login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A118057 a(n) = 8*n^2 - 4*n - 3. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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).

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A004767, A139098.

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

Sequence in context: A043382 A044123 A044504 * A327902 A020148 A037305

Adjacent sequences:  A118054 A118055 A118056 * A118058 A118059 A118060

KEYWORD

nonn,easy

AUTHOR

Charlie Marion, Apr 26 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 2 10:39 EDT 2022. Contains 355004 sequences. (Running on oeis4.)