Amrit Awasthi - cp's OEIS frontend

A374489 a(n) = floor(Sum_{k=n^4..(n+1)^4} k^(1/4)).

1, 26, 171, 628, 1685, 3726, 7231, 12776, 21033, 32770, 48851, 70236, 97981, 133238, 177255, 231376, 297041, 375786, 469243, 579140, 707301, 855646, 1026191, 1221048, 1442425, 1692626, 1974051, 2289196, 2640653, 3031110, 3463351, 3940256, 4464801, 5040058, 5669195

Offset: 0

Amrit Awasthi, Jul 09 2024

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A248698, A248575, A374384.

LinearRecurrence[{5,-10,10,-5,1},{1,26,171,628,1685},40] (* Harvey P. Dale, Nov 07 2024 *)

a(n) = 4*n^4+8*n^3+8*n^2+5*n+1.
From Stefano Spezia, Jul 09 2024: (Start)
G.f.: (1 + 21*x + 51*x^2 + 23*x^3)/(1 - x)^5.
E.g.f.: exp(x)*(1 + 25*x + 60*x^2 + 32*x^3 + 4*x^4). (End)

A374384 a(n) = floor(Sum_{k=n^3..(n+1)^3} k^(1/3)).

Original entry on oeis.org

1, 12, 51, 134, 281, 508, 835, 1278, 1857, 2588, 3491, 4582, 5881, 7404, 9171, 11198, 13505, 16108, 19027, 22278, 25881, 29852, 34211, 38974, 44161, 49788, 55875, 62438, 69497, 77068, 85171, 93822, 103041, 112844, 123251, 134278, 145945, 158268, 171267, 184958, 199361

Offset: 0

Amrit Awasthi, Jul 07 2024

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000384, A003215, A005804, A014105, A374489.

Table[Floor[Sum[(n^3+k)^(1/3),{k,0,3n^2+3n+1}]],{n,0,40}] (* Stefano Spezia, Jul 07 2024 *)

a(n) = 3*n^3+9*n^2\2+4*n+1; \\ Michel Marcus, Jul 09 2024

a(n) = floor(3*n^3+9*n^2/2+4*n+1).
a(2*n) = 24*n^3 + 18*n^2 + 8*n + 1.
a(2*n-1) = 24*n^3-18*n^2+8*n-2 for n > 0.
a(2*n) = A248575(2*n) + 4*n + 1.
a(2*n-1) = A248575(2*n-1) + 4*n - 2.
From Stefano Spezia, Jul 09 2024: (Start)
G.f.: (1 + 9*x + 17*x^2 + 7*x^3 + 2*x^3)/((1 - x)^4*(1 + x)).
E.g.f.: exp(x)*(1 + 11*x + 14*x^2 + 3*x^3). (End)

A350855 a(0) = 1, a(n) = (n+1)*a(n-1) + (n-2).

Original entry on oeis.org

1, 1, 3, 13, 67, 405, 2839, 22717, 204459, 2044597, 22490575, 269886909, 3508529827, 49119417589, 736791263847, 11788660221565, 200407223766619, 3607330027799157, 68539270528183999, 1370785410563679997, 28786493621837279955, 633302859680420159029, 14565965772649663657687

Offset: 0

Amrit Awasthi, Jan 19 2022

nonn

			a(1) = (1+1)*a(0) + (1-2) = 2-1 = 1.
a(2) = (2+1)*a(1) + (2-2) = 3.

Harvey P. Dale, Table of n, a(n) for n = 0..449

Cf. A020543.

Nest[Append[#1, (#2 + 1) #1[[-1]] + (#2 - 2)] & @@ {#, Length@ #} &, {1}, 20] (* Michael De Vlieger, Jan 19 2022 *)
nxt[{n_,a_}]:={n+1,a(n+2)+n-1}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Feb 03 2025 *)

a(n) = if (n, (n+1)*a(n-1) + (n-2), 1); \\ Michel Marcus, Jan 19 2022

terms = [1]
for n in range(1, 20):
    terms.append((n+1)*terms[-1]+n-2)
print(terms) # Gleb Ivanov, Jan 19 2022

a(n) ~ (6-2e)*(n+1)!.

E.g.f.: (exp(x)*(4*x-x^2-5)+6)/(x-1)^2. - Alois P. Heinz, Jan 19 2022

A345471 a(0) = a(1) = 1, a(n) is the smallest positive integer m >= a(n-1) + a(n-2) such that gcd(a(k),m) = 1 for all 1 < k <= n - 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 11, 17, 29, 47, 79, 127, 211, 343, 557, 907, 1469, 2377, 3847, 6229, 10079, 16319, 26399, 42719, 69119, 111841, 180967, 292811, 473779, 766607, 1240387, 2006999, 3247393, 5254397, 8501791, 13756189, 22258001, 36014191, 58272197, 94286389, 152558587

Offset: 0

Amrit Awasthi, Jun 20 2021

nonn

First differs from A073021 at a(12).

			a(5) = 11 because 11 is the smallest number greater than or equal to a(3) + a(4) = 5 + 3 = 8 which is coprime to all previous terms of the sequence.

Cf. A073021, A345020.

a[0] = a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + a[n - 2]}, While[! AllTrue[Range[2, n - 1], CoprimeQ[a[#], k] &], k++]; k]; Array[a, 40, 0] (* Amiram Eldar, Jun 20 2021 *)

A345020 a(0) = a(1) = 1, a(n) = largest natural number m <= a(n-1) + a(n-2) where gcd(m,a(k)) = 1 for all 1 < k <= n-1.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 361, 587, 947, 1531, 2477, 4007, 6481, 10487, 16963, 27449, 44393, 71837, 116227, 188063, 304289, 492343, 796627, 1288967, 2085593, 3374557, 5460139, 8834689, 14294827, 23129507, 37424333, 60553837, 97978169

Offset: 0

Amrit Awasthi, Jun 05 2021

nonn

First differs from A055500 at a(14).

			a(5) = 7 because 7 is the largest number less than or equal to a(4) + a(3) = 8 which is coprime to all the previous terms of sequence.

Robert Israel, Table of n, a(n) for n = 0..4781

Cf. A055500.

A[0]:= 1:
A[1]:= 1: P:= 1:
for n from 2 to 100 do
  for k from A[n-2]+A[n-1] by -1 do
    if igcd(k,P) = 1 then break fi
  od;
  A[n]:= k;
  P:= P*k;
od:
convert(A,list); # Robert Israel, Oct 23 2024

a[0] = a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + a[n - 2]}, While[! AllTrue[Range[2, n - 2], CoprimeQ[a[#], k] &], k--]; k]; Array[a, 50, 0] (* Amiram Eldar, Jun 05 2021 *)

A344496 a(0)=0; for n > 0, a(n) = a(n-1)n + n if n is odd, (a(n-1) + n)n otherwise.

Original entry on oeis.org

0, 1, 6, 21, 100, 505, 3066, 21469, 171816, 1546353, 15463630, 170099941, 2041199436, 26535592681, 371498297730, 5572474465965, 89159591455696, 1515713054746849, 27282834985443606, 518373864723428533, 10367477294468571060, 217717023183839992281, 4789774510044479830666

Offset: 0

Amrit Awasthi, May 21 2021

nonn

			a(0) = 0;
a(1) =  a(0)*1 + 1 =   0 + 1    =   1;
a(2) = (a(1)+2)* 2 =  (1 + 2)*2 =   6;
a(3) =  a(2)*3 + 3 = 6*3 + 3    =  21;
a(4) = (a(3)+4)* 4 = (21 + 4)*4 = 100.

Cf. A344262, A344495.

a:= proc(n) a(n):= n*a(n-1) + n^(2-(n mod 2)) end: a(0):= 0:
seq(a(n), n=0..22);  # Alois P. Heinz, May 21 2021

a[0] = 0; a[n_] := a[n] = n * (a[n - 1] + If[OddQ[n], 1, n]); Array[a, 30, 0] (* Amiram Eldar, May 21 2021 *)
Table[n*(-1 + 3*E*Gamma[n,1] + (n-1)*Subfactorial[n-2])/2, {n, 0, 30}] (* Vaclav Kotesovec, Jun 05 2021 *)

a(n) ~ n! * (3*exp(1)/2 + exp(-1)/2). - Vaclav Kotesovec, Jun 05 2021

A344935 a(0)=1; for n > 0, a(n) = n(a(n-1) + i^(n-1)) if n is odd, na(n-1) + i^n otherwise, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 6, 25, 130, 779, 5446, 43569, 392130, 3921299, 43134278, 517611337, 6728947394, 94205263515, 1413078952710, 22609263243361, 384357475137154, 6918434552468771, 131450256496906630, 2629005129938132601, 55209107728700784642, 1214600370031417262123

Offset: 0

Amrit Awasthi, Jun 03 2021

nonn

			a(0) = 1;
a(1) = 1*(a(0) + i^(1-1)) =  2;
a(2) = 2*a(1)  + i^2      =  3;
a(3) = 3*(a(2) + i^2)     =  6;
a(4) = 4*a(3)  + i^4      = 25.

Cf. A344262, A344419, A049470, A009001.

A344935 := proc(n)
    option remember ;
    if n = 0 then
        1;
    elif type(n,'odd') then
        n*(procname(n-1)+I^(n-1)) ;
    else
        n*procname(n-1)+I^n ;
    end if;
    simplify(%) ;
end proc:
seq(A344935(n),n=0..40) ; # R. J. Mathar, Aug 19 2022

a[0] = 1; a[n_] := a[n] = If[OddQ[n], n*(a[n - 1] + I^(n - 1)), n*a[n - 1] + I^n]; Array[a, 30, 0] (* Amiram Eldar, Jun 03 2021 *)

a(n) = if (n==0, 1, if (n%2, n*(a(n-1) + I^(n-1)), n*a(n-1) + I^n)); \\ Michel Marcus, Jun 05 2021

E.g.f.: (1+x)*cos(x)/(1-x).
Lim_{n->infinity} a(n)/n! = 2*cos(1) = 2*A049470.
D-finite with recurrence a(n) -n*a(n-1) +2*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Aug 19 2022

A344495 a(0)=1; for n>0 a(n)=(a(n-1) + n) * n if n is odd, a(n-1)*n + n otherwise.

Original entry on oeis.org

1, 2, 6, 27, 112, 585, 3516, 24661, 197296, 1775745, 17757460, 195332181, 2343986184, 30471820561, 426605487868, 6399082318245, 102385317091936, 1740550390563201, 31329907030137636, 595268233572615445, 11905364671452308920, 250012658100498487761, 5500278478210966730764

Offset: 0

Amrit Awasthi, May 21 2021

			a(0) = 1;
a(1) = (a(0)+1)*1 = (1+1)*1 = 2 ;
a(2) = a(1)*2+2 = (2*2)+2 = 6 ;
a(3) = (a(2)+3)*3 = (6+3)*3 = 9 ;

Cf. A344262.

a:= proc(n) a(n):= n*a(n-1) + n^(1+(n mod 2)) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 21 2021

a[0] = 1; a[n_] := a[n] = n*(a[n - 1] + If[OddQ[n], n, 1]); Array[a, 30, 0] (* Amiram Eldar, May 21 2021 *)

a(n) ~ n! * (1 + 3*exp(1)/2 - exp(-1)/2). - Vaclav Kotesovec, Jun 05 2021

A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.

Original entry on oeis.org

1, 2, 5, 18, 73, 370, 2221, 15554, 124433, 1119906, 11199061, 123189682, 1478276185, 19217590418, 269046265853, 4035693987810, 64571103804961, 1097708764684354, 19758757764318373, 375416397522049106, 7508327950440982121, 157674886959260624562

Offset: 0

Amrit Awasthi, May 13 2021

nonn

			a(0) = 1;
a(1) = (a(0)+1)*1 =  (1+1)*1 =   2;
a(2) = (a(1)*2)+1 =  (2*2)+1 =   5;
a(3) = (a(2)+1)*3 =  (5+1)*3 =  18;
a(4) = (a(3)*4)+1 = (18*4)+1 =  73;
a(5) = (a(4)+1)*5 = (73+1)*5 = 370.

Cf. A033540, A073743, A090805, A137204, A155521, A344317.

a:= proc(n) a(n):= n*a(n-1) + n^(n mod 2) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 14 2021

a[1] = 1; a[n_] := a[n] = If[OddQ[n], (n - 1)*a[n - 1] + 1, (n - 1)*(a[n - 1] + 1)]; Array[a, 25] (* Amiram Eldar, May 13 2021 *)

E.g.f.: (x+1)*cosh(x)/(1-x). - Alois P. Heinz, May 14 2021
Lim_{n->infinity} a(n)/n! = 2*cosh(1) = A137204 = 2*A073743. - Amrit Awasthi, May 15 2021
a(n) = A344317(n) - A155521(n-1) for n > 0. - Alois P. Heinz, May 18 2021

A343010 Integers k for which there exist three consecutive Fibonacci numbers a, b, and c such that abc = k*(a+b+c).

Original entry on oeis.org

0, 1, 3, 20, 52, 357, 935, 6408, 16776, 114985, 301035, 2063324, 5401852, 37024845, 96932303, 664383888, 1739379600, 11921885137, 31211900499, 213929548580, 560074829380, 3838809989301, 10050135028343, 68884650258840, 180342355680792, 1236084894669817

Offset: 1

Amrit Awasthi, Apr 02 2021

F(n-1)*F(n)*F(n+1) = k(n)*(F(n-1)+F(n)+F(n+1)). This implies that k(n)=(F(n-1)*F(n))/2. Now k(n) will be an integer only when n is of the form 3*m or 3*m+1. Therefore we get k = (F(3*m+-1)*F(3*m))/2.

			0 is a term because F(0)*F(1)*F(2)/(F(0)+F(1)+F(2)) is 0*1*1/(0+1+1) = 0.
1 is a term because F(2)*F(3)*F(4)/(F(2)+F(3)+F(4)) is 1*2*3/(1+2+3) = 1.
3 is a term because F(3)*F(4)*F(5)/(F(3)+F(4)+F(5)) is 2*3*5/(2+3+5) = 3.

Index entries for linear recurrences with constant coefficients, signature (0,17,0,17,0,-1).

Cf. A000045 (Fibonacci numbers), 1/2 times the even terms of sequence A001654.

Cf. A065563 (F(n-1)*F(n)*F(n+1)), A078642 (F(n-1)+F(n)+F(n+1)).

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> (k-> mul(F(k+j), j=0..2)/add(F(k+j), j=0..2))(floor(3*n/2)-1):
seq(a(n), n=1..30);  # Alois P. Heinz, Apr 02 2021

Select[Table[(Fibonacci[k-1]*Fibonacci[k]*Fibonacci[k+1])/(Fibonacci[k-1]+Fibonacci[k]+Fibonacci[k+1]),{k,37}],IntegerQ] (* or *)
b[k_]:=Fibonacci[3k-1]*Fibonacci[3k]/2; c[k_]:=Fibonacci[3k+1]*Fibonacci[3k]/2; Union[Table[b[k],{k,0,12}],Table[c[k],{k,0,12}]] (* Stefano Spezia, Apr 03 2021 *)

r(m)={fibonacci(m)*fibonacci(m-1)*fibonacci(m+1)/(fibonacci(m)+fibonacci(m-1)+fibonacci(m+1))}
{ for(m=2, 30, my(t=r(m)); if(!frac(t), print1(t, ", ")))} \\ Andrew Howroyd, Apr 02 2021

Union of the two sequences b(k) and c(k) defined respectively as F(3*k-1)*F(3*k)/2 and F(3*k+1)*F(3*k)/2.

G.f.: x^2*(1 + 3*x + 3*x^2 + x^3)/(1 - 17*x^2 - 17*x^4 + x^6). - Stefano Spezia, Apr 03 2021

cp's OEIS Frontend

User: Amrit Awasthi

A374489 a(n) = floor(Sum_{k=n^4..(n+1)^4} k^(1/4)).

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A374384 a(n) = floor(Sum_{k=n^3..(n+1)^3} k^(1/3)).

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A350855 a(0) = 1, a(n) = (n+1)*a(n-1) + (n-2).

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A345471 a(0) = a(1) = 1, a(n) is the smallest positive integer m >= a(n-1) + a(n-2) such that gcd(a(k),m) = 1 for all 1 < k <= n - 1.

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A345020 a(0) = a(1) = 1, a(n) = largest natural number m <= a(n-1) + a(n-2) where gcd(m,a(k)) = 1 for all 1 < k <= n-1.

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A344496 a(0)=0; for n > 0, a(n) = a(n-1)*n + n if n is odd, (a(n-1) + n)*n otherwise.

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A344935 a(0)=1; for n > 0, a(n) = n*(a(n-1) + i^(n-1)) if n is odd, n*a(n-1) + i^n otherwise, where i = sqrt(-1).

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A344495 a(0)=1; for n>0 a(n)=(a(n-1) + n) * n if n is odd, a(n-1)*n + n otherwise.

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A344496 a(0)=0; for n > 0, a(n) = a(n-1)n + n if n is odd, (a(n-1) + n)n otherwise.

A344935 a(0)=1; for n > 0, a(n) = n(a(n-1) + i^(n-1)) if n is odd, na(n-1) + i^n otherwise, where i = sqrt(-1).

A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.

A343010 Integers k for which there exist three consecutive Fibonacci numbers a, b, and c such that abc = k*(a+b+c).