cp's OEIS Frontend

A156859 The main column of a version of the square spiral.

%I A156859 #60 Apr 26 2024 05:08:36
%S A156859 0,3,7,14,22,33,45,60,76,95,115,138,162,189,217,248,280,315,351,390,
%T A156859 430,473,517,564,612,663,715,770,826,885,945,1008,1072,1139,1207,1278,
%U A156859 1350,1425,1501,1580,1660,1743,1827,1914,2002,2093,2185,2280,2376,2475,2575
%N A156859 The main column of a version of the square spiral.
%C A156859 This spiral is sometimes called an Ulam spiral, but square spiral is a better name. - _N. J. A. Sloane_, Jul 27 2018
%C A156859 It is easy to see that the only two primes in the sequence are 3, 7. Therefore the primes of the version of Ulam spiral are divided into four parts (see also A035608): northeast (NE), northwest (NW), southwest (SW), and southeast (SE).
%C A156859 Number of pairs (x,y) having x and y of opposite parity with x in {0,...,n} and y in {0,...,2n}. - _Clark Kimberling_, Jul 02 2012
%C A156859 Partial Sums of A014601(n). - _Wesley Ivan Hurt_, Oct 11 2013
%H A156859 E. Apricena, <a href="/A156859/a156859.png">A version of Ulam Spiral divided into four parts.</a>
%H A156859 Minh Nguyen, <a href="https://aquila.usm.edu/cgi/viewcontent.cgi?article=1774&amp;context=honors_theses">2-adic Valuations of Square Spiral Sequences</a>, Honors Thesis, Univ. of Southern Mississippi (2021).
%H A156859 Marco Ripà, <a href="http://vixra.org/abs/1508.0201">The n x n x n Points Problem Optimal Solution</a>, viXra.org.
%H A156859 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A156859 a(n) = n^2 + n + floor((n+1)/2) = A002378(n) + A004526(n+1) = A002620(n+1) + 3*A002620(n).
%F A156859 From _R. J. Mathar_, Feb 20 2009: (Start)
%F A156859 G.f.: x*(3+x)/((1+x)*(1-x)^3).
%F A156859 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End)
%F A156859 a(n-1) = floor(n/(e^(1/n)-1)). - _Richard R. Forberg_, Jun 19 2013
%F A156859 a(n) = A000290(n+1) + A004526(-n-1). - _Wesley Ivan Hurt_, Jul 15 2013
%F A156859 a(n) + a(n+1) = A014105(n+1). - _R. J. Mathar_, Jul 15 2013
%F A156859 a(n) = floor(A000384(n+1)/2). - _Bruno Berselli_, Nov 11 2013
%F A156859 E.g.f.: (x*(5 + 2*x)*cosh(x) + (1 + 5*x + 2*x^2)*sinh(x))/2. - _Stefano Spezia_, Apr 24 2024
%F A156859 Sum_{n>=1} 1/a(n) = 4/9 + 2*log(2) - Pi/3. - _Amiram Eldar_, Apr 26 2024
%p A156859 A156859:=n->n^2+n+floor((n+1)/2); seq(A156859(k), k=0..100); # _Wesley Ivan Hurt_, Oct 11 2013
%t A156859 Table[n^2 + n + Floor[(n+1)/2], {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 11 2013 *)
%Y A156859 Cf. A000290, A000384, A004526, A014601 (first differences), A115258.
%Y A156859 Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y A156859 Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y A156859 Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%K A156859 nonn,easy
%O A156859 0,2
%A A156859 Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 17 2009
%E A156859 More terms added by _Wesley Ivan Hurt_, Oct 11 2013