A059045 -id:A059045 - cp's OEIS frontend

A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).

1, 7, 34, 142, 547, 2005, 7108, 24604, 83653, 280483, 930022, 3055786, 9964519, 32285041, 104029576, 333612088, 1065406345, 3389929279, 10750918570, 33996147910, 107218620331, 337346390797, 1059110761804, 3318547053652, 10379285465677, 32408789311195, 101039166676078

Offset: 1

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 1..600
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 14.
Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, arXiv:1809.00122 [math.CA], 2018.
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A027261, A027471, A059045, A064017, A079272.

[((2*n - 1)*3^n + 1)/4: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

LinearRecurrence[{7, -15, 9}, {1, 7, 34}, 25] (* L. Edson Jeffery, May 08 2015 *)

From Henry Bottomley, Dec 18 2000: (Start)
a(n) = ((2*n-1)*3^n + 1)/4.
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n > 3.
a(n) = 1 + 2*3 + 3*3^2 + .. + n*3^(n-1).
a(n) = a(n-1) + A027471(n+1). (End)
G.f.: x/((1-x)*(1-3*x)^2). - Colin Barker, Jul 28 2012
a(n) = f^n(n)/2 with f(x) = 3*x-1. - Glen Gilchrist, Apr 10 2019
E.g.f.: exp(x)*(1 + exp(2*x)*(6*x - 1))/4. - Stefano Spezia, May 14 2024
a(n) = 6*a(n-1) - 9*a(n-2) + 1 for n > 2. - Elmo R. Oliveira, May 24 2025

A217785 Smallest integer s>n such that 1+2s+3s^2+...+n*s^{n-1} is prime.

Original entry on oeis.org

3, 12, 12, 9, 21, 12, 26, 23, 30, 24, 138, 33, 80, 32, 54, 192, 48, 40, 4500, 48, 50, 192, 30, 88, 32, 114, 178, 48, 45, 42, 356, 41, 53, 138, 174, 66, 44, 990, 120, 819, 2898, 112, 1052, 122, 164, 132, 108, 77, 540, 198, 106, 135, 237, 98, 234, 162, 83, 720, 3870, 135, 188, 1014, 94, 489, 180, 110, 204, 180, 107, 468, 1542, 508, 218, 608, 88, 102, 228, 140, 3890, 93, 361, 1848, 462, 99, 125, 390, 92, 237, 933, 172, 606, 303, 208, 924, 114, 266, 156, 410, 1330

Offset: 2

Zhi-Wei Sun, Mar 24 2013

nonn

Conjecture: For each n=2,3,... there are infinitely many primes of the form 1+2*s+...+n*s^{n-1}, where s is a positive integer; moreover, we have a(n)<12*n^2.
This is related to the following conjecture of the author: The polynomials s_n(x)=sum_{k=0}^n(k+1)x^k (n=1,2,3,...) are all irreducible over the field of rational numbers; moreover, s_n(x) is reducible modulo every prime if and only if n has the form 8k(k+1), where k is a positive integer.
Sum_{k=1..n} k*s^(k-1) = (1+n*s^(n+1)-s^n*(n+1))/(s-1)^2, see A059045. - R. J. Mathar, Mar 29 2013

			a(20)=4500<12*20^2=4800 since 4500 is the least integer s>20 with 1+2*s+3*s^2+...+20*s^{19} prime.

Zhi-Wei Sun and Charles R Greathouse IV, Table of n, a(n) for n = 2..1000 (first 450 terms from Sun)

Cf. A000040.

A[n_,x_]:=A[n,x]=Sum[(k+1)*x^k,{k,0,n-1}]
Do[Do[If[PrimeQ[A[n,s]]==True,Print[n," ",s];Goto[aa]],{s,n+1,12*n^2-1}];
Print[n," ",counterexample];Label[aa];Continue,{n,2,100}]

f(n,s)=my(t);forstep(k=n,1,-1,t=s*t+k);t
a(n)=my(s=n);while(!ispseudoprime(f(n,s++)),);s \\ Charles R Greathouse IV, Mar 25 2013

A108283 Triangle read by rows, generated from (..., 3, 2, 1).

Original entry on oeis.org

1, 1, 3, 1, 5, 6, 1, 7, 17, 10, 1, 9, 34, 49, 15, 1, 11, 57, 142, 129, 21, 1, 13, 86, 313, 547, 321, 28, 1, 15, 121, 586, 1593, 2005, 769, 36, 1, 17, 162, 985, 3711, 7737, 7108, 1793, 45, 1, 19, 209, 1534, 7465, 22461, 36409, 24604, 4097, 55, 1, 21, 262, 2257, 13539, 54121, 131836, 167481, 83653, 9217, 66

Offset: 1

Gary W. Adamson, May 30 2005

Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...).

			4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142.
First few rows of the triangle:
  1;
  1,  3;
  1,  5,  6;
  1,  7, 17,  10;
  1,  9, 34,  49,  15;
  1, 11, 57, 142, 129, 21;
  ...

Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917.

A108283 := proc(n,k)
    local x ;
    x := n-k+1 ;
    add( i*x^(i-1),i=1..k) ;
end proc:
seq(seq( A108283(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Sep 14 2016

T[, 1] := 1; T[n, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2016 *)

n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1.

T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - Jean-François Alcover, Sep 13 2016

More terms from Jean-François Alcover, Sep 13 2016

cp's OEIS Frontend

A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).

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A217785 Smallest integer s>n such that 1+2s+3s^2+...+n*s^{n-1} is prime.

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A108283 Triangle read by rows, generated from (..., 3, 2, 1).

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Maple

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cp's OEIS Frontend

A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A217785 Smallest integer s>n such that 1+2*s+3*s^2+...+n*s^{n-1} is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A108283 Triangle read by rows, generated from (..., 3, 2, 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A217785 Smallest integer s>n such that 1+2s+3s^2+...+n*s^{n-1} is prime.