cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Original entry on oeis.org

92, 138, 4278, 6532, 205252, 313398, 9847818, 15036572, 472490012, 721442058, 22669672758, 34614182212, 1087671802372, 1660759304118, 52185576841098, 79681832415452, 2503820016570332, 3823067196637578, 120131175218534838, 183427543606188292
Offset: 1

Views

Author

Colin Barker, May 07 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    LinearRecurrence[{0, 48, 0, -1}, {92, 138, 4278, 6532}, 30] (* Vincenzo Librandi, May 11 2015 *)
  • PARI
    Vec(-46*x*(x-1)*(x+2)*(2*x+1)/(x^4-48*x^2+1) + O(x^100))

Formula

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

Views

Author

Colin Barker, May 07 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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))

Formula

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

Views

Author

Colin Barker, May 07 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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))

Formula

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

Views

Author

Colin Barker, May 08 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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))

Formula

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

Views

Author

Colin Barker, May 08 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    LinearRecurrence[{0, 96, 0, -1}, {3854, 5170, 369890, 496226}, 50] (* Vincenzo Librandi, May 11 2015 *)
  • PARI
    Vec(-94*x*(x^3+x^2-55*x-41) / (x^4-96*x^2+1) + O(x^100))

Formula

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

Views

Author

Colin Barker, May 10 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Colin Barker, May 10 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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))

Formula

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

Views

Author

Colin Barker, May 10 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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).

A201633 Numbers k such that Sum_{j=0..3} (k + j)^2 is a triangular number.

Original entry on oeis.org

11, 28, 424, 1001, 14453, 34054, 491026, 1156883, 16680479, 39300016, 566645308, 1335043709, 19249260041, 45352186138, 653908196134, 1540639285031, 22213629408563, 52336383504964, 754609491695056, 1777896399883793, 25634509088223389, 60396141212544046
Offset: 1

Views

Author

Paul Weisenhorn, Jan 09 2013

Keywords

Comments

Sum_{j=0..3} (a(n)+j)^2 = u(n)*(u(n)+1)/2 = t(u(n)) with A201632(n) = u(n) give the Pell equation c(n)^2 - 32*d(n)^2 = 41. 2*u(n)+1 = c(n) and a(n) + 1.5 = d(n).
Also integers k such that k^2 + (k+1)^2 + (k+2)^2 + (k+3)^2 is equal to a hexagonal number. - Colin Barker, Dec 21 2014

Examples

			For n=3: a(3)=424; 424^2+425^2+426^2+427^2=724206.
u(3)=A201632(3)=1203; t(1203)=1203*1204/2=724206.

Links

Crossrefs

Cf. A201632, A218395, A252747.

Programs

  • Mathematica
    LinearRecurrence[{1,34,-34,-1,1},{11,28,424,1001,14453},30] (* Harvey P. Dale, Apr 16 2013 *)
  • PARI
    Vec(x*(x^4+x^3-22*x^2-17*x-11)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)) + O(x^30)) \\ Colin Barker, Dec 21 2014
  • Python
    from functools import cache
@cache
def a(n):
    if n < 6: return [11, 28, 424, 1001, 14453][n-1]
    return a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5)
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Nov 28 2021

Formula

G.f.: (11*x+17*x^2+22*x^3-x^4-x^5)/((1-x)*(1-34*x^2+x^4)). [corrected by Georg Fischer, May 11 2019]
a(n+4) = 34*a(n-2) - a(n-4) + 48; r=sqrt(2).
a(n+5) = a(n+4) + 34*a(n+3) - 34*a(n+2) - a(n+1) + a(n).
Eigenvalues ej: {1,(3+2r),-(3+2r),(3-2*r),-(3-2*r)}.
a(n+1) = (k1*e1+k2*e2^n+k3*e3^n+k4*e4^n+k5*e5^n)/16 for k1=-24, k2=70+50r, k3=30+21r, k4=70-50r, k5=30-21r.

Extensions

More terms from Colin Barker, Dec 21 2014

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

Views

Author

Colin Barker, May 10 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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)).
Showing 1-10 of 13 results. Next

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