A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Author

Paul Curtz, Aug 15 2015

Keywords

• nonn
• easy

Comments

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.

b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.

Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:

0, 1/0, 1/0, 1/0, 1/0, 1/0, 1/0, 1/0, ...

-1/0, 0, 3/4, 8/9, 15/16, 24/25, 35/36, 48/49, ...

-1/0, -3/4, 0, 5/36, 3/16, 21/100, 2/9, 45/196, ...

-1/0, -8/9, -5/36, 0, 7/144, 16/225, 1/12, 40/441, ...

-1/0, -15/16, -3/16, -7/144, 0, 9/400, 5/144, 33/784, ...

-1/0, -24/25, -21/100, -16/225, -9/400, 0, 11/900, 24/1225, ...

-1/0, -35/36, -2/9, -1/12, -5/144, -11/900, 0, 13/1764, ...

-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764, 0, ... .

The numerators are almost A165795(n).

Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).

A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).

c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .

c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).

The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
• T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

Programs

Magma

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

Maple

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

Mathematica

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

PARI

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

PARI

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

PARI

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

Python

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

Sage

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

Formula

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).

a(2*k) = 1 + k^2.

a(2*k+1) = 5 + 4*k*(k+1).

a(2*k+1) = 4*a(2*k) + 4*k + 1.

a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019

a(-n) = a(n).

a(n+2) = a(n) + A144433(n) (or A195161(n+1)).

a(n) = A168077(n) + period 2: repeat 1, 4.

a(n) = A171621(n) + period 2: repeat 2, 8.

From Colin Barker, Aug 15 2015: (Start)

a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.

a(n) = n^2/4 + 1 for n even;

a(n) = n^2 + 4 for n odd.

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.

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

E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015

Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

Extensions

New name by Peter Luschny, Mar 18 2022