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Search: a257826 -id:a257826
Displaying 1-3 of 3 results found.
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A257828
Positive integers whose square is the sum of 97 consecutive squares.
+10
11
679, 1545404, 3675742735, 81619738879, 194132514608060, 461744104375531831, 10253011689091642135, 24386783991798773338556, 58003955471481693294113311, 1287975802673112210113634031, 3063449905150311732357259611836, 7286414311424213782299531873117895
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers x in the solutions to 2*x^2-194*y^2-18624*y-599072 = 0.
LINKS
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).
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).
EXAMPLE
679 is in the sequence because 679^2 = 461041 = 15^2+16^2+...+111^2.
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 *)
PROG
(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))
(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
CROSSREFS
Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257826, A257827.
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 10 2015
STATUS
approved
A257825
Positive integers whose square is the sum of 74 consecutive squares.
+10
3
2257, 2849, 21941, 27713, 604765, 763865, 16669573, 21054961, 162316669, 205018517, 4474051285, 5651073085, 123321498797, 155764598629, 1200818695321, 1516726961053, 33099030801665, 41806637918965, 912332431430633, 1152346479602381, 8883656545668089
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers x in the solutions to 2*x^2-148*y^2-10804*y-264698 = 0.
LINKS
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).
FORMULA
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)).
EXAMPLE
2257 is in the sequence because 2257^2 = 5094049 = 225^2+226^2+...+298^2.
MATHEMATICA
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 *)
PROG
(PARI) 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))
(Magma) 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
CROSSREFS
Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823, A257824, A257826-A257828.
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 10 2015
STATUS
approved
A257827
Positive integers whose square is the sum of 96 consecutive squares.
+10
3
652, 724, 788, 1012, 1828, 2372, 2596, 2908, 6164, 6908, 7564, 9836, 17996, 23404, 25628, 28724, 60988, 68356, 74852, 97348, 178132, 231668, 253684, 284332, 603716, 676652, 740956, 963644, 1763324, 2293276, 2511212, 2814596, 5976172, 6698164, 7334708
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Positive integers x in the solutions to 2*x^2-192*y^2-18240*y-580640 = 0.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-1).
FORMULA
a(n) = 10*a(n-8) -a(n-16).
G.f.: -4*x*(89*x^15 +83*x^14 +79*x^13 +71*x^12 +71*x^11 +79*x^10 +83*x^9 +89*x^8 -727*x^7 -649*x^6 -593*x^5 -457*x^4 -253*x^3 -197*x^2 -181*x-163) / (x^16 -10*x^8 +1).
EXAMPLE
652 is in the sequence because 652^2 = 425104 = 13^2+14^2+...+108^2.
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, -1}, {652, 724, 788, 1012, 1828, 2372, 2596, 2908, 6164, 6908, 7564, 9836, 17996, 23404, 25628, 28724}, 40] (* Vincenzo Librandi, May 11 2015 *)
PROG
(PARI)
Vec(-4*x*(89*x^15 +83*x^14 +79*x^13 +71*x^12 +71*x^11 +79*x^10 +83*x^9 +89*x^8 -727*x^7 -649*x^6 -593*x^5 -457*x^4 -253*x^3 -197*x^2 -181*x -163) / (x^16-10*x^8+1) + O(x^100))
(Magma) I:=[652, 724, 788, 1012, 1828, 2372, 2596, 2908, 6164, 6908, 7564, 9836, 17996, 23404, 25628, 28724]; [n le 16 select I[n] else 10*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, May 11 2015
CROSSREFS
Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257826, A257828.
KEYWORD
nonn,easy
AUTHOR
Colin Barker, May 10 2015
STATUS
approved
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