A118057 a(n) = 8*n^2 - 4*n - 3.

1, 21, 57, 109, 177, 261, 361, 477, 609, 757, 921, 1101, 1297, 1509, 1737, 1981, 2241, 2517, 2809, 3117, 3441, 3781, 4137, 4509, 4897, 5301, 5721, 6157, 6609, 7077, 7561, 8061, 8577, 9109, 9657, 10221, 10801, 11397, 12009, 12637, 13281, 13941, 14617

Offset: 1

Author

Charlie Marion, Apr 26 2006

Keywords

• nonn
• easy

Comments

In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(1)=A001652(1)=3 and z(1)=A001653(1)=5; cf. A000290, A118058-A118061.

Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 57, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011

Examples

			a(3)=8*3^2-4*3-3=57, a(4)=8*4^2-4*4-3=109 and 57+58+...+86=87+...+108.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A004767, A139098.

Cf. A000290, A001652, A001653, A118058-A118061.

Programs

Magma

[8*n^2-4*n-3 : n in [1..60]]; // Wesley Ivan Hurt, Jan 28 2021

Mathematica

Table[8n^2-4n-3,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,57},50] (* Harvey P. Dale, Sep 18 2012 *)

PARI

a(n)=8*n^2-4*n-3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+18*x-3*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012

a(n)+(a(n)+1)+...+(a(n)+8n+5)=(a(n)+8n+6)+...+a(n+1)-1; a(n+1)-1=a(n)+16n+3.

a(n)+(a(n)+1)+...+(a(n)+8n+5)=(4n-1)(4n+1)(4n+3); e.g., 21+22+...+56=693=7*9*11.

a(n) = 16*n+a(n-1)-12 (with a(1)=1). - Vincenzo Librandi, Nov 13 2010

a(n) = A139098(n) - A004767(n). - Omar E. Pol, Sep 18 2012