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 A156859 The main column of a version of the square spiral. 31
 0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, 162, 189, 217, 248, 280, 315, 351, 390, 430, 473, 517, 564, 612, 663, 715, 770, 826, 885, 945, 1008, 1072, 1139, 1207, 1278, 1350, 1425, 1501, 1580, 1660, 1743, 1827, 1914, 2002, 2093, 2185, 2280, 2376, 2475, 2575 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This spiral is sometimes called an Ulam spiral, but square spiral is a better name. - N. J. A. Sloane, Jul 27 2018 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). a(n) = [b(n) U c(n)], being b(n) = 4*n^2 + 3*n and c(n) = 4*n^2 + 7*n + 3 with n >= 0. - Paolo P. Lava, Mar 05 2009 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 Partial Sums of A014601(n). - Wesley Ivan Hurt, Oct 11 2013 LINKS E. Apricena, A version of Ulam Spiral divided into four parts. Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021). Marco RipĂ , The n x n x n Points Problem Optimal Solution, viXra.org. Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(n) = n^2 + n + floor((n+1)/2) = A002378(n) + A004526(n+1) = A002620(n+1) + 3*A002620(n). From R. J. Mathar, Feb 20 2009: (Start) G.f.: x*(3+x)/((1+x)*(1-x)^3). a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End) a(n) = -n - 1 + Sum_{j=0..n}Sum_{k=0..j} (2 + (-1)^k). - Paolo P. Lava, Mar 05 2009 a(n-1) = floor(n/(e^(1/n)-1)). - Richard R. Forberg, Jun 19 2013 a(n) = A000290(n+1) + A004526(-n-1). - Wesley Ivan Hurt, Jul 15 2013 a(n) + a(n+1) = A014105(n+1). - R. J. Mathar, Jul 15 2013 a(n) = floor(A000384(n+1)/2). - Bruno Berselli, Nov 11 2013 MAPLE P:=proc(i) local a, j, k, n; for n from 0 by 1 to i do a:=sum(sum(2-(-1)^k, k=0..j), j=0..n)-n-1; print(a); od; end: P(100); # Paolo P. Lava, Mar 05 2009 A156859:=n->n^2+n+floor((n+1)/2); seq(A156859(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013 MATHEMATICA Table[n^2 + n + Floor[(n+1)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 11 2013 *) CROSSREFS Cf. A000384, A014601 (first differences), A115258. Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951. 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. 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. Sequence in context: A249341 A310278 A154772 * A173209 A331240 A146931 Adjacent sequences:  A156856 A156857 A156858 * A156860 A156861 A156862 KEYWORD nonn,easy AUTHOR Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 17 2009 EXTENSIONS More terms added by Wesley Ivan Hurt, Oct 11 2013 STATUS approved

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Last modified June 30 15:29 EDT 2022. Contains 354943 sequences. (Running on oeis4.)