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A144929 Numbers n such that there exists an integer k with (n+1)^3 - n^3 = 7*k^2. 7
1, 166, 18313, 2014318, 221556721, 24369225046, 2680393198393, 294818882598238, 32427396692607841, 3566718817304264326, 392306642506776468073, 43150163956928107223758, 4746125728619585018145361, 522030679984197423888766006, 57418628672533097042746115353, 6315527123298656477278183922878 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

LINKS

Colin Barker, Table of n, a(n) for n = 1..450

Index entries for linear recurrences with constant coefficients, signature (111,-111,1).

FORMULA

a(n+2) = 110*a(n+1)-a(n)+54.

a(n) = -(1/2)+(3/4)*{[55+12*sqrt(21)]^n+[55-12*sqrt(21)]^n}+(1/6)*sqrt(21)*{[55+12*sqrt(21)]^n-[55-12*sqrt(21)]^n }, with n>=0. - Paolo P. Lava, Nov 25 2008

a(n) = 111*a(n-1)-111*a(n-2)+a(n-3), with n>3. - Harvey P. Dale, Jun 11 2011

G.f.: x*(-1-55*x+2*x^2) / ((x-1)*(x^2-110*x+1)). - Harvey P. Dale, Jun 11 2011

a(n) = (A350967(n)-3)/6. - N. J. A. Sloane, Mar 03 2022

EXAMPLE

a(1) = 1 because 2^3-1^3 = 7 = 7*1^2.

MATHEMATICA

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 *)

PROG

(PARI) Vec(x*(-1-55*x+2*x^2)/((x-1)*(x^2-110*x+1)) + O(x^20)) \\ Colin Barker, Jul 14 2016

CROSSREFS

Cf. A144927, A144928, A144930, A350967, A350968.

Sequence in context: A204963 A011815 A188413 * A281934 A212325 A163398

Adjacent sequences: A144926 A144927 A144928 * A144930 A144931 A144932

KEYWORD

nonn,easy

AUTHOR

Richard Choulet, Sep 25 2008

STATUS

approved

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Last modified November 30 19:14 EST 2022. Contains 358453 sequences. (Running on oeis4.)