A299645 - OEIS

%I #32 Mar 18 2022 05:20:26

%S 0,3,13,22,42,57,87,108,148,175,225,258,318,357,427,472,552,603,693,

%T 750,850,913,1023,1092,1212,1287,1417,1498,1638,1725,1875,1968,2128,

%U 2227,2397,2502,2682,2793,2983,3100,3300,3423,3633,3762,3982,4117,4347,4488,4728,4875

%N Numbers of the form m*(8*m + 5), where m is an integer.

%C Equivalently, numbers k such that 32*k + 25 is a square. This means that 4*a(n) + 3 is a triangular number.

%C Interleaving of A139277 and A139272 (without 0).

%H Colin Barker, <a href="/A299645/b299645.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 O.g.f.: x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2).

%F E.g.f.: (1 + 2*x - (1 - 8*x^2)*exp(2*x))*exp(-x)/4.

%F a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

%F a(n) = (8*n*(n - 1) - (2*n - 1)*(-1)^n - 1)/4 = (2*n + (-1)^n - 1)*(4*n - 3*(-1)^n - 2)/4. Therefore, 3 and 13 are the only prime numbers in this sequence.

%F a(n) + a(n+1) = 4*n^2 for even n, otherwise a(n) + a(n+1) = 4*n^2 - 1.

%F From _Amiram Eldar_, Mar 18 2022: (Start)

%F Sum_{n>=2} 1/a(n) = 8/25 + (sqrt(2)-1)*Pi/5.

%F Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/5 - sqrt(2)*log(2*sqrt(2)+3)/5 - 8/25. (End)

%p seq((exp(I*Pi*x)*(1-2*x)+8*(x-1)*x-1)/4, x=1..50); # _Peter Luschny_, Feb 27 2018

%t Table[(8 n (n - 1) - (2 n - 1) (-1)^n - 1)/4, {n, 1, 50}]

%o (PARI) vector(50, n, nn; (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4)

%o (PARI) concat(0, Vec(x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Feb 27 2018

%o (Sage) [(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4 for n in (1..50)]

%o (Maxima) makelist((8*n*(n-1)-(2*n-1)*(-1)^n-1)/4, n, 1, 50);

%o (GAP) List([1..50], n -> (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4);

%o (Magma) [(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4: n in [1..50]];

%o (Python) [(8*n*(n-1)-(2*n-1)*(-1)**n-1)/4 for n in range(1, 60)]

%o (Julia) [div((8n*(n-1)-(2n-1)*(-1)^n-1), 4) for n in 1:50] # _Peter Luschny_, Feb 27 2018

%Y Cf. A139272, A139277.

%Y Subsequence of A011861, A047222.

%Y Cf. numbers of the form m*(8*m + h): A154260 (h=1), A014494 (h=2), A274681 (h=3), A046092 (h=4), this sequence (h=5), 2*A074377 (h=6), A274979 (h=7).

%K nonn,easy

%O 1,2

%A _Bruno Berselli_, Feb 26 2018