Original entry on oeis.org
9, 999, 109881, 12085911, 1329340329, 146215350279, 16082359190361, 1768913295589431, 194564380155647049, 21400312903825585959, 2353839855040658808441, 258900983741568643342551, 28476754371717510108872169, 3132184079905184543332596039, 344511772035198582256476692121
Offset: 1
- E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185.
A144929
Numbers n such that there exists an integer k with (n+1)^3 - n^3 = 7*k^2.
Original entry on oeis.org
1, 166, 18313, 2014318, 221556721, 24369225046, 2680393198393, 294818882598238, 32427396692607841, 3566718817304264326, 392306642506776468073, 43150163956928107223758, 4746125728619585018145361, 522030679984197423888766006, 57418628672533097042746115353, 6315527123298656477278183922878
Offset: 1
a(1) = 1 because 2^3-1^3 = 7 = 7*1^2.
- E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185.
- P.-F. Teilhet, Query 2228, L'Intermédiaire des Mathématiciens, 11 (1904), 44-45. - N. J. A. Sloane, Mar 08 2022
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RecurrenceTable[{a[1]==1,a[2]==166,a[n]==54+110a[n-1]-a[n-2]},a[n],{n,20}] (* or *) LinearRecurrence[{111,-111,1},{1,166,18313},20] (* Harvey P. Dale, Jun 11 2011 *)
Rest@ CoefficientList[Series[x (-1 - 55 x + 2 x^2)/((x - 1) (x^2 - 110 x + 1)), {x, 0, 16}], x] (* or *) Last /@ Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && (n + 1)^3 - n^3 == 7 k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 16}] // Union (* Michael De Vlieger, Jul 14 2016 *)
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Vec(x*(-1-55*x+2*x^2)/((x-1)*(x^2-110*x+1)) + O(x^20)) \\ Colin Barker, Jul 14 2016
A144927
Numbers n such that there exists an integer k with (n+7)^3-n^3=k^2.
Original entry on oeis.org
7, 1162, 128191, 14100226, 1550897047, 170584575322, 18762752388751, 2063732178187666, 226991776848254887, 24967031721129850282, 2746146497547435276511, 302051147698496750566306, 33222880100337095127017527, 3654214759889381967221362042
Offset: 1
a(1)=7 because 14^3-7^3=49^2.
-
Last /@ Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && (n + 7)^3 - n^3 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 14}] (* or *)
Rest@ CoefficientList[Series[7 x (-1 - 55 x + 2 x^2)/((x - 1) (x^2 - 110 x + 1)), {x, 0, 14}], x] (* Michael De Vlieger, Jul 14 2016 *)
LinearRecurrence[{111,-111,1},{7,1162,128191},20] (* Harvey P. Dale, Jul 05 2024 *)
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Vec(7*x*(-1-55*x+2*x^2)/((x-1)*(x^2-110*x+1)) + O(x^20)) \\ Colin Barker, Jul 14 2016
A145693
Numbers X such that there exists Y in N with X^2=21*Y^2+7.
Original entry on oeis.org
14, 1526, 167846, 18461534, 2030600894, 223347636806, 24566209447766, 2702059691617454, 297201999868472174, 32689517925840321686, 3595549769842566913286, 395477785164756520139774, 43498960818353374648461854, 4784490212233706454810664166
Offset: 1
a(1)=14 because the first relation is 14^2=21*3^2+7.
-
I:=[14,1526]; [n le 2 select I[n] else 110*Self(n-1)-Self(n-2): n in [1..15]]; // Vincenzo Librandi, Oct 21 2014
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CoefficientList[Series[14 (1 - x)/(x^2 - 110 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, Oct 21 2014 *)
LinearRecurrence[{110,-1},{14,1526},20] (* Harvey P. Dale, Sep 19 2024 *)
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Vec(-14*x*(x-1)/(x^2-110*x+1) + O(x^20)) \\ Colin Barker, Oct 21 2014
Original entry on oeis.org
4, 499, 54940, 6042955, 664670164, 73107675139, 8041179595180, 884456647794715, 97282190077823524, 10700156451912792979, 1176919927520329404220, 129450491870784321671275, 14238377185858755054436084, 1566092039952592271666298019, 172255886017599291128238346060
Offset: 1
- E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185.
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