A112524 a(n) = a(n-1) + 2*n^2 with a(1) = 1.

1, 9, 27, 59, 109, 181, 279, 407, 569, 769, 1011, 1299, 1637, 2029, 2479, 2991, 3569, 4217, 4939, 5739, 6621, 7589, 8647, 9799, 11049, 12401, 13859, 15427, 17109, 18909, 20831, 22879, 25057, 27369, 29819, 32411, 35149, 38037, 41079, 44279, 47641

Offset: 1

Author

Dennis Farr (dfarr(AT)comcast.net), Dec 13 2005

Keywords

• easy
• nonn

Comments

This is the total number of operations or total storage if a process first replaces a square array by an array one smaller, repeatedly down to 1 and then regrows the array to the original size.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Condensation
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A006331.

Programs

Magma

[n*(n+1)*(2*n+1)/3 - 1: n in [1..40]]; // G. C. Greubel, Jan 12 2022

Maple

a[1]:=1: for n from 2 to 50 do a[n]:=a[n-1]+2*n^2 od: seq(a[n],n=1..50); # Emeric Deutsch, Feb 13 2006
a:=n->sum(k^2, k=1..n):seq(a(n)+sum(k^2, k=2..n), n=1...40); # Zerinvary Lajos, Jun 11 2008

Mathematica

Table[n*(n+1)*(2n+1)/3 - 1, {n, 50}] (* Stefan Steinerberger, Mar 11 2006 *)
2*Accumulate[Range[50]^2]-1 (* or *) LinearRecurrence[{4,-6,4,-1},{1,9,27,59},50] (* Harvey P. Dale, Dec 03 2012 *)

Sage

[n*(n+1)*(2*n+1)/3 - 1 for n in (1..40)] # G. C. Greubel, Jan 12 2022

Formula

Twice the sum of the first n square numbers - 1 = n*(n + 1)*(2n + 1)/3 - 1. - Stefan Steinerberger, Mar 11 2006

From R. J. Mathar, Sep 09 2008: (Start)

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

a(n) = A006331(n) - 1. (End)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(1)=1, a(2)=9, a(3)=27, a(4)=59. - Harvey P. Dale, Dec 03 2012

E.g.f.: ( 3 + (-3 + 6*x + 9*x^2 + 2*x^3)*exp(x) )/3. - G. C. Greubel, Jan 12 2022

Extensions

Definition corrected by Alexandre Wajnberg, Jan 02 2006

More terms from Emeric Deutsch, Feb 13 2006

More terms from Stefan Steinerberger, Mar 11 2006