2]]&] (* Ivan N. Ianakiev, May 04 2015 *)
LinearRecurrence[{1, 0, 0, 102, -102, 0, 0, -1, 1}, {3, 29, 75, 432, 998, 3624, 8310, 44717, 102443}, 50] (* Vincenzo Librandi, May 05 2015 *)

for(n=0,10^8,issquare(binomial(n+14,3)-binomial(n+1,3))&&print1(n","))

Vec(x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)) + O(x^100)) \\ Colin Barker, May 04 2015

G.f.: x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)). - Colin Barker, May 04 2015

A202391 Indices of the smallest of four consecutive triangular numbers summing up to a square.

Original entry on oeis.org

5, 39, 237, 1391, 8117, 47319, 275805, 1607519, 9369317, 54608391, 318281037, 1855077839, 10812186005, 63018038199, 367296043197, 2140758220991, 12477253282757, 72722761475559, 423859315570605, 2470433131948079

Offset: 1

Max Alekseyev, Dec 18 2011

nonn

Positive integers n such that A000217(n) + A000217(n + 1) + A000217(n + 2) + A000217(n + 3) is a square (=A075870(n+1)^2).

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A176541, A176542, A165517, A116476, A075870

a(n) = A002315(n) - 2.
G.f.: x*(1+x)*(x-5) / ( (x-1)*(1-6*x+x^2) ). - R. J. Mathar, Dec 19 2011
a(n+2) = 6*a(n+1) - a(n) + 8; a(n+3) = 7*a(n+2) - 7*a(n+1) + a(n); a(n+1) = (-4 + (7 + 5*r)*(3 + 2*r)^n + (7 - 5*r)*(3 - 2*r)^n)/2 where r = sqrt(2). - Paul Weisenhorn, Jan 13 2013

A257707 Numbers n such that T(n) + T(n+1) + ... + T(n+22) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

56, 470, 1094, 7856, 128534, 201539, 3293081, 23435699, 53805155, 382911281, 6256309475, 9809462822, 160274811896, 1140616029542, 2618697452438, 18636292598096, 304494582579398, 477426555904883, 7800575092244921, 55513782134933123, 127452004956911987

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-23*y^2-529*y-4048 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,48670,-48670,0,0,0,0,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13), A257708 (length 25), A257709 (length 27), A257710 (length 37).

LinearRecurrence[{1, 0, 0, 0, 0, 48670, -48670, 0, 0, 0, 0, -1, 1}, {56, 470, 1094, 7856, 128534, 201539, 3293081, 23435699, 53805155, 382911281, 6256309475, 9809462822, 160274811896}, 50] (* Vincenzo Librandi, May 05 2015 *)

Vec(x*(10*x^12 +3*x^11 +66*x^10 +414*x^9 +624*x^8 +6762*x^7 -366022*x^6 -73005*x^5 -120678*x^4 -6762*x^3 -624*x^2 -414*x -56) / ((x -1)*(x^12 -48670*x^6 +1)) + O(x^100))

G.f.: x*(10*x^12 +3*x^11 +66*x^10 +414*x^9 +624*x^8 +6762*x^7 -366022*x^6 -73005*x^5 -120678*x^4 -6762*x^3 -624*x^2 -414*x -56) / ((x -1)*(x^12 -48670*x^6 +1)).

A257708 Numbers n such that T(n) + T(n+1) + ... + T(n+24) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

25, 55, 208, 382, 1273, 2287, 7480, 13390, 43657, 78103, 254512, 455278, 1483465, 2653615, 8646328, 15466462, 50394553, 90145207, 293721040, 525404830, 1711931737, 3062283823, 9977869432, 17848298158, 58155284905, 104027505175, 338953840048, 606316732942

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-25*y^2-625*y-5200 = 0.

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

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257709 (length 27), A257710 (length 37).

LinearRecurrence[{1, 6, -6, -1, 1}, {25, 55, 208, 382, 1273}, 50] (* Vincenzo Librandi, May 05 2015 *)

Vec(x*(x^2+4*x+5)*(2*x^2-2*x-5)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)) + O(x^100))

G.f.: x*(x^2+4*x+5)*(2*x^2-2*x-5) / ((x-1)*(x^2-2*x-1)*(x^2+2*x-1)).

A257709 Numbers n such that T(n) + T(n+1) + ... + T(n+26) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160, 2734, 5159, 6525, 11415, 12296, 21502, 27184, 51189, 64711, 113117, 121838, 212968, 269214, 506839, 640693, 1119863, 1206192, 2108286, 2665064, 5017309, 6342327, 11085621, 11940190, 20870000

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-27*y^2-729*y-6552 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,10,-10,0,0,0,0,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257708 (length 25), A257710 (length 37).

I:=[8,14,39,53,103,112,206,264,509,647,1141,1230, 2160]; [n le 13 select I[n] else Self(n-1)+10*Self(n-6)-10*Self(n-7)-Self(n-12)+Self(n-13): n in [1..40]]; // Vincenzo Librandi, May 05 2015

LinearRecurrence[{1, 0, 0, 0, 0, 10, -10, 0, 0, 0, 0, -1, 1}, {8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160}, 50] (* Vincenzo Librandi, May 05 2015 *)
Position[Total/@Partition[Accumulate[Range[70000]],27,1],?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* The program generates the first 22 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* _Harvey P. Dale, Jul 27 2021 *)

Vec(x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)) + O(x^100))

G.f.: x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)).

A257710 Numbers n such that T(n) + T(n+1) + ... + T(n+36) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

5, 32, 291, 661, 4102, 8515, 13685, 113558, 182368, 377701, 2290342, 5027232, 30483491, 63130838, 101378488, 840238915, 1349295285, 2794368792, 16944086651, 37191598501, 225516999142, 467042067835, 749998177365, 6216087516438, 9982086472888, 20672740082341

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-37*y^2-1369*y-16872 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,7398,-7398,0,0,0,0,0,0,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257708 (length 25), A257709 (length 27).

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 7398, -7398, 0, 0, 0, 0, 0, 0, -1, 1}, {5, 32, 291, 661, 4102, 8515, 13685, 113558, 182368, 377701, 2290342, 5027232, 30483491, 63130838, 101378488, 840238915, 1349295285}, 50] (* Vincenzo Librandi, May 05 2015 *)

Vec(x*(5*x^16 +27*x^15 +10*x^14 +27*x^13 +259*x^12 +370*x^11 +3441*x^10 +4413*x^9 -31820*x^8 -99873*x^7 -5170*x^6 -4413*x^5 -3441*x^4 -370*x^3 -259*x^2 -27*x -5) / ((x -1)*(x^8 -86*x^4 -1)*(x^8 +86*x^4 -1)) + O(x^100))

G.f.: x*(5*x^16 +27*x^15 +10*x^14 +27*x^13 +259*x^12 +370*x^11 +3441*x^10 +4413*x^9 -31820*x^8 -99873*x^7 -5170*x^6 -4413*x^5 -3441*x^4 -370*x^3 -259*x^2 -27*x -5) / ((x -1)*(x^8 -86*x^4 -1)*(x^8 +86*x^4 -1)).

A254443 Numbers n such that T(n) + T(n+1) + ... + T(n+21) is a square, where T(m) = A000217(m) is the m-th triangular number.

Original entry on oeis.org

35, 75, 911, 1707, 18383, 34263, 366947, 683751, 7320755, 13640955, 146048351, 272135547, 2913646463, 5429070183, 58126881107, 108309268311, 1159623975875, 2160756296235, 23134352636591, 43106816656587, 461527428756143, 859975576835703, 9207414222486467

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-22*y^2-484*y-3542 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13).

Vec(x*(9*x^4+4*x^3-136*x^2-40*x-35)/((x-1)*(x^4-20*x^2+1)) + O(x^100))

G.f.: x*(9*x^4+4*x^3-136*x^2-40*x-35) / ((x-1)*(x^4-20*x^2+1)).

cp's OEIS Frontend

A116476 Numbers n such that T(n) + T(n+1) + ... + T(n+10) is a square, where T(m) = A000217(m) is the m-th triangular number.

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A176541 Numbers n such that there exist n consecutive triangular numbers which sum to a square.

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A257293 Numbers n such that T(n) + T(n+1) + ... + T(n+12) is a square, where T = A000217 (triangular numbers).

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A202391 Indices of the smallest of four consecutive triangular numbers summing up to a square.

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A257707 Numbers n such that T(n) + T(n+1) + ... + T(n+22) is a square, where T = A000217 (triangular numbers).

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A257708 Numbers n such that T(n) + T(n+1) + ... + T(n+24) is a square, where T = A000217 (triangular numbers).

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A257709 Numbers n such that T(n) + T(n+1) + ... + T(n+26) is a square, where T = A000217 (triangular numbers).

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A257710 Numbers n such that T(n) + T(n+1) + ... + T(n+36) is a square, where T = A000217 (triangular numbers).

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