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A261327 Numerators of 1 + n^2/4. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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, 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., 47 (Nov 09 1981), 1400-1403.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

FORMULA

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)).

A195161(n+1) = 1, 8, 3, 16, 5, 24, 7, 32, ... .

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

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 *)

PROG

(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

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

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

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.

Sequence in context: A283944 A294684 A013946 * A330613 A085436 A277710

Adjacent sequences:  A261324 A261325 A261326 * A261328 A261329 A261330

KEYWORD

nonn,easy,less

AUTHOR

Paul Curtz, Aug 15 2015

STATUS

approved

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Last modified August 11 23:45 EDT 2020. Contains 336434 sequences. (Running on oeis4.)