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A062717 Numbers m such that 6*m+1 is a perfect square. 21
0, 4, 8, 20, 28, 48, 60, 88, 104, 140, 160, 204, 228, 280, 308, 368, 400, 468, 504, 580, 620, 704, 748, 840, 888, 988, 1040, 1148, 1204, 1320, 1380, 1504, 1568, 1700, 1768, 1908, 1980, 2128, 2204, 2360, 2440, 2604, 2688, 2860, 2948, 3128, 3220, 3408, 3504 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
X values of solutions to the equation 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007
Arithmetic averages of the k triangular numbers 0, 1, 3, 6, ..., (k-1)*k/2 that take integer values. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 [edited by Jon E. Schoenfield, Jan 10 2015]
Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
a(n) are integers produced by Sum_{i = 1..k-1} i*(k-i)/k, for some k > 0. Values for k are given by A007310 = sqrt(6*a(n)+1), the square roots of those perfect squares. - Richard R. Forberg, Feb 16 2015
Equivalently, numbers of the form 2*h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the sixth comment of A152749). - Bruno Berselli, Feb 02 2017
LINKS
FORMULA
G.f.: 4*x^2*(1 + x + x^2) / ( (1+x)^2*(1-x)^3 ).
a(2*k) = k*(6*k+2), a(2*k+1) = 6*k^2 + 10*k + 4. - Mohamed Bouhamida, Nov 06 2007
a(n) = n^2 - n + 2*ceiling((n-1)/2)^2. - Gary Detlefs, Feb 23 2010
From Bruno Berselli, Nov 28 2010: (Start)
a(n) = (6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4.
6*a(n) + 1 = A007310(n)^2. (End)
E.g.f.: (3*x^2*exp(x) - x*exp(-x) + sinh(x))/2. - Ilya Gutkovskiy, Jun 18 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 21 2021
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = (9-sqrt(3)*Pi)/6.
Sum_{n>=2} (-1)^n/a(n) = 3*(log(3)-1)/2. (End)
MAPLE
seq(n^2+n+2*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010
MATHEMATICA
Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* Harvey P. Dale, Mar 10 2013 *)
PROG
(PARI) je=[]; for(n=0, 7000, if(issquare(6*n+1), je=concat(je, n))); je
(PARI) { n=0; for (m=0, 10^9, if (issquare(6*m + 1), write("b062717.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 09 2009
(Magma) [(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // Wesley Ivan Hurt, Apr 21 2021
(Python)
def A062717(n): return (n*(3*n + 4) + 1 if n&1 else n*(3*n + 2))>>1 # Chai Wah Wu, Jan 31 2023
CROSSREFS
Equals 4 * A001318.
Cf. A007310.
Diagonal of array A323674. - Sally Myers Moite, Feb 03 2019
Sequence in context: A087254 A160726 A191483 * A084922 A180794 A047185
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Jul 14 2001
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)