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A156859 The main column of a version of the square spiral. 32

%I #52 Feb 09 2024 10:12:12

%S 0,3,7,14,22,33,45,60,76,95,115,138,162,189,217,248,280,315,351,390,

%T 430,473,517,564,612,663,715,770,826,885,945,1008,1072,1139,1207,1278,

%U 1350,1425,1501,1580,1660,1743,1827,1914,2002,2093,2185,2280,2376,2475,2575

%N The main column of a version of the square spiral.

%C This spiral is sometimes called an Ulam spiral, but square spiral is a better name. - _N. J. A. Sloane_, Jul 27 2018

%C 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 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 Partial Sums of A014601(n). - _Wesley Ivan Hurt_, Oct 11 2013

%H E. Apricena, <a href="/A156859/a156859.png">A version of Ulam Spiral divided into four parts.</a>

%H 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 Marco RipĂ , <a href="http://vixra.org/abs/1508.0201">The n x n x n Points Problem Optimal Solution</a>, viXra.org.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = n^2 + n + floor((n+1)/2) = A002378(n) + A004526(n+1) = A002620(n+1) + 3*A002620(n).

%F From _R. J. Mathar_, Feb 20 2009: (Start)

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

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

%F a(n-1) = floor(n/(e^(1/n)-1)). - _Richard R. Forberg_, Jun 19 2013

%F a(n) = A000290(n+1) + A004526(-n-1). - _Wesley Ivan Hurt_, Jul 15 2013

%F a(n) + a(n+1) = A014105(n+1). - _R. J. Mathar_, Jul 15 2013

%F a(n) = floor(A000384(n+1)/2). - _Bruno Berselli_, Nov 11 2013

%p A156859:=n->n^2+n+floor((n+1)/2); seq(A156859(k), k=0..100); # _Wesley Ivan Hurt_, Oct 11 2013

%t Table[n^2 + n + Floor[(n+1)/2], {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 11 2013 *)

%Y Cf. A000384, A014601 (first differences), A115258.

%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

%Y 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 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 nonn,easy

%O 0,2

%A Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 17 2009

%E More terms added by _Wesley Ivan Hurt_, Oct 11 2013

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)