A020739 -id:A020739 - cp's OEIS frontend

A139570 a(n) = 2n(n+3).

0, 8, 20, 36, 56, 80, 108, 140, 176, 216, 260, 308, 360, 416, 476, 540, 608, 680, 756, 836, 920, 1008, 1100, 1196, 1296, 1400, 1508, 1620, 1736, 1856, 1980, 2108, 2240, 2376, 2516, 2660, 2808, 2960, 3116, 3276, 3440, 3608, 3780, 3956, 4136, 4320, 4508, 4700, 4896

Offset: 0

Omar E. Pol, May 19 2008

Numbers n such that 2*n + 9 is a square. - Vincenzo Librandi, Nov 24 2010

a(n) appears also as the fourth member of the quartet [p0(n), p1(n), p2(n), a(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n), and p2(n) = A054000(n+1). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A001105, A020739, A022998, A028552, A046092, A054000, A067728, A147973.

CoefficientList[Series[4 x (2 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 23 2014 *)

a(n)=2*n*(n+3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*A028552(n) = 2*n^2 + 6*n = n*(2*n+6).
a(n) = a(n-1) + 4*n + 4 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Paul Curtz, Mar 27 2011: (Start)
a(n) = A022998(n)*A022998(n+3).
a(n) = 4*A000096(n). (End)
G.f.: 4*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 31 2011
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=1} 1/a(n) = 11/36.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 5/36. (End)
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: 2*exp(x)*x*(4 + x).
a(n) = n*A020739(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A009056 Numbers >= 3.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

David W. Wilson

Same as Pisot sequences E(3,4), P(3,4), T(3,4). See A008776 for definitions of Pisot sequences.
Non-Fermat-exponents, positive integers n such that there are no solutions in positive integers of the equation a^n + b^n = c^n. - Tanya Khovanova, Jul 09 2011
Sums of twin primes. - Charles R Greathouse IV, Jun 21 2012

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

Cf. A009005, A020739, A055998.

Range[3,80] (* Harvey P. Dale, Sep 26 2017 *)

a(n)=n+2 \\ Charles R Greathouse IV, Jun 21 2012

a(n) = n + 2.
From R. J. Mathar, May 26 2008: (Start)
O.g.f.: x*(3-2*x)/(1-x)^2.
a(n) = A009005(n-1), n > 2. (End)
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(x + 2) - 2.
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = A055998(n) - A055998(n-1) = A020739(n-1)/2. (End)

A020744 Pisot sequences P(8,10), T(8,10).

Original entry on oeis.org

8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138

Offset: 0

David W. Wilson

Conjecturally, even sums of four primes. - Charles R Greathouse IV, Feb 16 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A005843, A020739. See A008776 for definitions of Pisot sequences.

Cf. A020705, A028563.

LinearRecurrence[{2,-1},{8,10},70] (* Harvey P. Dale, Jul 19 2015 *)
P[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Ceiling[a[n - 1]^2/a[n - 2] - 1/2]; Table[a[n], {n, 0, z}]]; P[8, 10, 65] (* or *)
T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[8, 10, 65] (* Michael De Vlieger, Aug 08 2016 *)

a(n)=2*n+8 \\ Charles R Greathouse IV, Feb 16 2012

pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 8, 10) \\ Colin Barker, Aug 08 2016

a(n) = 2*n + 8.
a(n) = 2*a(n-1) - a(n-2).
From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: 2*(4 - 3*x)/(1 - x)^2.
E.g.f.: 2*(4 + x)*exp(x).
a(n) = 2*A020705(n) = A028563(n+1) - A028563(n). (End)

A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.

Original entry on oeis.org

1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124

Offset: 1

Jaroslav Krizek, Aug 20 2018

nonn

This is the lexicographically earliest increasing sequence such that n-1 divides the sum of the first n terms.
Sequence b(n) of the sums of the first n+1 terms, Sum_{k=1..n+1} a(k): 3, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, ...
Sequence c(n) of quotients when a(n) is calculated, (Sum_{k=1..n+1} a(k) ) / n is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

			a(1) = 1 because 1 divides the sum of the first 2 (i.e., 1 + 1) terms (a(1) + a(2)) for whatever term a(2) > a(1).
a(2) = 2 because 2 is the smallest number > a(1) and 2 divides the sum of the first 3 (i.e., 2 + 1) terms (a(1) + a(2) + a(3)) for whatever term a(3) > a(2) such that 2 divides the sum a(1) + a(2) + a(3); the smallest number > a(2) with this property for a(3) is 3.
a(3) = 3.
a(4) = 6 because 6 is the smallest number > a(3) such that 3 divides the sum of the first 4 (i.e., 3 + 1) terms.
a(5) = 8 because 8 is the smallest number > a(4) such that 4 divides the sum of the first 5 (i.e., 4 + 1) terms.

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

Cf. A005843, A020739, A316156.

CoefficientList[Series[(1 + 2 x^3 - x^4)/(-1 + x)^2 , {x, 0, 50}], x] (* Stefano Spezia, Sep 16 2018 *)
Join[{1, 2, 3}, LinearRecurrence[{2, -1}, {6, 8}, 60]] (* Jean-François Alcover, Dec 30 2018 *)

a(1)=1, a(2)=2, a(3)=3, a(n)=2(n-1) for n >= 4.
a(n) = A005843(n-1) for n >= 4. - Antti Karttunen, Sep 16 2018
G.f.: (1 + 2*x^3 - x^4)/(-1 + x)^2. - Stefano Spezia, Sep 16 2018
a(n) = A020739(n-4) = 2*(n - 4) + 6 for n >= 4. - Georg Fischer, Jan 19 2019

Definition clarified by Georg Fischer and Alois P. Heinz, Jan 19 2019

cp's OEIS Frontend

A139570 a(n) = 2*n*(n+3).

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A009056 Numbers >= 3.

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A020744 Pisot sequences P(8,10), T(8,10).

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A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.

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A139570 a(n) = 2n(n+3).