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

%I A261327 #117 Jan 15 2023 19:49:36
%S A261327 1,5,2,13,5,29,10,53,17,85,26,125,37,173,50,229,65,293,82,365,101,445,
%T A261327 122,533,145,629,170,733,197,845,226,965,257,1093,290,1229,325,1373,
%U A261327 362,1525,401,1685,442,1853,485,2029,530,2213,577,2405,626,2605,677
%N A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).
%C A261327 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.
%C A261327 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.
%C A261327 Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
%C A261327    0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
%C A261327 -1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
%C A261327 -1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
%C A261327 -1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
%C A261327 -1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
%C A261327 -1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
%C A261327 -1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
%C A261327 -1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
%C A261327 The numerators are almost A165795(n).
%C A261327 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).
%C A261327 A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
%C A261327 c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
%C A261327 c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
%C A261327 The final digit of a(n) is neither 4 nor 8. - _Paul Curtz_, Jan 30 2019
%H A261327 Colin Barker, <a href="/A261327/b261327.txt">Table of n, a(n) for n = 0..1000</a>
%H A261327 Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, <a href="http://www.editions-ellipses.fr/product_info.php?products_id=10284">Le monde des fractales La Nature trans-échelles</a>, 478p., ellipses, Paris, 2015, page 220.
%H A261327 T. A. Witten, Jr. and L. M. Sander, <a href="http://dx.doi.org/10.1103/PhysRevLett.47.1400">Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom</a>, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
%H A261327 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A261327 a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
%F A261327 a(2*k) = 1 + k^2.
%F A261327 a(2*k+1) = 5 + 4*k*(k+1).
%F A261327 a(2*k+1) = 4*a(2*k) + 4*k + 1.
%F A261327 a(4*k+2) = A069894(k). - _Paul Curtz_, Jan 30 2019
%F A261327 a(-n) = a(n).
%F A261327 a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
%F A261327 a(n) = A168077(n) + period 2: repeat 1, 4.
%F A261327 a(n) = A171621(n) + period 2: repeat 2, 8.
%F A261327 From _Colin Barker_, Aug 15 2015: (Start)
%F A261327 a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
%F A261327 a(n) = n^2/4 + 1 for n even;
%F A261327 a(n) = n^2 + 4 for n odd.
%F A261327 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
%F A261327 G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
%F A261327 E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - _Robert Israel_, Aug 18 2015
%F A261327 Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - _Amiram Eldar_, Mar 22 2022
%p A261327 A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # _Wesley Ivan Hurt_, Aug 15 2015
%t A261327 LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* _Vincenzo Librandi_, Aug 15 2015 *)
%t A261327 a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* _Peter Luschny_, Mar 18 2022 *)
%o A261327 (PARI) vector(60, n, n--; numerator(1+n^2/4)) \\ _Michel Marcus_, Aug 15 2015
%o A261327 (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
%o A261327 (PARI) a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ _Charles R Greathouse IV_, Oct 16 2015
%o A261327 (Magma) [Numerator(1+n^2/4): n in [0..60]]; // _Vincenzo Librandi_, Aug 15 2015
%o A261327 (Sage) [numerator(1+n^2/4) for n in (0..60)] # _G. C. Greubel_, Feb 09 2019
%o A261327 (Python) [(n*n+4)//4**((n+1)%2) for n in range(60)] # _Gennady Eremin_, Mar 18 2022
%Y A261327 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.
%Y A261327 Cf. A069894.
%K A261327 nonn,easy
%O A261327 0,2
%A A261327 _Paul Curtz_, Aug 15 2015
%E A261327 New name by _Peter Luschny_, Mar 18 2022