A165795 -id:A165795 - 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

A172157 Triangle T(n,m) = numerator of 1/n^2 - 1/m^2, read by rows, with T(n,0) = -1.

Original entry on oeis.org

-1, -1, -3, -1, -8, -5, -1, -15, -3, -7, -1, -24, -21, -16, -9, -1, -35, -2, -1, -5, -11, -1, -48, -45, -40, -33, -24, -13, -1, -63, -15, -55, -3, -39, -7, -15, -1, -80, -77, -8, -65, -56, -5, -32, -17, -1, -99, -6, -91, -21, -3, -4, -51, -9, -19, -1, -120, -117, -112

Offset: 1

Paul Curtz, Jan 27 2010

The triangle obtained by negating the values of the triangle A120072 and adding a row T(n,0) = -1.

			The full array of numerators starts in row n=1 with columns m>=0 as:
-1...0...3...8..15..24..35..48..63..80..99. A005563
-1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
-1..-8..-5...0...7..16...1..40..55...8..91. A061039
-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The triangle is the portion below the main diagonal, left from the zeros, 0<=m<n.

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A172370, A174233, A165795.

T[n_, 0] := -1; T[n_, k_] := 1/n^2 - 1/k^2; Table[Numerator[T[n, k]], {n, 1, 100}, {k, 0, n - 1}] // Flatten (* G. C. Greubel, Sep 19 2018 *)

A172370 Mirrored triangle A120072 read by rows.

Original entry on oeis.org

3, 5, 8, 7, 3, 15, 9, 16, 21, 24, 11, 5, 1, 2, 35, 13, 24, 33, 40, 45, 48, 15, 7, 39, 3, 55, 15, 63, 17, 32, 5, 56, 65, 8, 77, 80, 19, 9, 51, 4, 3, 21, 91, 6, 99, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 25, 48, 69, 88, 105, 120, 133, 144

Offset: 2

Paul Curtz, Feb 01 2010

A table of numerators of 1/n^2 - 1/m^2 extended to negative m looks as follows, stacked such that values of common m are aligned
and the central column of -1 is defined for m=0:
.............................0..-1...0...3...8..15..24..35..48..63..80..99. A005563
.........................0..-3..-1..-3...0...5...3..21...2..45..15..77...6. A061037
.....................0..-5..-8..-1..-8..-5...0...7..16...1..40..55...8..91. A061039
.................0..-7..-3.-15..-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
.............0..-9.-16.-21.-24..-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
.........0.-11..-5..-1..-2.-35..-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
.....0.-13.-24.-33.-40.-45.-48..-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
.0.-15..-7.-39..-3.-55.-15.-63..-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The row-reversed variant of A120072 appears (negated) after the leftmost 0.
Equals A061035 with the first column removed. - Georg Fischer, Jul 26 2023

			The table starts
   3
   5   8
   7   3  15
   9  16  21  24
  11   5   1   2  35
  13  24  33  40  45  48
  15   7  39   3  55  15  63
  17  32   5  56  65   8  77  80
  19   9  51   4   3  21  91   6  99

G. C. Greubel, Rows n = 2..100 of triangle, flattened

Lower diagonal gives: A070262, A061037(n+2).

Cf. A061035, A172157, A165795.

[[Numerator(1/(n-k)^2 -1/n^2): k in [1..n-1]]: n in [2..20]]; // G. C. Greubel, Sep 20 2018

Table[Numerator[1/(n-k)^2 -1/n^2], {n, 2, 20}, {k, 1, n-1}]//Flatten (* G. C. Greubel, Sep 20 2018 *)

for(n=2,20, for(k=1,n-1, print1(numerator(1/(n-k)^2 -1/n^2), ", "))) \\ G. C. Greubel, Sep 20 2018

T(n,m) = numerator of 1/(n-m)^2 - 1/n^2, n >= 2, 1 <= m < n. - R. J. Mathar, Nov 23 2010

Comment rewritten and offset set to 2 by R. J. Mathar, Nov 23 2010

A173651 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2-1/m^2, or -1 if m=0.

Original entry on oeis.org

-1, -1, 0, -1, -3, 3, -1, -8, 0, 8, -1, -15, -5, 5, 15, -1, -24, -3, 0, 3, 24, -1, -35, -21, -7, 7, 21, 35, -1, -48, -2, -16, 0, 16, 2, 48, -1, -63, -45, -1, -9, 9, 1, 45, 63, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, -1, -120, -6, -8, -3, -24, 0, 24, 3, 8, 6, 120, -1, -143, -117, -91, -65, -39, -13, 13, 39, 65, 91, 117, 143, -1, -168, -35, -112, -21, -56, -7, 0, 7, 56, 21, 112, 35, 168, -1, -195, -165, -5, -105, -3, -5, -15

Offset: 1

Paul Curtz, Nov 24 2010

This is triangle A165507 with an additional column T(n,0)= -1.

The triangle is obtained from the infinite array shown in the comment in A172370 by starting in its row n, column 0 and reading diagonally upwards up to row n=1, column n-1.

			The triangle starts in row n=1 with columns 0<=m<n as
-1;
-1,0,
-1,-3,3,
-1,-8,0,8,
-1,-15,-5,5,15,
-1,-24,-3,0,3,24,
-1,-35,-21,-7,7,21,35,
-1,-48,-2,-16,0,16,2,48,
-1,-63,-45,-1,-9,9,1,45,63,

Cf. A165795.

A173651 := proc(n,m) if m = 0 then -1 ; else 1/(n-m)^2-1/m^2 ; numer(%) ; end if; end proc:

cp's OEIS Frontend

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

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A172157 Triangle T(n,m) = numerator of 1/n^2 - 1/m^2, read by rows, with T(n,0) = -1.

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A172370 Mirrored triangle A120072 read by rows.

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A173651 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2-1/m^2, or -1 if m=0.

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