A257767 -id:A257767 - 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).

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).

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).

A269449 The first of 33 consecutive positive integers the sum of the squares of which is a square.

Original entry on oeis.org

7, 27, 60, 181, 227, 612, 1085, 1985, 3492, 9047, 11161, 28860, 50607, 91987, 161276, 416685, 513883, 1327652, 2327541, 4230121, 7415908, 19159167, 23628161, 61043836, 107016983, 194494283, 340971196, 880905701, 1086382227, 2806689508, 4920454381, 8942507601

Offset: 1

Colin Barker, Feb 27 2016

Positive integers y in the solutions to 2*x^2-66*y^2-2112*y-22880 = 0.
All sequences of this type (i.e. sequences with fixed offset k, and a discernible pattern: k=0...32 for this sequence, k=0..1 for A001652, k=0...10 for A106521) can be extended using a formula such as x(n) = a*x(n-p) - x(n-2p) + b, where a and b are various constants, and p is the period of the series. Alternatively 'p' can be considered the number of concurrent series. - Daniel Mondot, Aug 08 2016
Numbers x such that 11440+33*x*(32+x)is a square. - Harvey P. Dale, Oct 18 2020

			7 is in the sequence because sum(k=7, 39, k^2) = 20449 = 143^2.

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

Cf. A001032, A001652, A094196, A106521, A257767, A269447, A269448, A269451.

Rest@ CoefficientList[Series[x (7 + 20 x + 33 x^2 + 121 x^3 + 46 x^4 + 385 x^5 + 151 x^6 - 20 x^7 - 11 x^8 - 11 x^9 - 2 x^10 - 11 x^11 - 4 x^12)/((1 - x) (1 - 46 x^6 + x^12)), {x, 0, 32}], x] (* Michael De Vlieger, Aug 08 2016 *)
LinearRecurrence[{1,0,0,0,0,46,-46,0,0,0,0,-1,1},{7,27,60,181,227,612,1085,1985,3492,9047,11161,28860,50607},50] (* Harvey P. Dale, Oct 18 2020 *)

Vec(x*(7 +20*x +33*x^2 +121*x^3 +46*x^4 +385*x^5 +151*x^6 -20*x^7 -11*x^8 -11*x^9 -2*x^10 -11*x^11 -4*x^12) / ((1 -x)*(1 -46*x^6 +x^12)) + O(x^40))

G.f.: x*(7 +20*x +33*x^2 +121*x^3 +46*x^4 +385*x^5 +151*x^6 -20*x^7 -11*x^8 -11*x^9 -2*x^10 -11*x^11 -4*x^12) / ((1 -x)*(1 -46*x^6 +x^12)).

a(1)=7, a(2)=27, a(3)=60, a(4)=181, a(5)=227, a(6)=612, a(7)=1085, a(8)=1985, a(9)=3492, a(10)=9047, a(11)=11161, a(12)=28860, a(n)=46*a(n-6)-a(n-12)+704. - Daniel Mondot, Aug 08 2016

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)).

A257825 Positive integers whose square is the sum of 74 consecutive squares.

Original entry on oeis.org

2257, 2849, 21941, 27713, 604765, 763865, 16669573, 21054961, 162316669, 205018517, 4474051285, 5651073085, 123321498797, 155764598629, 1200818695321, 1516726961053, 33099030801665, 41806637918965, 912332431430633, 1152346479602381, 8883656545668089

Offset: 1

Colin Barker, May 10 2015

Positive integers x in the solutions to 2*x^2-148*y^2-10804*y-264698 = 0.

			2257 is in the sequence because 2257^2 = 5094049 = 225^2+226^2+...+298^2.

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

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823, A257824, A257826-A257828.

I:=[2257,2849,21941,27713,604765,763865,16669573, 21054961,162316669,205018517,4474051285,5651073085]; [n le 12 select I[n] else 7398*Self(n-6)-Self(n-12): n in [1..40]]; // Vincenzo Librandi, May 11 2015

 LinearRecurrence[{0, 0, 0, 0, 0, 7398, 0, 0, 0, 0, 0, -1}, {2257, 2849, 21941, 27713, 604765, 763865, 16669573, 21054961, 162316669, 205018517, 4474051285, 5651073085}, 40] (* Vincenzo Librandi, May 11 2015 *)

Vec(-37*x*(5*x^11+5*x^10+61*x^9+77*x^8+593*x^7+749*x^6-20645*x^5-16345*x^4-749*x^3-593*x^2-77*x-61) / ((x^6-86*x^3-1)*(x^6+86*x^3-1)) + O(x^100))

a(n) = 7398*a(n-6)-a(n-12).

G.f.: -37*x*(5*x^11+5*x^10+61*x^9+77*x^8+593*x^7+749*x^6-20645*x^5-16345*x^4-749*x^3-593*x^2-77*x-61) / ((x^6-86*x^3-1)*(x^6+86*x^3-1)).

cp's OEIS Frontend

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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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.

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