A180274 -id:A180274 - cp's OEIS frontend

A257761 Positive integers whose square is the sum of 23 consecutive squares.

92, 138, 4278, 6532, 205252, 313398, 9847818, 15036572, 472490012, 721442058, 22669672758, 34614182212, 1087671802372, 1660759304118, 52185576841098, 79681832415452, 2503820016570332, 3823067196637578, 120131175218534838, 183427543606188292

Offset: 1

Colin Barker, May 07 2015

Positive integers x in the solutions to 2*x^2-46*y^2-1012*y-7590 = 0.

			92 is in the sequence because 92^2 = 8464 = 7^2+8^2+...+29^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,48,0,-1).

Cf. A001032, A001653, A180274, A218395, A257765, A257767, A257780, A257781, A257823-A257828.

I:=[92,138,4278,6532]; [n le 4 select I[n] else 48*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 48, 0, -1}, {92, 138, 4278, 6532}, 30] (* Vincenzo Librandi, May 11 2015 *)

Vec(-46*x*(x-1)*(x+2)*(2*x+1)/(x^4-48*x^2+1) + O(x^100))

a(n) = 48*a(n-2)-a(n-4).

G.f.: -46*x*(x-1)*(x+2)*(2*x+1) / (x^4-48*x^2+1).

A257765 Positive integers whose square is the sum of 26 consecutive squares.

Original entry on oeis.org

195, 1599, 2379, 19695, 163059, 242619, 2008695, 16630419, 24744759, 204867195, 1696139679, 2523722799, 20894445195, 172989616839, 257394980739, 2131028542695, 17643244777899, 26251764312579, 217344016909695, 1799437977728859, 2677422564902319

Offset: 1

Colin Barker, May 07 2015

Positive integers x in the solutions to 2*x^2-52*y^2-1300*y-11050 = 0.

			195 is in the sequence because 195^2 = 38025 = 25^2+26^2+...+50^2.

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

Cf. A001032, A001653, A180274, A218395, A257761, A257767, A257780, A257781, A257823-A257828.

I:=[195,1599,2379,19695,163059,242619 ]; [n le 6 select I[n] else 102*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 102, 0, 0, -1}, {195, 1599, 2379, 19695, 163059, 242619}, 30] (* Vincenzo Librandi, May 11 2015 *)
Select[Sqrt[#]&/@Total/@Partition[Range[10^6]^2,26,1],IntegerQ] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Mar 10 2024 *)

Vec(-39*x*(x^5+x^4+5*x^3-61*x^2-41*x-5) / (x^6-102*x^3+1) + O(x^100))

a(n) = 102*a(n-3)-a(n-6).

G.f.: -39*x*(x^5+x^4+5*x^3-61*x^2-41*x-5) / (x^6-102*x^3+1).

A257767 Positive integers whose square is the sum of 33 consecutive squares.

Original entry on oeis.org

143, 253, 440, 1133, 1397, 3608, 6325, 11495, 20152, 52063, 64207, 165880, 290807, 528517, 926552, 2393765, 2952125, 7626872, 13370797, 24300287, 42601240, 110061127, 135733543, 350670232, 614765855, 1117284685, 1958730488, 5060418077, 6240790853

Offset: 1

Colin Barker, May 07 2015

Positive integers x in the solutions to 2*x^2-66*y^2-2112*y-22880 = 0.

			143 is in the sequence because 143^2 = 20449 = 7^2+8^2+...+39^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,46,0,0,0,0,0,-1).

Cf. A001032, A001653, A180274, A218395, A257761, A257765, A257780, A257781, A257823-A257828.

I:=[143,253,440,1133,1397,3608,6325,11495,20152, 52063,64207,165880]; [n le 12 select I[n] else 46*Self(n-6)-Self(n-12): n in [1..30]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 0, 0, 0, 46, 0, 0, 0, 0, 0, -1}, {143, 253, 440, 1133, 1397, 3608, 6325, 11495, 20152, 52063, 64207, 165880}, 50] (* Vincenzo Librandi, May 08 2015 *)

Vec(-11*x*(8*x^11+5*x^10+5*x^9+8*x^8+13*x^7+23*x^6-328*x^5-127*x^4-103*x^3-40*x^2-23*x-13) / (x^12-46*x^6+1) + O(x^100))

a(n) = 46*a(n-6)-a(n-12).

G.f.: -11*x*(8*x^11+5*x^10+5*x^9+8*x^8+13*x^7+23*x^6-328*x^5-127*x^4-103*x^3-40*x^2-23*x-13) / (x^12-46*x^6+1).

A257781 Positive integers whose square is the sum of 50 consecutive squares.

Original entry on oeis.org

245, 385, 495, 655, 795, 1055, 1365, 2205, 2855, 3795, 4615, 6135, 7945, 12845, 16635, 22115, 26895, 35755, 46305, 74865, 96955, 128895, 156755, 208395, 269885, 436345, 565095, 751255, 913635, 1214615, 1573005, 2543205, 3293615, 4378635, 5325055, 7079295

Offset: 1

Colin Barker, May 08 2015

Positive integers x in the solutions to 2*x^2-100*y^2-4900*y-80850 = 0.

			245 is in the sequence because 245^2 = 60025 = 7^2+8^2+...+56^2.

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

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257823-A257828.

I:=[245,385,495,655,795,1055,1365,2205,2855,3795, 4615,6135]; [n le 12 select I[n] else 6*Self(n-6)-Self(n-12): n in [1..40]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -1}, {245, 385, 495, 655, 795, 1055, 1365, 2205, 2855, 3795, 4615, 6135}, 50] (* Vincenzo Librandi, May 11 2015 *)
Select[Sqrt[Total/@Partition[Range[10^6]^2,50,1]],IntegerQ] (* Harvey P. Dale, Aug 07 2025 *)

Vec(-5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)) + O(x^100))

a(n) = 6*a(n-6)-a(n-12).

G.f.: -5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)).

A257780 Positive integers whose square is the sum of 47 consecutive squares.

Original entry on oeis.org

3854, 5170, 369890, 496226, 35505586, 47632526, 3408166366, 4572226270, 327148465550, 438886089394, 31402844526434, 42128492355554, 3014345926072114, 4043896380043790, 289345806058396510, 388171923991848286, 27774183035679992846, 37260460806837391666

Offset: 1

Colin Barker, May 08 2015

Positive integers x in the solutions to 2*x^2-94*y^2-4324*y-67022 = 0.

			3854 is in the sequence because 3854^2 = 14853316 = 539^2+540^2+...+585^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,96,0,-1).

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257781, A257823-A257828.

I:=[3854,5170,369890,496226 ]; [n le 4 select I[n] else 96*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 96, 0, -1}, {3854, 5170, 369890, 496226}, 50] (* Vincenzo Librandi, May 11 2015 *)

Vec(-94*x*(x^3+x^2-55*x-41) / (x^4-96*x^2+1) + O(x^100))

a(n) = 96*a(n-2)-a(n-4).

G.f.: -94*x*(x^3+x^2-55*x-41) / (x^4-96*x^2+1).

A257823 Positive integers whose square is the sum of 59 consecutive squares.

Original entry on oeis.org

413, 531, 8673, 11269, 426511, 554187, 9192849, 11944727, 452101247, 587437689, 9744411267, 12661399351, 479226895309, 622683396153, 10329066750171, 13421071367333, 507980056926293, 660043812484491, 10948801010769993, 14226322987973629, 538458381114975271

Offset: 1

Colin Barker, May 10 2015

Positive integers x in the solutions to 2*x^2-118*y^2-6844*y-133458 = 0.

			413 is in the sequence because 413^2 = 170569 = 22^2+23^2+...+80^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1060,0,0,0,-1).

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257824-A257828.

I:=[413,531,8673,11269,426511,554187,9192849, 11944727]; [n le 8 select I[n] else 1060*Self(n-4)-Self(n-8): n in [1..30]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 0, 1060, 0, 0, 0, -1}, {413, 531, 8673, 11269, 426511, 554187, 9192849, 11944727}, 30] (* Vincenzo Librandi, May 11 2015 *)

Vec(-59*x*(x-1)*(7*x^6+16*x^5+163*x^4+354*x^3+163*x^2+16*x+7) / (x^8-1060*x^4+1) + O(x^100))

a(n) = 1060*a(n-4)-a(n-8).

G.f.: -59*x*(x-1)*(7*x^6+16*x^5+163*x^4+354*x^3+163*x^2+16*x+7) / (x^8-1060*x^4+1).

A257828 Positive integers whose square is the sum of 97 consecutive squares.

Original entry on oeis.org

679, 1545404, 3675742735, 81619738879, 194132514608060, 461744104375531831, 10253011689091642135, 24386783991798773338556, 58003955471481693294113311, 1287975802673112210113634031, 3063449905150311732357259611836, 7286414311424213782299531873117895

Offset: 1

Colin Barker, May 10 2015

Positive integers x in the solutions to 2*x^2-194*y^2-18624*y-599072 = 0.

			679 is in the sequence because 679^2 = 461041 = 15^2+16^2+...+111^2.

Colin Barker, Table of n, a(n) for n = 1..370
Index entries for linear recurrences with constant coefficients, signature (0,0,125619266,0,0,-1).

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257826, A257827.

I:=[679,1545404,3675742735,81619738879, 194132514608060,461744104375531831]; [n le 6 select I[n] else 125619266*Self(n-3)-Self(n-6): n in [1..20]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 125619266, 0, 0, -1}, {679, 1545404, 3675742735, 81619738879, 194132514608060, 461744104375531831}, 30] (* Vincenzo Librandi, May 11 2015 *)
Rest[CoefficientList[Series[-679x(x-1)(x^4+2277x^3+5415742x^2+ 2277x+1)/ (x^6-125619266x^3+1),{x,0,15}],x]] (* Harvey P. Dale, Aug 02 2021 *)

Vec(-679*x*(x-1)*(x^4+2277*x^3+5415742*x^2+2277*x+1) / (x^6-125619266*x^3+1) + O(x^100))

a(n) = 125619266*a(n-3)-a(n-6).

G.f.: -679*x*(x-1)*(x^4+2277*x^3+5415742*x^2+2277*x+1) / (x^6-125619266*x^3+1).

A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 15, 17, 18, 20, 21, 22, 25, 27, 28, 30, 32, 38, 44, 50, 52, 55, 58, 60, 64, 65, 67, 73, 74, 76, 83, 87, 91, 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200, 204

Offset: 1

Zhining Yang, Jan 19 2011

That is, numbers n such that Sum_{i=n..k} i^2 is a square for some k > n.
The paper by Bremner, Stroeker, and Tzanakis describes how they found all n <= 100 by solving elliptic curves. Their solutions are the same as the terms in this sequence. They also show that there are only a finite number of sums of squares beginning with n^2 that sum to a square. For example, starting with 3^2, there are only 3 ways to sum consecutive squares to produce a square: 3^2 + 4^2, 3^2 + ... + 580^2, and 3^2 + ... + 963^2. See A184762, A184763, A184885, and A184886 for more results from their paper.
This sequence is more difficult than A001032, which has the possible lengths of the sequences of consecutive squares that sum to a square. Be careful adding terms to this sequence; a simple search may miss some terms. An elliptic curve needs to be solved for each number.
It is conjectured that the sequence continues 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200. - Jean-François Alcover, Sep 17 2013. Conjecture confirmed (see the Schoenfield link below). - Jon E. Schoenfield, Nov 22 2013

			30 is in the sequence because 30^2 + 31^2 + 32^2 + ... + 197^2 + 198^2 = 1612^2.

Jon E. Schoenfield, Table of n, a(n) for n = 1..123 (includes a derivation of the elliptic curves and Magma code used to find the terms)
A. Bremner, R. J. Stroeker, N. Tzanakis, On Sums of Consecutive Squares, J. Number Theory 62 (1997), 39-70.
K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Masoto Kuwata, Jaap Top, An elliptic surface related to sums of consecutive squares, Exposition. Math. 12 (1994) 181-192

Cf. A180259, A180273, A180274, A180465.

Cf. A075404, A075405, A075406.

jmax[11] = jmax[74] = 10^5; jmax[n_ /; n > 91] = 10^6; jmax[] = 10^4; Reap[For[n = 1, n <= 200, n++, s = n^2; For[j = n+1, j <= jmax[n], j++, s += j^2; If[IntegerQ[Sqrt[s]], Sow[n]; Print[n, "(", j, ", ", Sqrt[s], ")"]; Break[]]]]][[2, 1]] (* _Jean-François Alcover, Sep 17 2013, translated and adapted from Pari *)

for(n=1, 100,s=n^2;for(j=n+1,999999,s+=j^2; if(issquare(s), print1(n, "(",j,",",sqrtint(s),")");break())))

Numbers n such that A075404(n) > 0.

Example simplified by Jon E. Schoenfield, Sep 18 2013

More terms from Jon E. Schoenfield, Nov 22 2013

A257826 Positive integers whose square is the sum of 88 consecutive squares.

Original entry on oeis.org

2222, 2530, 39358, 55990, 872938, 994598, 15506810, 22059818, 343935350, 391869082, 6109643782, 8691512302, 135509654962, 154395423710, 2407184143298, 3424433787170, 53390460119678, 60831405072658, 948424442815630, 1349218220632678, 21035705777498170

Offset: 1

Colin Barker, May 10 2015

Positive integers x in the solutions to 2*x^2-176*y^2-15312*y-446600 = 0.

			2222 is in the sequence because 2222^2 = 4937284 = 192^2+193^2+...+279^2.

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

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257825, A257827, A257828.

I:=[2222,2530,39358,55990,872938,994598,15506810, 22059818]; [n le 8 select I[n] else 394*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 0, 394, 0, 0, 0, -1}, {2222, 2530, 39358, 55990, 872938, 994598, 15506810, 22059818}, 40] (* Vincenzo Librandi, May 11 2015 *)

Vec(-22*x*(11*x^7+11*x^6+101*x^5+115*x^4-2545*x^3-1789*x^2-115*x-101) / (x^8-394*x^4+1) + O(x^100))

a(n) = 394*a(n-4)-a(n-8).

G.f.: -22*x*(11*x^7+11*x^6+101*x^5+115*x^4-2545*x^3-1789*x^2-115*x-101) / (x^8-394*x^4+1).

A257824 Positive integers whose square is the sum of 73 consecutive squares.

Original entry on oeis.org

4088, 23360, 1582640, 9047912, 18642443912, 106578370640, 7220791811360, 41281080400088, 85056113063608088, 486263602888235360, 32944848197744794640, 188344846763231651912, 388068345740467131839912, 2218576715650261475158640, 150310804012507009263599360

Offset: 1

Colin Barker, May 10 2015

Positive integers x in the solutions to 2*x^2-146*y^2-10512*y-254040 = 0.

			4088 is in the sequence because 4088^2 = 16711744 = 442^2+443^2+...+514^2.

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4562498,0,0,0,-1).

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823, A257825-A257828.

I:=[4088,23360,1582640,9047912,18642443912, 106578370640,7220791811360,41281080400088]; [n le 8 select I[n] else 4562498*Self(n-4)-Self(n-8): n in [1..20]]; // Vincenzo Librandi, May 11 2015

LinearRecurrence[{0, 0, 0, 4562498, 0, 0, 0, -1}, {4088, 23360, 1582640, 9047912, 18642443912, 106578370640, 7220791811360, 41281080400088}, 40] (* Vincenzo Librandi, May 11 2015 *)

Vec(-584*x*(x-1)*(7*x^6+47*x^5+2757*x^4+18250*x^3+2757*x^2+47*x+7) / ((x^4-2136*x^2-1)*(x^4+2136*x^2-1)) + O(x^100))

a(n) = 4562498*a(n-4)-a(n-8).

G.f.: -584*x*(x-1)*(7*x^6+47*x^5+2757*x^4+18250*x^3+2757*x^2+47*x+7) / ((x^4-2136*x^2-1)*(x^4+2136*x^2-1)).

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A257761 Positive integers whose square is the sum of 23 consecutive squares.

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A257765 Positive integers whose square is the sum of 26 consecutive squares.

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A257767 Positive integers whose square is the sum of 33 consecutive squares.

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A257781 Positive integers whose square is the sum of 50 consecutive squares.

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A257780 Positive integers whose square is the sum of 47 consecutive squares.

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A257823 Positive integers whose square is the sum of 59 consecutive squares.

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A257828 Positive integers whose square is the sum of 97 consecutive squares.