A061050 -id:A061050 - cp's OEIS frontend

A061037 Numerator of 1/4 - 1/n^2.

0, 5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165, 12, 221, 63, 285, 20, 357, 99, 437, 30, 525, 143, 621, 42, 725, 195, 837, 56, 957, 255, 1085, 72, 1221, 323, 1365, 90, 1517, 399, 1677, 110, 1845, 483, 2021, 132, 2205, 575, 2397, 156, 2597, 675

Offset: 2

N. J. A. Sloane, May 26 2001

From Balmer spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
a(-2) = 0, a(-1) = a(1) = -3. - Paul Curtz, Feb 19 2011
Can be thought of as 4 interlocking sequences, each of the form a(n) = 3a(n - 1) - 3a(n - 2) + a(n - 3). - Charles R Greathouse IV, May 27 2011

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Harry J. Smith, Table of n, a(n) for n=2..1000
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg.
Wikipedia, Balmer series.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A061038 (denominators), A061035-A061050, A126252, A028347.

import Data.Ratio ((%), numerator)
a061037 n = numerator (1%4 - 1%n^2)  -- Reinhard Zumkeller, Dec 17 2011

[ Numerator(1/4-1/n^2): n in [2..52] ]; // Bruno Berselli, Feb 10 2011

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 4] &, 51, 0]
f[n_] := Numerator[(n - 2) (n + 2)/(4 n^2)]; Array[f, 51, 2] (* Or *)
a[n_] := 3 a[n - 4] - 3 a[n - 8] + a[n - 12]; a[1] = -3; a[2] = 0; a[3] = 5; a[4] = 3; a[5] = 21; a[6] = 2; a[7] = 45; a[8] = 15; a[9] = 77; a[10] = 6; a[11] = 117; a[12] = 35; Array[a, 51, 2] (* Robert G. Wilson v *)
Numerator[1/4-1/Range[2,60]^2] (* Harvey P. Dale, Aug 18 2011 *)

a(n) = { numerator(1/4 - 1/n^2) } \\ Harry J. Smith, Jul 17 2009

G.f.: x^2(-3x^11-x^10-3x^9+14x^7+6x^6+30x^5+2x^4+21x^3+3x^2+5x)/(1-x^4)^3.
a(4n+2) = n(n+1), a(2n+3) = (2n+1)(2n+5), a(4n+4) = (2n+1)(2n+3). - Ralf Stephan, Jun 10 2005
a(n+2) = A060819(n) * A060819(n+4).
a(n) = (n^2-4)*(3*i^n+3*(-i)^n-27*(-1)^n+37)/64, where i is the imaginary unit. - Bruno Berselli, Feb 10 2011
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). - Paul Curtz, Feb 28 2011
a(n+2) = n*(n+4)/(period 4: 16, 1, 4, 1 = A146160(n)) = A028347(n+2) / A146160(n). - Paul Curtz, Mar 24 2011 [edited by Franklin T. Adams-Watters, Mar 25 2011]
a(n) = (n^2-4) / gcd(4*n^2, (n^2-4)). - Colin Barker, Jan 13 2014
Sum_{n>=3} 1/a(n) = 11/6. - Amiram Eldar, Aug 12 2022

A061039 Numerator of 1/9 - 1/n^2.

Original entry on oeis.org

0, 7, 16, 1, 40, 55, 8, 91, 112, 5, 160, 187, 8, 247, 280, 35, 352, 391, 16, 475, 520, 7, 616, 667, 80, 775, 832, 11, 952, 1015, 40, 1147, 1216, 143, 1360, 1435, 56, 1591, 1672, 65, 1840, 1927, 224, 2107, 2200, 85, 2392, 2491, 32, 2695, 2800, 323, 3016, 3127

Offset: 3

N. J. A. Sloane, May 26 2001

The denominators are given in A061040.

From Paschen spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A061035..A061050.

import Data.Ratio ((%), numerator)
a061039 n = numerator $ 1%9 - 1%n ^ 2  -- Reinhard Zumkeller, Jan 03 2012

A061039:=n->numer(1/9-1/n^2): seq(A061039(n), n=3..80); # Wesley Ivan Hurt, Apr 12 2017

Table[Numerator[1/9 - 1/n^2], {n, 3, 60}] (* Stefan Steinerberger, Apr 16 2006 *)

a(n)=numerator(1/9-1/n^2) \\ Charles R Greathouse IV, Nov 20 2012

from math import gcd
def A061039(n): return (n**2-9)//gcd(n**2-9,9*n**2) # Chai Wah Wu, Apr 02 2021

a(n) <= n^2 - 9; if n is not divisible by 3 then a(n) = n^2 - 9. - Stefan Steinerberger, Apr 16 2006
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n > 83. - Colin Barker, Oct 09 2016
a(n) = (n^2 - 9)/9^2 if n == 3 or 24 (mod 27), a(n) = (n^2 - 9)/(3*9) if n == 6 or 24 or 15 or 21 (mod 27), a(n) = (n^2 - 9)/9 if n == 0  (mod 9) and n^2 - 9 otherwise. From  the period length 27 sequence gcd(n^2 - 9, 9*n^2). - Wolfdieter Lang, Mar 15 2018

A061041 Numerator of 1/16 - 1/n^2.

Original entry on oeis.org

0, 9, 5, 33, 3, 65, 21, 105, 1, 153, 45, 209, 15, 273, 77, 345, 3, 425, 117, 513, 35, 609, 165, 713, 3, 825, 221, 945, 63, 1073, 285, 1209, 5, 1353, 357, 1505, 99, 1665, 437, 1833, 15, 2009, 525, 2193, 143, 2385, 621, 2585, 21, 2793, 725

Offset: 4

N. J. A. Sloane, May 26 2001

From Brackett spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 4..10000
Frederick Sumner Brackett, Visible and Infra-Red Radiation of Hydrogen Astrophysical Journal, 56, No. 3 (1922) pp. 154-161.
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0, -6,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0, -10,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).

Cf. A061035 - A061050, A064038, A078371.

import Data.Ratio ((%), numerator)
a061041 n = numerator (1%16 - 1%n^2)  -- Reinhard Zumkeller, May 30 2012

[Numerator(1/16 -1/n^2): n in [4..60]]; // G. C. Greubel, Apr 19 2023

A061041:=n->numer(1/16 - 1/n^2); seq(A061041(n), n=4..60); # Wesley Ivan Hurt, Jan 28 2014

Numerator/@(1/16-1/Range[4,60]^2)  (* Harvey P. Dale, Mar 24 2011 *)

a(n)=numerator(1/16-1/n^2) \\ Charles R Greathouse IV, Aug 17 2011

[numerator(1/16 -1/n^2) for n in range(4,61)] # G. C. Greubel, Apr 19 2023

a(4*n+6) = A078371(n). - Paul Curtz, Oct 05 2008
a(n) = 3*a(n-8) - 6*a(n-16) + 10*a(n-24) - 12*a(n-32) + 12*a(n-40) - 10*a(n-48) + 6*a(n-56) - 3*a(n-64) + a(n-72). - Charles R Greathouse IV, Aug 17 2011
a(n) = (n^2-16) / gcd(16*n^2, n^2-16). - Franklin T. Adams-Watters, Sep 25 2011, corrected by Colin Barker, Jan 13 2014.

A061035 Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

Original entry on oeis.org

0, 0, 3, 0, 5, 8, 0, 7, 3, 15, 0, 9, 16, 21, 24, 0, 11, 5, 1, 2, 35, 0, 13, 24, 33, 40, 45, 48, 0, 15, 7, 39, 3, 55, 15, 63, 0, 17, 32, 5, 56, 65, 8, 77, 80, 0, 19, 9, 51, 4, 3, 21, 91, 6, 99, 0, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 0, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 0

Offset: 1

N. J. A. Sloane, May 26 2001

Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

			Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
0
3/4, 0
8/9, 5/36, 0
15/16, 3/16, 7/144, 0
24/25, 21/100, 16/225, 9/400, 0
35/36, 2/9, 1/12, 5/144, 11/900, 0

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 77.

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula

Cf. A061036. Rows give A061037-A061050.

Cf. A126252.

import Data.Ratio ((%), numerator)
a061035 n k = a061035_tabl !! (n-1) !! (k-1)
a061035_row = map numerator . balmer where
   balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
a061035_tabl = map a061035_row [1..]
-- Reinhard Zumkeller, Apr 12 2012

t[m_, n_] := Numerator[1/m^2 - 1/n^2]; Table[t[m, n], {n, 1, 12}, {m, n, 1, -1}] // Flatten(* Jean-François Alcover, Oct 17 2012 *)

More terms from Naohiro Nomoto, Jul 15 2001

A061047 Numerator of 1/49 - 1/n^2.

Original entry on oeis.org

0, 15, 32, 51, 72, 95, 120, 3, 176, 207, 240, 275, 312, 351, 8, 435, 480, 527, 576, 627, 680, 15, 792, 851, 912, 975, 1040, 1107, 24, 1247, 1320, 1395, 1472, 1551, 1632, 5, 1800, 1887, 1976, 2067, 2160, 2255, 48, 2451, 2552, 2655, 2760, 2867

Offset: 7

N. J. A. Sloane, May 26 2001

a(n) = (n+7)^2-49 = n*(n+14) = A098848(n), except a(7p). The corresponding series of atomic transitions is named Hansen-Strong. It comes after Lyman (1906-1914), Balmer (1885), Paschen (1908), Brackett (1922), Pfund (1924) and Humphreys series (1952 not 1953, justified later). - Paul Curtz, Oct 07 2008

Vincenzo Librandi, Table of n, a(n) for n = 7..1000
Peter Hansen and John Strong, Seventh Series of Atomic Hydrogen, Applied Optics, vol 12, 2, (Feb 1973), pp. 429-430. [From _Paul Curtz_, Oct 07 2008]

Cf. A061035-A061050.

Cf. A061039, A061043, A045944, A098848.

[Numerator(1/49-1/n^2): n in [7..60]]; // Vincenzo Librandi, Sep 07 2016

Table[Numerator[1/49-1/n^2],{n,7,70}] (* Harvey P. Dale, Apr 26 2016 *)

a(n) = numerator(1/49 - 1/n^2); \\ Michel Marcus, Aug 15 2013

Edited by M. F. Hasler, Nov 17 2014

A061043 Numerator of 1/25 - 1/n^2.

Original entry on oeis.org

0, 11, 24, 39, 56, 3, 96, 119, 144, 171, 8, 231, 264, 299, 336, 3, 416, 459, 504, 551, 24, 651, 704, 759, 816, 7, 936, 999, 1064, 1131, 48, 1271, 1344, 1419, 1496, 63, 1656, 1739, 1824, 1911, 16, 2091, 2184, 2279, 2376, 99, 2576, 2679, 2784

Offset: 5

N. J. A. Sloane, May 26 2001

From Pfund spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

a(n) = (n+5)^2-25 = n*(n+10) except a(5p) for p positive. Second (with m=5) of this kind after A061039, Paschen (m=3) and before A061047, Hansen-Strong (m=7). For the fourth, what is the value of m in 1/m^2-1/n^2? m=9? - Paul Curtz, Nov 01 2008

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 5..10000
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula

Cf. A061035-A061050.

import Data.Ratio ((%), numerator)
a061043 = numerator . (1 % 25 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

Numerator[1/25-1/Range[5,60]^2] (* Harvey P. Dale, Jan 28 2013 *)

a(n)=numerator(1/25 - 1/n^2) \\ Charles R Greathouse IV, Feb 06 2017

from fractions import Fraction
def a(n): return (Fraction(1, 25) - Fraction(1, n*n)).numerator
print([a(n) for n in range(5, 54)]) # Michael S. Branicky, Nov 19 2021

A129194 a(n) = (n/2)^2*(3 - (-1)^n).

Original entry on oeis.org

0, 1, 2, 9, 8, 25, 18, 49, 32, 81, 50, 121, 72, 169, 98, 225, 128, 289, 162, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 450, 961, 512, 1089, 578, 1225, 648, 1369, 722, 1521, 800, 1681, 882, 1849, 968, 2025, 1058, 2209, 1152, 2401, 1250, 2601, 1352

Offset: 0

Paul Barry, Apr 02 2007

The numerator of the integral is 2,1,2,1,2,1,...; the moments of the integral are 2/(n+1)^2. See 2nd formula.
The sequence alternates between twice a square and an odd square, A001105(n) and A016754(n).
Partial sums of the positive elements give the absolute values of A122576. - Omar E. Pol, Aug 22 2011
Partial sums of the positive elements give A212760. - Omar E. Pol, Dec 28 2013
Conjecture: denominator of 4/n - 2/n^2. - Wesley Ivan Hurt, Jul 11 2016
Multiplicative because both A000290 and A040001 are. - Andrew Howroyd, Jul 25 2018

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 1, Sect. 7, Problem 73.

Vincenzo Librandi, Table of n, a(n) for n = 0..960
Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J.
Int. Seq. 18 (2015) # 15.3.7.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A001105, A007434, A010713, A016742, A016754, A040001.

Cf. A061038, A061040, A061050, A105398, A129204, A152020, A222171, A309337.

[n^2*(3-(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

A129194:=n->n^2*(3-(-1)^n)/4: seq(A129194(n), n=0..80); # Wesley Ivan Hurt, Jul 11 2016

Table[n^2*(3-(-1)^n)/4, {n,0,60}] (* Wesley Ivan Hurt, Jul 11 2016 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,1,2,9,8,25},60] (* Harvey P. Dale, Dec 27 2023 *)

a(n) = lcm(2, n^2)/2; \\ Andrew Howroyd, Jul 25 2018

[n^2*(1+(n%2))/2 for n in range(61)] # G. C. Greubel, Apr 04 2023

G.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/(1-x^2)^3.
a(n+1) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*n*t)(-((Pi-t)/i)^2)), i=sqrt(-1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5. - Paul Curtz, Mar 07 2011
a(n) is the numerator of the coefficient of x^4 in the Maclaurin expansion of exp(-n*x^2). - Francesco Daddi, Aug 04 2011
O.g.f. as a Lambert series: x*Sum_{n >= 1} J_2(n)*x^n/(1 + x^n), where J_2(n) denotes the Jordan totient function A007434(n). See Pólya and Szegő. - Peter Bala, Dec 28 2013
From Ilya Gutkovskiy, Jul 11 2016: (Start)
E.g.f.: x*((2*x + 1)*sinh(x) + (x + 2)*cosh(x))/2.
Sum_{n>=1} 1/a(n) = 5*Pi^2/24. [corrected by Amiram Eldar, Sep 11 2022] (End)
a(n) = A000290(n) / A040001(n). - Andrew Howroyd, Jul 25 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - Amiram Eldar, Sep 11 2022
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} (-1)^(1 + gcd(i,j,n)) = Sum_{d | n} (-1)^(d+1) * J_2(n/d), that is, the Dirichlet convolution of the pair of multiplicative functions f(n) = (-1)^(n+1) and the Jordan totient function J_2(n) = A007434(n). Hence this sequence is multiplicative. Cf. A193356 and A309337.
Dirichlet g.f.: (1 - 2/2^s)*zeta(s-2). (End)
a(n) = Sum_{1 <= i, j <= n} (-1)^(n + gcd(i, n)*gcd(j, n)) = Sum_{d|n, e|n} (-1)^(n+e*d) * phi(n/d)*phi(n/e). - Peter Bala, Jan 22 2024

More terms from Michel Marcus, Dec 28 2013

A061049 Numerator of 1/64 - 1/n^2.

Original entry on oeis.org

0, 17, 9, 57, 5, 105, 33, 161, 3, 225, 65, 297, 21, 377, 105, 465, 1, 561, 153, 665, 45, 777, 209, 897, 15, 1025, 273, 1161, 77, 1305, 345, 1457, 3, 1617, 425, 1785, 117, 1961, 513, 2145, 35, 2337, 609, 2537, 165, 2745, 713, 2961, 3, 3185, 825

Offset: 8

N. J. A. Sloane, May 26 2001

The 495 initial terms are the same as A000265(n * (n+16)), n > 0. - Simon Strandgaard, Oct 30 2021

G. C. Greubel, Table of n, a(n) for n = 8..1000

Cf. A061035-A061050.

Cf. also A191871.

Table[ Numerator[1/64 - 1/n^2], {n, 8, 58}] (* Jean-François Alcover, May 31 2013 *)

for(n=8,50, print1(numerator(1/64 - 1/n^2), ", ")) \\ G. C. Greubel, Jul 07 2017

A061045 Numerator of 1/36 - 1/n^2.

Original entry on oeis.org

-35, -2, -1, -5, -11, 0, 13, 7, 5, 4, 85, 1, 133, 10, 7, 55, 253, 2, 325, 91, 5, 28, 493, 5, 589, 40, 77, 187, 805, 2, 925, 247, 13, 70, 1189, 35, 1333, 88, 55, 391, 1645, 4, 1813, 475, 221, 130, 2173, 7, 2365, 154, 95, 667, 2773, 20, 2989, 775, 119, 208, 3445, 11, 3685, 238, 437, 1015, 4189, 10

Offset: 1

N. J. A. Sloane, May 26 2001

Sixth case of Rydberg's formula. From Humphrey's spectrum of hydrogen. See A045944, A000567, A061043, A061046, A061047. - Paul Curtz, Dec 08 2008

			The fractions are -35/36, -2/9, -1/12, -5/144, -11/900, 0, 13/1764, 7/576, 5/324, 4/225, 85/4356, 1/48, 133/6084, 10/441, 7/300, 55/2304, 253/10404, 2/81, 325/12996, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Curtis J. Humphreys, The Sixth Series in the Hydrogen Spectrum, Journal of the Optical Society of America, 1952, 42, p. 432.

A061046 gives denominators.

Cf. A000567, A045944, A061035 - A061050.

import Data.Ratio ((%), numerator)
a061045 = numerator . (1 % 36 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

[Numerator(1/6^2 -1/n^2): n in [1..80]]; // G. C. Greubel, Feb 24 2023

Numerator[(1/36-1/Range[100]^2)] (* Harvey P. Dale, Mar 17 2013 *)

def A061045(n): return ((n^2-36)/(6*n)^2).numerator()
[A061045(n) for n in range(1,81)] # G. C. Greubel, Feb 24 2023

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

cp's OEIS Frontend

A061037 Numerator of 1/4 - 1/n^2.

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A061039 Numerator of 1/9 - 1/n^2.

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A061041 Numerator of 1/16 - 1/n^2.

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A061035 Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

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A061047 Numerator of 1/49 - 1/n^2.

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A061043 Numerator of 1/25 - 1/n^2.

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