A122430 -id:A122430 - cp's OEIS frontend

A056109 Fifth spoke of a hexagonal spiral.

1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321, 386, 457, 534, 617, 706, 801, 902, 1009, 1122, 1241, 1366, 1497, 1634, 1777, 1926, 2081, 2242, 2409, 2582, 2761, 2946, 3137, 3334, 3537, 3746, 3961, 4182, 4409, 4642, 4881, 5126, 5377, 5634, 5897, 6166, 6441

Offset: 0

Henry Bottomley, Jun 09 2000

Squared distance from (0,0,-1) to (n,n,n) in R^3. - James R. Buddenhagen, Jun 15 2013

			Illustration of initial terms:
.
.                                                o
.                           o                 o o o o
.            o           o o o o           o o o o o o o
.   o     o o o o     o o o o o o o     o o o o o o o o o o
.            o           o o o o           o o o o o o o
.                           o                 o o o o
.                                                o
.
.   1        6              17                   34
- _Aaron David Fairbanks_, Feb 16 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms
Tanya Khovanova, Recursive Sequences
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Leo Tavares, Triple Diamond Illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008810, A122430 (prime terms).
Other spokes: A003215, A056105, A056106, A056107, A056108.
Other spirals: A054552.
Cf. A000290.

List([0..50],n->3*n^2+2*n+1); # Muniru A Asiru, Oct 07 2018

[3*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Mar 15 2013

seq(coeff(series(n!*(exp(x)*(3*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 07 2018

Table[3 n^2 + 2 n + 1, {n, 0, 100}] (* Vincenzo Librandi, Mar 15 2013 *)
LinearRecurrence[{3,-3,1},{1,6,17},60] (* Harvey P. Dale, Mar 28 2019 *)

{a(n) = 3*n^2 + 2*n + 1}; /* Michael Somos, Aug 03 2006 */

Vec((1+3*x+2*x^2)/(1-3*x+3*x^2-x^3)+O(x^100)) \\ Stefano Spezia, Oct 17 2018

a(n) = 3n^2+2n+1 = a(n-1)+6n-1 = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+4n = A056106(n)+3n = A056107(n)+2n = A056108(n)+n = A003215(n)-n.
G.f.: (1+3*x+2*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
G.f.: (1 + x) * (1 + 2*x) / (1 - x)^3. - Michael Somos, Feb 04 2012
a(n) = A008810(3*n + 1) = A056105(-n). - Michael Somos, Aug 03 2006
E.g.f.: exp(x)*(1 + 5*x + 3*x^2). - Stefano Spezia, Oct 06 2018
a(n) = A000290(n+1) + 2*A000290(n). - Leo Tavares, May 29 2023
a(n) = A069894(n) - A000290(n+1). - Jarrod G. Sage, Jul 19 2024

A216894 n - (sum of prime factors of n) is a positive square.

Original entry on oeis.org

1, 6, 22, 54, 68, 164, 166, 336, 388, 454, 588, 854, 886, 1086, 1122, 1124, 1636, 1710, 1828, 2182, 2356, 2468, 2702, 2960, 3046, 3048, 3708, 3748, 3770, 4036, 4054, 4655, 5106, 5394, 5636, 6502, 7108, 7368, 7956, 8324, 9170, 9188, 9412, 9438, 9471, 9726

Offset: 1

Michel Lagneau, Sep 19 2012

nonn

Contains 4p for odd prime p if 3p-2 is a square, in particular if p is in A122430. - Robert Israel, Apr 13 2014

			54 = 2*3^3 and 54 -(2+3) = 49 is a square, hence 54 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..2121

Cf. A008472.

filter:= proc(x) local y;
    y:= x - add(i, i=numtheory:-factorset(x));
    y > 0 and issqr(y)
end proc;
N:= 10000;  # to get all entries <= N
A216894:= select(filter,[$1..N]); # Robert Israel, Apr 13 2014

nspfQ[n_]:=Module[{c=n-Total[Transpose[FactorInteger[n]][[1]]]},c>0 && IntegerQ[ Sqrt[c]]]; Join[{1},Select[Range[10000],nspfQ]] (* Harvey P. Dale, Feb 12 2015 *)

cp's OEIS Frontend

A056109 Fifth spoke of a hexagonal spiral.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI