A061042 -id:A061042 - cp's OEIS frontend

A147792 A quadrisection of A061042.

1, 64, 18, 256, 50, 576, 49, 1024, 81, 1600, 242, 2304, 338, 3136, 225, 4096, 289, 5184, 722, 6400, 882, 7744, 529, 9216, 625, 10816, 1458, 12544, 1682, 14400, 961, 16384, 1089, 18496, 2450, 20736, 2738, 23104, 1521, 25600, 1681, 28224, 3698, 30976, 4050, 33856

Offset: 0

Paul Curtz, Nov 13 2008

Related to the fractions that describe each fourth transition in the Brackett spectrum of hydrogen.

Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, -6, 0, 10, 0, -12, 0, 12, 0, -10, 0, 6, 0, -3, 0, 1).

a(n)=A061042(4n+4).

Conjecture: a(n) = 3*a(n-2) -6*a(n-4) +10*a(n-6) -12*a(n-8) +12*a(n-10) -10*a(n-12) +6*a(n-14) - 3*a(n-16) +a(n-18). - R. J. Mathar, Nov 14 2008

Index in formula corrected, extended by R. J. Mathar, Nov 14 2008

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

Original entry on oeis.org

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

A171522 Denominator of 1/n^2-1/(n+2)^2.

Original entry on oeis.org

0, 9, 16, 225, 144, 1225, 576, 3969, 1600, 9801, 3600, 20449, 7056, 38025, 12544, 65025, 20736, 104329, 32400, 159201, 48400, 233289, 69696, 330625, 97344, 455625, 132496, 613089, 176400, 808201, 230400, 1046529, 295936, 1334025, 374544, 1677025, 467856

Offset: 0

Paul Curtz, Dec 11 2009

This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):
0,   0,   0,   0,   0,   0,   0,   0... A000004
1,   4,   9,  16,  25,  36,  49,  64... A000290
1,  36,  16, 100,   9, 196,  64, 324... A061038
1, 144, 225,  12, 441, 576,  81, 900... A061040
1, 400, 144, 784,  64,1296, 400,1936... A061042
1, 900 1225,1600,2025, 100,3025,3600... A061044
1,1764, 576, 324, 225,4356,  48,6084... A061046
1,3136,3969,4900,5929,7056,8281, 196... A061048.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A105371. Bisections: A060300, A069075.

A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
seq(A171522(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

a[n_] := If[EvenQ[n], (n*(n+2))^2/4, (n*(n+2))^2]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jun 13 2017 *)

concat(0, Vec(x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5) + O(x^100))) \\ Colin Barker, Nov 05 2014

a(n) = (A066830(n+1))^2.
a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - Colin Barker, Nov 05 2014
G.f.: x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5). - Colin Barker, Nov 05 2014

Edited and extended by R. J. Mathar, Dec 15 2009

A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 16, 36, 36, 16, 25, 16, 1, 16, 25, 36, 100, 144, 144, 100, 36, 49, 9, 225, 1, 225, 9, 49, 64, 196, 12, 400, 400, 12, 196, 64, 81, 64, 441, 144, 1, 144, 441, 64, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 25, 81, 64, 1225, 1, 1225, 64, 81, 25, 121

Offset: 1

Paul Curtz, Sep 19 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			.1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
.4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
.9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035

T:= (k,n)-> denom(1/min (n,k)^2 -1/max (n, k)^2):
seq(seq(T(k, d-k), k=1..d-1), d=2..12);

T[n_, k_] := Denominator[1/Min[n, k]^2 - 1/Max[n, k]^2];
Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 04 2020 *)

T(n,k) = A165727(n,k).

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

A017114 a(n) = (8*n + 4)^2.

Original entry on oeis.org

16, 144, 400, 784, 1296, 1936, 2704, 3600, 4624, 5776, 7056, 8464, 10000, 11664, 13456, 15376, 17424, 19600, 21904, 24336, 26896, 29584, 32400, 35344, 38416, 41616, 44944, 48400, 51984, 55696, 59536, 63504, 67600, 71824, 76176, 80656, 85264, 90000, 94864, 99856

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006752, A016754, A016826, A017113, A061042.

[(8*n+4)^2: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

LinearRecurrence[{3, -3, 1},{16, 144, 400},30] (* Ray Chandler, Aug 04 2015 *)
(8*Range[0,40]+4)^2 (* Harvey P. Dale, Aug 13 2024 *)

a(n)=(8*n+4)^2 \\ Charles R Greathouse IV, Jun 17 2017

From Paul Curtz, Nov 07 2008: (Start)
a(n) = 16*A016754(n).
a(n+2) = A061042(2n+1), from Brackett spectrum of hydrogen. (End)
G.f.: -16*(1 + 6*x + x^2)/(x-1)^3. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^2.
a(n) = 2^2*A016826(n).
Sum_{n>=0} 1/a(n) = Pi^2/128.
Sum_{n>=0} (-1)^n/a(n) = G/16, where G is Catalan's constant (A006752). (End)

A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 4, 0, 0, 9, 1, 9, 0, 0, 16, 36, 36, 16, 0, 0, 25, 16, 1, 16, 25, 0, 0, 36, 100, 144, 144, 100, 36, 0, 0, 49, 9, 225, 1, 225, 9, 49, 0, 0, 64, 196, 12, 400, 400, 12, 196, 64, 0, 0, 81, 64, 441, 144, 1, 144, 441, 64, 81, 0, 0, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 0

Offset: 0

Paul Curtz, Sep 25 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			0,  0,   0,   0,    0,    0,    0,    0,    0,    0, ... A000004
0,  1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
0,  4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
0,  9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
0, 16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
0, 25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
0, 36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
0, 49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
0, 64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
0, 81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1034

Cf. A165441 (top row and left column removed)

T:= (k,n)-> `if` (n=0 or k=0, 0, denom (1/min (n,k)^2 -1/max (n, k)^2)):
seq (seq (T (k, d-k), k=0..d), d=0..11);

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

A174683 Denominator of 1/16 - 1/n^2.

Original entry on oeis.org

0, 16, 16, 144, 16, 400, 144, 784, 64, 1296, 400, 1936, 18, 2704, 784, 3600, 256, 4624, 1296, 5776, 50, 7056, 1936, 8464, 576, 10000, 2704, 11664, 49, 13456, 3600, 15376, 1024, 17424, 4624, 19600, 81, 21904, 5776, 24336, 1600, 26896, 7056, 29584, 242, 32400, 8464, 35344, 2304, 38416

Offset: 0

Paul Curtz, Nov 30 2010

The value of a(n)=0 is substituted at the pole n=0.

Extends the Bracket spectrum to negative quantum numbers in the fashion of A061038 (1/4-1/n^2) and A181759 (1/9-1/n^2).

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf A174680 (numerators).

Table[If[n == 0, 0, If[n == 4, 16, Denominator[(n^2 - 16)/(4*n)^2]]], {n, 0, 100}] (* G. C. Greubel, Sep 16 2018 *)
Table[Which[n==0,0,n==4,16,True,Denominator[(n^2-16)/(16n^2)]],{n,0,100}] (* Harvey P. Dale, Dec 13 2024 *)

for(n=0,100, print1(if(n==0,0, if(n==4,16, denominator((n^2 - 16)/(4*n)^2))), ", ")) \\ G. C. Greubel, Sep 16 2018

a(n) = A061042(n), n>=4.

a(n) = LCM[n^2 - 16, 16*n^2]/(n^2 - 16), for n>=5. - G. C. Greubel, Sep 16 2018

Removed a(-4)-a(-1) since a(-n)=a(n) by G. C. Greubel, Sep 16 2018

A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

Original entry on oeis.org

9, 9, 81, 18, 225, 81, 441, 72, 729, 225, 1089, 162, 1521, 441, 2025, 288, 2601, 729, 3249, 450, 3969, 1089, 4761, 648, 5625, 1521, 6561, 882, 7569, 2025, 8649, 1152, 9801, 2601, 11025, 1458, 12321, 3249, 13689, 1800, 15129, 3969, 16641, 2178, 18225

Offset: 1

Paul Curtz, Nov 20 2008

nonn

The associated terms of the n-th main series of the Hydrogen energy spectrum are A000290(3), A061038(6), A061040(9), A061042(12), A061044(15), A061046(18), A061048(21), A061050(24), etc.

All numbers are multiples of 9.

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

Cf. A143025 with a similar principle of construction.

Cf. A291050.

Denominator/@(8/(9Range[50]^2))  (* Harvey P. Dale, Mar 15 2011 *)

Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Sep 14 2022

Stratified definition, corrected indices, extended, R. J. Mathar, Dec 10 2008

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.

Original entry on oeis.org

0, 0, 4, 0, 9, 36, 0, 16, 16, 144, 0, 25, 100, 225, 400, 0, 36, 9, 12, 144, 900, 0, 49, 196, 441, 784, 1225, 1764, 0, 64, 64, 576, 64, 1600, 576, 3136, 0, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 0, 100, 25, 900, 400, 100, 225, 4900, 1600, 8100, 0, 121, 484, 1089

Offset: 0

Paul Curtz, Mar 17 2010

Companion to A172157 (numerators). Hence -1/0; -1/0,-3/4; -1/0,-8/9,-5/36; -1/0,-15/16,-3/16,-7/144; -1/0,-24/25,-21/100,-16/225,-9/400; for 1) (-1/0, A005563/A000290(n+1))=A067998(n+1)/A000290 Lyman; 2) -1/0, -3/4, A061037/A061038 Balmer ; 3) -1/0, -8/9, -5/36, A061039/A061040)=A171709(n+3)/ Paschen; 4) (-1/0, -15/16, -3/16, -7/144, A061041/A061042 Brackett; .. .

a(n)= 0, (mix 0 or A000004 , n-th row of A120073)

A222740 Denominators of 1/16 - 1/(4 + 8*n)^2.

Original entry on oeis.org

1, 18, 50, 49, 81, 242, 338, 225, 289, 722, 882, 529, 625, 1458, 1682, 961, 1089, 2450, 2738, 1521, 1681, 3698, 4050, 2209, 2401, 5202, 5618, 3025, 3249, 6962, 7442, 3969, 4225, 8978, 9522, 5041, 5329, 11250, 11858, 6241

Offset: 0

Paul Curtz, May 29 2013

Denominators of the reduced fractions A064038(n)/a(n) = 0/1, 1/18, 3/50, 3/49, 5/81, 15/242, 21/338, 14/225, 18/289, ... .

Also, A064038 and a(n) are related to the sequence of period 4: repeat 1, 2, 2, 1.

			a(0) = 1*1, a(1) = 2*9 = 18, a(2) = 2*25 = 50, a(3) = 1*49 = 49.
a(0) = 16*0 + 1 = 1, a(1) = 16*1 + 2 = 18, a(2) = 16*3 + 2 = 50, a(3) = 16*3 + 1 = 49.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Table[1/16-1/(4+8n)^2,{n,0,40}]//Denominator (* or *) LinearRecurrence[ {3,-6,10,-12,12,-10,6,-3,1},{1,18,50,49,81,242,338,225,289},40] (* Harvey P. Dale, Aug 30 2021 *)

a(n) = A014695(n) * A016754(n).
a(n) = 16*A064038(n+1) + A014695(n).
a(n) = A061042(4+8*n).
a(2n+2) - a(2n+1) = 32*A026741(n+1).
G.f.: ( -1 - 15*x - 2*x^2 + 3*x^3 - 66*x^4 + 3*x^5 - 2*x^6 - 15*x^7 - x^8 ) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Jun 04 2013
a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))*(2*n+1)^2/2. - Wesley Ivan Hurt, Oct 04 2018

cp's OEIS Frontend

A147792 A quadrisection of A061042.

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

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A171522 Denominator of 1/n^2-1/(n+2)^2.

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A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

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A017114 a(n) = (8*n + 4)^2.

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A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

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A174683 Denominator of 1/16 - 1/n^2.

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