A171522 -id:A171522 - cp's OEIS frontend

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

Paul Curtz, Aug 15 2015

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

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).

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.

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

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

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 *)

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

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

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

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

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

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

New name by Peter Luschny, Mar 18 2022

A171621 Numerator of 1/4 - 1/n^2, each fourth term multiplied by 4.

Original entry on oeis.org

0, 5, 3, 21, 8, 45, 15, 77, 24, 117, 35, 165, 48, 221, 63, 285, 80, 357, 99, 437, 120, 525, 143, 621, 168, 725, 195, 837, 224, 957, 255, 1085, 288, 1221, 323, 1365, 360, 1517, 399, 1677, 440, 1845, 483, 2021, 528

Offset: 2

Paul Curtz, Dec 13 2009

These are the square roots of the fifth column of the array of denominators mentioned in A171522.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A005563, A010121, A061037, A078371, A171522.

[-(-5+3*(-1)^n)*(-4+n^2)/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018

A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
seq(A171621(n),n=2..90) ; # R. J. Mathar, Apr 02 2011

Table[-(-5+3*(-1)^n)*(-4+n^2)/8, {n,0,100}] (* G. C. Greubel, Sep 19 2018 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,5,3,21,8,45},50] (* Harvey P. Dale, Nov 01 2019 *)

concat(0, Vec(x^3*(-5-3*x-6*x^2+x^3+3*x^4)/((x-1)^3*(1+x)^3) + O(x^100))) \\ Colin Barker, Nov 03 2014

a(n) = A061037(n) * A010121(n+2).
a(2n+2) = A005563(n). a(2n+3) = A078371(n).
G.f.: x^3*(-5-3*x-6*x^2+x^3+3*x^4) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 02 2011
a(n) = -(-5+3*(-1)^n)*(-4+n^2)/8. - Colin Barker, Nov 03 2014
Sum_{n>=3} 1/a(n) = 13/12. - Amiram Eldar, Aug 11 2022

A147560 a(n) = 4*A046162(n+1).

Original entry on oeis.org

0, 4, 16, 12, 64, 100, 48, 196, 256, 108, 400, 484, 192, 676, 784, 300, 1024, 1156, 432, 1444, 1600, 588, 1936, 2116, 768, 2500, 2704, 972, 3136, 3364, 1200, 3844, 4096, 1452, 4624, 4900, 1728, 5476, 5776, 2028, 6400, 6724, 2352, 7396, 7744, 2700, 8464

Offset: 0

Paul Curtz, Nov 07 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000

Companion to A144437.
Cf. A046162.
Cf. A171522. [R. J. Mathar, Dec 15 2009]

[4*Numerator(n^2/(n^2+3*n+3)): n in [0..70]]; // G. C. Greubel, Oct 27 2022

A046162 := proc(n) (n-1)^2/(n^2+n+1) ; numer(%) ; end proc: A147560 := proc(n) 4*A046162(n+1) ; end proc: seq(A147560(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

a[n_] := 4 * Numerator[n^2/(n^2 + 3*n + 3)]; Array[a, 50, 0] (* Amiram Eldar, Aug 14 2022 *)

[4*numerator(n^2/(n^2 +3*n +3)) for n in range(71)] # G. C. Greubel, Oct 27 2022

a(n) = 4*numerator(n^2/(n^2 + 3*n + 3)).
Sum_{n>=1} 1/a(n) = 11*Pi^2/216. - Amiram Eldar, Aug 14 2022
G.f.: 4*x*(1 + 4*x + 3*x^2 + 13*x^3 + 13*x^4 + 3*x^5 + 4*x^6 + x^7)/(1-x^3)^3. - G. C. Greubel, Oct 27 2022

More terms from R. J. Mathar, Dec 15 2009

A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 2, 0, 1, 8, 3, 4, 5, 0, 1, 16, 3, 8, 5, 3, 0, 1, 32, 3, 16, 5, 6, 7, 0, 1, 64, 3, 32, 5, 12, 7, 4, 0, 1, 128, 3, 64, 5, 24, 7, 8, 9, 0, 1, 256, 3, 128, 5, 48, 7, 16, 9, 5, 0, 1, 512, 3, 256, 5, 96, 7, 32, 9, 10, 11, 0, 1, 1024, 3, 512, 5, 192, 7, 64, 9, 20, 11, 6, 0, 1, 2048, 3, 1024, 5

Offset: 0

Paul Curtz, Nov 18 2009

The array is constructed multiplying the even-indexed A026741(k) by 2^n, and keeping the odd-indexed A026471(k) as they are.

Connections to the hydrogen spectrum: The squares of the second row are T(1,k)^2 = A001477(k)^2 = A000290(k) which are the denominators of the Lyman lines (see A171522). The squares of the row T(2,k) are in A154615, denominators of the Balmer series. Row T(3,k) is related to A106833 and A061038.

			The array starts in row n=0 with columns k>=0 as:
0,1,1,3,2,5,3,7,4, A026741
0,1,2,3,4,5,6,7,8, A001477
0,1,4,3,8,5,12,7,16, A022998
0,1,8,3,16,5,24,7,32, A144433
0,1,16,3,32,5,48,7,64,
0,1,32,3,64,5,96,7,128,

A168068 := proc(n,k) if type(k,'odd') then k; else 2^(n-1)*k ; end if; end proc: # R. J. Mathar, Jan 22 2011

A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 8, 1, 0, 7, 3, 15, 1, 0, 9, 16, 21, 24, 1, 0, 11, 5, 1, 2, 35, 1, 0, 13, 24, 33, 40, 45, 48, 1, 0, 15, 7, 39, 3, 55, 15, 63, 1, 0, 17, 32, 5, 56, 65, 8, 77, 80

Offset: 0

Paul Curtz, Dec 04 2010

T(n,0) is set to zero at the pole m=0. T(n,n) is otherwise set to 1 at the pole n=m.
This is the triangle A061035 augmented by a diagonal of 1's.
Essentially the same information is in A120072, A166492, A172157 and A174233.

			The triangle starts in row n=0 with columns 0<=m<=n as:
.1.
.0..1.
.0..3..1.
.0..5..8..1.
.0..7..3.15..1.
.0..9.16.21.24..1.
.0.11..5..1..2.35..1.
.0.13.24.33.40.45.48..1.
.0.15..7.39..3.55.15.63..1.
.0.17.32..5.56.65..8.77.80..1.
.0.19..9.51..4..3.21.91..6.99..1.

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A172157, A166925, A171522 (denominators)

T[n_, n_] := 1; T[n_, k_] := 1/(n - k)^2 - 1/n^2; Table[Numerator[T[n, k]], {n, 0, 20}, {k, 0, n}] // Flatten  (* G. C. Greubel, Sep 19 2018 *)

A171638 Denominator of 1/(n-2)^2 - 1/(n+2)^2.

Original entry on oeis.org

0, 25, 9, 441, 64, 2025, 225, 5929, 576, 13689, 1225, 27225, 2304, 48841, 3969, 81225, 6400, 127449, 9801, 190969, 14400, 275625, 20449, 385641, 28224, 525625, 38025, 700569, 50176, 915849, 65025, 1177225, 82944, 1490841

Offset: 2

Paul Curtz, Dec 13 2009

Fifth column of an array of denominators related to the energies of the hydrogen spectrum, mentioned in A171522. At n=2, the defining formula has a pole and is replaced by 0 to conform with A171621 and A099761.

G. C. Greubel, Table of n, a(n) for n = 2..5000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A099761, A171621.

[0] cat [Denominator((1/(n-2)^2 -1/(n+2)^2)): n in [3..350]]; // Bruno Berselli, Apr 05 2011

A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
A171638 := proc(n) A171621(n)^2 ; end proc:
seq(A171638(n),n=2..90) ; # R. J. Mathar, Apr 02 2011

Table[If[n == 2, 0, Denominator[1/(n-2)^2 - 1/(n+2)^2]], {n, 2, 50}] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,25,9,441,64,2025,225,5929,576,13689},50] (* Harvey P. Dale, Sep 07 2021 *)

for(n=2,100, print1(if(n==2,0, denominator(1/(n-2)^2 - 1/(n+2)^2)), ", ")) \\ G. C. Greubel, Sep 20 2018

a(n) = (A171621(n))^2.
a(2*n+2) = A099761(n).
G.f.: -((x(25+9*x+316*x^2+19*x^3+70*x^4-5*x^5-36*x^6+x^7+9*x^8))/((-1+x)^5 (1+x)^5)). - Harvey P. Dale, Sep 07 2021
Sum_{n>=3} 1/a(n) = 19*Pi^2/192 - 115/144. - Amiram Eldar, Aug 14 2022

cp's OEIS Frontend

A174381 Triangle for denominators of half extended Rydberg-Ritz spectrum of the hydrogenic spectra. a(n) is an antidiagonal writing of array in A171522 without first column.

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

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A171621 Numerator of 1/4 - 1/n^2, each fourth term multiplied by 4.

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A147560 a(n) = 4*A046162(n+1).

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A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

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A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

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A171638 Denominator of 1/(n-2)^2 - 1/(n+2)^2.

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