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A219190 Numbers of the form n*(5*n+1), where n = 0,-1,1,-2,2,-3,3,... 5
0, 4, 6, 18, 22, 42, 48, 76, 84, 120, 130, 174, 186, 238, 252, 312, 328, 396, 414, 490, 510, 594, 616, 708, 732, 832, 858, 966, 994, 1110, 1140, 1264, 1296, 1428, 1462, 1602, 1638, 1786, 1824, 1980, 2020, 2184, 2226, 2398, 2442, 2622, 2668, 2856, 2904, 3100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equivalently, numbers m such that 20*m+1 is a square.

Also, integer values of h*(h+1)/5.

More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:

. generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);

. n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;

. first differences: (n-1)*((-1)^n*(k-2)+k)/2;

. b(2n+1)-b(2n) = 2*n (independent from k);

. (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.

LINKS

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

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

FORMULA

G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.

a(n) = 2*A057569(n) = (1/5)*A008851(n+1)*A047208(n).

a(1)=0, a(2)=4, a(3)=6, a(4)=18, a(5)=22, a(n)=a(n-1)+2*a(n-2)- 2*a(n-3)- a(n-4)+a (n-5). - Harvey P. Dale, Jan 21 2015

MATHEMATICA

Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]

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

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 4, 6, 18, 22}, 50] (* Harvey P. Dale, Jan 21 2015 *)

PROG

(MAGMA) k:=5; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..25]];

(MAGMA) I:=[0, 4, 6, 18, 22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

CROSSREFS

Subsequence of A011858.

Cf. A090771: square roots of 20*a(n)+1 (see the first comment).

Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.

Cf. similar sequences listed in A219257.

Sequence in context: A102020 A125133 A109310 * A120391 A064217 A026623

Adjacent sequences:  A219187 A219188 A219189 * A219191 A219192 A219193

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Nov 14 2012

STATUS

approved

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)