A062717 - OEIS

%I #97 Jan 31 2023 14:53:23

%S 0,4,8,20,28,48,60,88,104,140,160,204,228,280,308,368,400,468,504,580,

%T 620,704,748,840,888,988,1040,1148,1204,1320,1380,1504,1568,1700,1768,

%U 1908,1980,2128,2204,2360,2440,2604,2688,2860,2948,3128,3220,3408,3504

%N Numbers m such that 6*m+1 is a perfect square.

%C X values of solutions to the equation 6*X^3 + X^2 = Y^2. - _Mohamed Bouhamida_, Nov 06 2007

%C 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]

%C Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - _Reinhard Zumkeller_, Feb 21 2011

%C 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

%C 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

%H Harry J. Smith, <a href="/A062717/b062717.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F G.f.: 4*x^2*(1 + x + x^2) / ( (1+x)^2*(1-x)^3 ).

%F a(2*k) = k*(6*k+2), a(2*k+1) = 6*k^2 + 10*k + 4. - _Mohamed Bouhamida_, Nov 06 2007

%F a(n) = n^2 - n + 2*ceiling((n-1)/2)^2. - _Gary Detlefs_, Feb 23 2010

%F From _Bruno Berselli_, Nov 28 2010: (Start)

%F a(n) = (6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4.

%F 6*a(n) + 1 = A007310(n)^2. (End)

%F E.g.f.: (3*x^2*exp(x) - x*exp(-x) + sinh(x))/2. - _Ilya Gutkovskiy_, Jun 18 2016

%F 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

%F From _Amiram Eldar_, Mar 11 2022: (Start)

%F Sum_{n>=2} 1/a(n) = (9-sqrt(3)*Pi)/6.

%F Sum_{n>=2} (-1)^n/a(n) = 3*(log(3)-1)/2. (End)

%p seq(n^2+n+2*ceil(n/2)^2,n=0..48); # _Gary Detlefs_, Feb 23 2010

%t Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* _Harvey P. Dale_, Mar 10 2013 *)

%o (PARI) je=[]; for(n=0,7000, if(issquare(6*n+1),je=concat(je,n))); je

%o (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

%o (Magma) [(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // _Wesley Ivan Hurt_, Apr 21 2021

%o (Python)

%o def A062717(n): return (n*(3*n + 4) + 1 if n&1 else n*(3*n + 2))>>1 # _Chai Wah Wu_, Jan 31 2023

%Y Equals 4 * A001318.

%Y Cf. A005563, A046092, A001082, A002378, A036666.

%Y Cf. A160757, A000217. - _Vladimir Joseph Stephan Orlovsky_, Aug 05 2009

%Y Cf. A007310.

%Y Diagonal of array A323674. - _Sally Myers Moite_, Feb 03 2019

%K nonn,easy

%O 1,2

%A _Jason Earls_, Jul 14 2001