Gary Detlefs - cp's OEIS frontend

A382813 Denominator of the n-th partial sum of the squares of the harmonic numbers.

1, 4, 18, 144, 200, 400, 4900, 78400, 635040, 6350400, 64033200, 768398400, 9275666400, 8657288640, 16232416200, 519437318400, 2779951574400, 16679709446400, 60213751101504, 3823095308032, 1216439416192, 26761667156224, 1769615240705312, 127412297330782464, 3062795608913040000

Offset: 1

Gary Detlefs, Apr 05 2025

All terms for n>1 are even.

			The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3) = 18.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001008, A002805, A382812 (numerators).

Cf. A027611, A027612.

H2:= n-> add(harmonic(k)^2, k = 1..n): seq(denom(H2(n)), n=1..25);

a(n) = denominator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

a(n) = denominator((n+1)*H(n)^2-(2*n+1)*H(n)+2*n), where H(n) is the n-th harmonic number.

a(n) = denominator((S(n)*H(n)^2+(2*n-2*S(n)+1)*H(n) - 2*n)/(H(n) - 1)), where S(n) = the n-th partial sum of H(n).

A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers.

Original entry on oeis.org

1, 13, 119, 1577, 3233, 8867, 141563, 2844129, 28119709, 335676251, 3968696491, 55023970333, 758025067309, 799020611041, 1676892996083, 59597395635137, 351844709221043, 2314823924364859, 9114392136427625, 628176680098075, 216039223801697, 5117413095318143, 363066107054194281, 27957386425926920257

Offset: 1

Gary Detlefs, Apr 05 2025

			The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3)=119.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001008, A002805, A382813 (denominators).

Cf. A027611, A027612

H2:= n-> add(harmonic(k)^2, k = 1..n): seq(numer(H2(n)), n=1..25);

Accumulate[HarmonicNumber[Range[30]]^2]//Numerator (* Harvey P. Dale, Aug 10 2025 *)

a(n) = numerator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

a(n) = numerator((n+1)*H(n)^2-(2*n+1)*H(n) + 2*n), where H(n) is the n-th harmonic number.

a(n) = numerator((S(n)*H(n)^2 + (2*n - 2*S(n) + 1)*H(n)-2*n)/(H(n)-1)), where S(n) is the n-th partial sum of H(n).

A374192 Number of iterations required to reach 1 or 10 in a modified Collatz trajectory where x -> x/2 if x is even, x -> x+1 if x is odd and not divisible by 3 and x -> 3x+1 if x is odd and divisible by 3; or a(n) = -1 if 1 or 10 is never reached.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 3, 7, 0, 4, 3, 6, 5, 7, 4, 9, 8, 2, 1, 7, 5, 5, 4, 8, 7, 11, 6, 9, 8, 6, 5, 11, 10, 10, 9, 4, 3, 12, 2, 9, 8, 7, 6, 13, 6, 6, 5, 10, 9, 16, 8, 13, 12, 8, 7, 10, 10, 10, 9, 8, 7, 9, 6, 13, 12, 12, 11, 11, 11, 11, 10, 6, 5, 14, 4, 14, 13, 4, 3

Offset: 1

Gary Detlefs, Jun 30 2024

nonn

It is conjectured that this iteration always reaches one of two cycles [1,2] or [10,5,6,3].

			n=1 takes a(1) = 0 iterations to reach 1 or 10, since it's already 1.
n=9 takes a(9) = 7 iterations to reach 1, by 9,28,14,7,8,4,2,1.
n=15 takes a(15) = 7 iterations to reach 10, by 15,46,23,24,12,6,3,10.

A366314 a(n) = a(n-1) + 3a(n-2) + 9a(n-3) with a(0)=0, a(1)=1, a(2)=4.

Original entry on oeis.org

0, 1, 4, 7, 28, 85, 232, 739, 2200, 6505, 19756, 59071, 176884, 531901, 1594192, 4781851, 14351536, 43044817, 129136084, 387434359, 1162245964, 3486773797, 10460420920, 31380955987, 94143182920, 282429839161, 847287991804, 2541866155567, 7625598683428, 22876789076365, 68630380526752

Offset: 0

Gary Detlefs, Oct 06 2023

This sequence could be considered a companion sequence to A103770, in that both sequences are manifested in the averaging of a Tribonacci sequence with initial seeds of x, y, z.
If f(n) is a third order recurrence with f(0)=x, f(1)=y, f(2)=z, and f(n) = (f(n-1)+f(n-2)+f(n-3))/3,n>2, then
  f(n) =(A103770(n-2)*z + a(n-1)*y + A103770(n-3)x)/3^(n-2).
In the general case, these "averaging" sequences will approach a limit of (x+2*y+3*z)/6.
a(n) mod 9 repeats [1,4,7] from offset 1 = A100402(n-1)...

			Starting with initial terms of x, y, z, the sequence proceeds:
  (z + y + x)/3
  (4z + 4y +x)/9
  (16z + 7y + 4x)/27
  (37z + 28y + 16x)/81
  (121z + 85y + 37x)/243
  ....

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,9).

Cf. A103770.

a:= proc(n) option remember; if n < 3 then n^2 else a(n-1)+3*a(n-2)+9*a(n-3) fi end: seq(a(n), n=0..30);

LinearRecurrence[{1,3,9},{0,1,4},50] (* Paolo Xausa, Nov 14 2023 *)

From Stefano Spezia, Oct 15 2023: (Start)
G.f.: x*(1 + 3*x)/((1 - 3*x)*(1 + 2*x + 3*x^2)).
a(n) = (4*3^n + (-2 - sqrt(2)*i)*(-1 - sqrt(2)*i)^n + i*(-1 + sqrt(2)*i)^n*(2*i + sqrt(2)))/12, where i denotes the imaginary unit. (End)

A350747 Number of iterations required to terminate trajectory mapping described in A349824.

Original entry on oeis.org

0, 1, 0, 0, 8, 0, 9, 0, 7, 4, 8, 0, 3, 0, 7, 8, 7, 0, 6, 0, 1, 2, 2, 0, 5, 2, 1, 0, 0, 0, 0, 0, 6, 0, 7, 6, 4, 0, 6, 7, 3, 0, 5, 0, 2, 1, 6, 0, 8, 1, 5, 4, 5, 0, 3, 7, 6, 3, 11, 0, 9, 0, 10, 8, 11, 5, 9, 0, 9, 6, 6, 0, 10

Offset: 0

Gary Detlefs, Jan 13 2022

nonn

From Wolfdieter Lang, Feb 09 2022: (Start)
Conjecture from A349824: the iteration f: n -> A349824(n) becomes periodic for each n >= 0.
a(n) gives the number of steps from n to reach the first member of the periodic part. There are the two length 2 periods: (33,28) and (28,33). (End)
It appears that the only nonprime values of n for which a(n) = 0 are {0, 27, 28, 30, 33}.

			For n = 6, the trajectory is 6, 10, 14, 18, 24, 36, 40, 44, 45, 33, ... so a(6) = 9.
For n = 24, the trajectory is 24, 36, 40, 44, 45, 33, ... so a(24) = 5.

Cf. A349824, A001414, A001222.

with(numtheory): A001222:= n -> bigomega(n):
A001414:= proc(n) local e, j; e:=ifactors(n)[2]; add(e[j][1] * e[j][2],j= 1..nops(e)) end proc :
B := n-> A001414(n) * A001222(n):
g:= proc(n) if isprime(n) or n=0 or n=27 or n=28 or n=30 or n=33 then return 0 else return 1 fi end proc:
F:= proc(n) local v,i; v:=n;if n = 1 then return 1 else if g(n)=0 then return 0 else for i from 0 to 100 do v:= B(v);if  v=27 or v=28 or v=30 or v=33 then return i+1; i:=100 fi od fi fi end proc :
Seq(F(n), n=0..100)

More terms from Jinyuan Wang, Jan 15 2022

Edited by Wolfdieter Lang, Feb 09 2022

A343689 a(1)=0, a(2)=1, a(n) = (4n-2)a(n-1) + a(n-2), n > 2.

Original entry on oeis.org

0, 1, 10, 141, 2548, 56197, 1463670, 43966297, 1496317768, 56904041481, 2391466059970, 110064342800101, 5505608606065020, 297412929070311181, 17255455494684113518, 1070135653599485349297, 70646208593060717167120, 4946304737167849687047697, 366097196759013937558696698

Offset: 1

Gary Detlefs, Apr 26 2021

nonn

This sequence is one of the two "basis" sequences for sequences having the form s(a,b,1)=a, s(a,b,2)=b, s(n)=(4*n-2)*s(a,b,n-1) + s(a,b,n-2), the second being A343688. s(a,b,n) = a*A343688(n) + b*a(n).
Of specific interest is s(3,19,n) and s(1,7,n) which produce the odd terms of A340737 and A340738 respectively and whose quotient converges to e.
a(n) mod n = 1 for even n and n-2 for odd n (empirical).

			a(4)=14*10+1, a(5)=18*141+10...

Harvey P. Dale, Table of n, a(n) for n = 1..366

Cf. A340737, A340738, A343688.

e := proc(a, b, n) option remember; if n = 1 then a; else if n = 2 then b; else (4*n - 2)*e(a, b, n - 1) + e(a, b, n - 2); end if; end if; end proc
for n from 1 to 20 do print(e(0,1,n)) od

RecurrenceTable[{a[1]==0,a[2]==1,a[n]==(4n-2)a[n-1]+a[n-2]},a,{n,20}] (* Harvey P. Dale, Dec 17 2021 *)

a(1)=0, a(1)=1, a(n) = (4*n-2)*a(n-1) + a(n-2).

A343688 a(1)=1, a(2)=0, a(n) = (4n-2)a(n-1) + a(n-2), n > 2.

Original entry on oeis.org

1, 0, 1, 14, 253, 5580, 145333, 4365570, 148574713, 5650204664, 237457170601, 10928680052310, 546671459786101, 29531187508501764, 1713355546952888413, 106257575098587583370, 7014713312053733390833, 491136189418859924941680, 36351092730307688179075153

Offset: 1

Gary Detlefs, Apr 26 2021

nonn

This sequence is one of the two "basis" sequences for sequences having the form s(a,b,1)=a, s(a,b,2)=b, s(n) = (4*n-2)*s(a,b,n-1) + s(a,b,n-2), the second being A343689. s(a,b,n) = a*a(n) + b*A343689(n).
Of specific interest is s(3,19,n) and s(1,7,n) which produce the odd terms of A340737 and A340738 respectively and whose quotient converges to e.
It is of interest to note that a(n)*A343689(n+1) - a(n+1)*A343689(n) = (-1)^(n+1), a(n)*A343689(n+2) - a(n+2)*A343689(n) = (4*n+6)*(-1)^(n+1) and a(n)*A343689(n+3) - a(n+3)*A343689(n) =((4*n+8)^2-3)* (-1)^(n+1)
a(n) mod n = n-6 for even n > 4 and 13 for odd n > 13 (empirical).

			a(4)=14*1+0, a(5)=18*14+1, ...

Cf. A340737, A340738, A343689.

e := proc(a, b, n) option remember; if n = 1 then a; else if n = 2 then b; else (4*n - 2)*e(a, b, n - 1) + e(a, b, n - 2); end if; end if; end proc;
for n from 1 to 20 do print(e(1,0,n)) od

a[1]=1;a[2]=0;a[n_]:=a[n]=(4n-2)a[n-1]+a[n-2];Array[a,20] (* Giorgos Kalogeropoulos, Apr 27 2021 *)

a(1)=1, a(2)=0, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.

A340738 Denominator of a sequence of fractions converging to e.

Original entry on oeis.org

1, 2, 7, 18, 71, 252, 1001, 4540, 18089, 99990, 398959, 2602278, 10391023, 78132152, 312129649, 2658297528, 10622799089, 101072656170, 403978495031, 4247085597370, 16977719590391, 195445764537012, 781379079653017, 9775727355457908, 39085931702241241, 528050767520083262, 2111421691000680031

Offset: 1

Gary Detlefs, Jan 18 2021

This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the denominators filtered out of this sequence are a(1)..a(8).

The convergence is conjectured.

			Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...

Numerators are listed in A340737.

Cf. A007676/A007677.

e:=proc(a,b,n)option remember; e(a,b,1):=a; e(a,b,2):=b; if n>2 and n mod 2 =1 then 2*e(a,b,n-1)+n*e(a,b,n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a,b,n-1)/2 -(e(a,b,n-2)+(n-2)*e(a,b,n-3)/2) fi fi end
seq(e(1,2,n), n = 1..20)
# code to print the sequence of fractions and error
for n from 1` to 20 do print(e(3,5,n)/e(1,2,n), evalf(exp(1)-e(3,5,n)/e(1,2,n)) od

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)

a(1) = 1, a(2) = 2; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.

A340737 Numerators of a sequence of fractions converging to e.

Original entry on oeis.org

3, 5, 19, 49, 193, 685, 2721, 12341, 49171, 271801, 1084483, 7073725, 28245729, 212385209, 848456353, 7226001865, 28875761731, 274743964621, 1098127402131, 11544775603241, 46150226651233, 531276670190245, 2124008553358849, 26573182030311229, 106246577894593683, 1435390805853694145

Offset: 1

Gary Detlefs, Jan 18 2021

This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the numerators filtered out of this sequence are a(1)..a(8).

The convergence is conjectured.

			Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...

Denominators are listed in A340738.

Cf. A007676/A007677.

e:=proc(a,b,n)option remember; e(a,b,1):=a; e(a,b,2):=b; if n>2 and n mod 2 =1 then 2*e(a,b,n-1)+n*e(a,b,n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a,b,n-1)/2 -(e(a,b,n-2)+(n-2)*e(a,b,n-3)/2) fi fi end
seq(e(3,5,n), n = 1..20)
# code to print the sequence of fractions and error
for n from 1` to 20 do print(e(3,5,n)/e(1,2,n), evalf(exp(1)-e(3,5,n)/e(1,2,n)) od

a[1] = 3; a[2] = 5; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)

a(1) = 3, a(2) = 5; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.

A338192 Sum of Fibonacci and tribonacci numbers: a(n) = A000073(n) + A000045(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 15, 26, 45, 78, 136, 238, 418, 737, 1304, 2315, 4123, 7365, 13193, 23694, 42655, 76958, 139126, 251974, 457112, 830501, 1510930, 2752175, 5018581, 9160293, 16734631, 30595694, 55976389, 102474674, 187700488, 343973242, 630623826, 1156594669

Offset: 0

Gary Detlefs, Oct 15 2020

In general, the sum of a second-order sequence with signature (a,b) and a third-order sequence with signature (x,y,z) will be a fifth-order sequence with signature (a+x,-x*a+b+y, -y*a+z-b*x,-a*z-b*y,-b*z). In this instance, a=b=x=y=z=1 resulting in a signature of (2,1,-1,-2,-1).

Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-2,-1).

Cf. A000045, A000073.

LinearRecurrence[{2, 1, -1, -2, -1}, {0, 1, 2, 3, 5}, 50] (* Amiram Eldar, Oct 15 2020 *)

a(n) = A000073(n) + A000045(n).
a(n) = 2*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) for n > 4 with a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5.
G.f.: x*(1 - 2*x^2 - 2*x^3)/(1 - 2*x - x^2 + x^3 + 2*x^4 + x^5). - Stefano Spezia, Oct 15 2020

cp's OEIS Frontend

User: Gary Detlefs

A382813 Denominator of the n-th partial sum of the squares of the harmonic numbers.

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A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers.

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A374192 Number of iterations required to reach 1 or 10 in a modified Collatz trajectory where x -> x/2 if x is even, x -> x+1 if x is odd and not divisible by 3 and x -> 3x+1 if x is odd and divisible by 3; or a(n) = -1 if 1 or 10 is never reached.

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A366314 a(n) = a(n-1) + 3*a(n-2) + 9*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

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A350747 Number of iterations required to terminate trajectory mapping described in A349824.

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A343689 a(1)=0, a(2)=1, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.

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A343688 a(1)=1, a(2)=0, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.

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A340738 Denominator of a sequence of fractions converging to e.

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A366314 a(n) = a(n-1) + 3a(n-2) + 9a(n-3) with a(0)=0, a(1)=1, a(2)=4.

A343689 a(1)=0, a(2)=1, a(n) = (4n-2)a(n-1) + a(n-2), n > 2.

A343688 a(1)=1, a(2)=0, a(n) = (4n-2)a(n-1) + a(n-2), n > 2.