A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...

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

Offset: 1

Author

Bruno Berselli, Nov 14 2012

Keywords

• nonn
• easy

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).

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.

Programs

Magma

k:=5; f:=func; [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

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 *)

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

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

Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - Amiram Eldar, Mar 15 2022

a(n) = A132356(n-1)/2, n >= 1. - Bernard Schott, Mar 15 2022