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A219391 Numbers n such that 21*n+1 is a square. 4
0, 3, 8, 19, 23, 40, 55, 80, 88, 119, 144, 183, 195, 240, 275, 328, 344, 403, 448, 515, 535, 608, 663, 744, 768, 855, 920, 1015, 1043, 1144, 1219, 1328, 1360, 1475, 1560, 1683, 1719, 1848, 1943, 2080, 2120, 2263, 2368, 2519, 2563, 2720, 2835, 3000, 3048, 3219 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equivalently, numbers in increasing order of the form m*(21*m+2) or m*(21*m+16)+3, where m = 0,-1,1,-2,2,-3,3,...

LINKS

Bruno Berselli, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

FORMULA

G.f.: x^2*(3 + 5*x + 11*x^2 + 4*x^3 + 11*x^4 + 5*x^5 + 3*x^6)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).

a(n) = a(-n+1) = (42*n*(n-1) + 2*i^(n*(n+1))*(6*n + (-1)^n-3) + 7*(-1)^n*(2*n-1) + 11)/32, where i=sqrt(-1).

MAPLE

A219391:=proc(q)

local n;

for n from 1 to q do if type(sqrt(21*n+1), integer) then print(n);

fi; od; end:

A219391(1000); # Paolo P. Lava, Feb 19 2013

MATHEMATICA

Select[Range[0, 3300], IntegerQ[Sqrt[21 # + 1]] &]

CoefficientList[Series[x (3 + 5 x + 11 x^2 + 4 x^3 + 11 x^4 + 5 x^5 + 3 x^6)/((1 + x)^2 (1 - x)^3 (1 + x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(MAGMA) [n: n in [0..3300] | IsSquare(21*n+1)];

(Maxima) makelist((42*n*(n-1)+2*%i^(n*(n+1))*(6*n+(-1)^n-3)+7*(-1)^n*(2*n-1)+11)/32, n, 1, 50);

(MAGMA) I:=[0, 3, 8, 19, 23, 40, 55, 80, 88]; [n le 9 select I[n] else Self(n-1)+2*Self(n-4)-2*Self(n-5)-Self(n-8)+Self(n-9): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

CROSSREFS

Cf. similar sequences listed in A219257.

Cf. A219721 (square roots of 21*a(n)+1).

Subsequence of A047528.

Sequence in context: A023371 A124086 A091109 * A123982 A171308 A113535

Adjacent sequences:  A219388 A219389 A219390 * A219392 A219393 A219394

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Nov 20 2012

STATUS

approved

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Last modified March 27 13:47 EDT 2017. Contains 284176 sequences.