A016814 a(n) = (4*n + 1)^2.

1, 25, 81, 169, 289, 441, 625, 841, 1089, 1369, 1681, 2025, 2401, 2809, 3249, 3721, 4225, 4761, 5329, 5929, 6561, 7225, 7921, 8649, 9409, 10201, 11025, 11881, 12769, 13689, 14641, 15625, 16641, 17689, 18769, 19881, 21025, 22201, 23409, 24649, 25921, 27225, 28561, 29929

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A bisection of A016754. Sequence arises from reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..200 from Ivan Panchenko).
• Leo Tavares, Illustration: Square trapeziums
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), this sequence (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Cf. A001539, A006752, A016742, A016802, A016813.

Cf. A016826, A016838, A017126, A068467, A087197.

Cf. A272399, A014105.

Programs

Magma

[(4*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

Maple

A016814:=n->(4*n+1)^2; seq(A016814(k), k=0..100); # Wesley Ivan Hurt, Nov 02 2013

Mathematica

(4*Range[0,40] +1)^2 (* or *) LinearRecurrence[{3,-3,1}, {1,25,81}, 40] (* Harvey P. Dale, Nov 20 2012 *)
Accumulate[32Range[0, 47] - 8] + 9 (* Alonso del Arte, Aug 19 2017 *)

PARI

a(n)=(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[(4*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

Formula

a(n) = a(n-1) + 32*n - 8, n > 0. - Vincenzo Librandi, Dec 15 2010

From George F. Johnson, Sep 28 2012: (Start)

G.f.: (1 + 22*x + 9*x^2)/(1 - x)^3.

a(n+1) = a(n) + 16 + 8*sqrt(a(n)).

a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).

a(n-1)*a(n+1) = (a(n) - 16)^2 ; a(n+1) - a(n-1) = 16*sqrt(a(n)).

a(n) = A016754(2*n) = (A016813(n))^2. (End)

Sum_{n>=0} 1/a(n) = G/2 + Pi^2/16, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020

Product_{n>=1} (1 - 1/a(n)) = 2*Gamma(5/4)^2/sqrt(Pi) = 2 * A068467^2 * A087197. - Amiram Eldar, Feb 01 2021

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A017078(n).

a(2*n+1) = A017126(n).

E.g.f.: (1 + 24*x + 16*x^2)*exp(x). (End)

a(n) = A272399(n+1) - A014105(n). - Leo Tavares, Dec 24 2023