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A033951
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Write 1,2,... in clockwise spiral; sequence gives numbers on positive x axis.
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51
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1, 8, 23, 46, 77, 116, 163, 218, 281, 352, 431, 518, 613, 716, 827, 946, 1073, 1208, 1351, 1502, 1661, 1828, 2003, 2186, 2377, 2576, 2783, 2998, 3221, 3452, 3691, 3938, 4193, 4456, 4727, 5006, 5293, 5588, 5891, 6202, 6521, 6848, 7183, 7526, 7877, 8236
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Ulam's spiral (S spoke of A054552). - Robert G. Wilson v, Oct 31 2011
a(n) is the first term in a sum of 2*n + 1 consecutive integers that equals (2*n + 1)^3. - Patrick J. McNab, Dec 24 2016
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..1000
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 4*n^2 + 3*n + 1.
G.f.: (1 + 5*x + 2*x^2)/(1-x)^3.
A014848(2n+1) = a(n).
Equals A132774 * [1, 2, 3, ...]; = binomial transform of [1, 7, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 28 2007
a(n) = A016754(n) - n. - Reinhard Zumkeller, May 17 2009
a(n) = a(n-1) + 8*n-1 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
a(0)=1, a(1)=8, a(2)=23, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 07 2015
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EXAMPLE
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Spiral begins:
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65--66--67--68--69--70--71--72--73
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64 37--38--39--40--41--42--43 74
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63 36 17--18--19--20--21 44 75
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62 35 16 5---6---7 22 45 76
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61 34 15 4 1 8 23 46 77
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60 33 14 3---2 9 24 47
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59 32 13--12--11--10 25 48
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58 31--30--29--28--27--26 49
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57--56--55--54--53--52--51--50
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MAPLE
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A033951:=n->4*n^2 + 3*n + 1: seq(A033951(n), n=0..100); # Wesley Ivan Hurt, Feb 11 2017
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MATHEMATICA
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lst={}; Do[p=4*n^2+3*n+1; AppendTo[lst, p], {n, 1, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 23}, 60] (* Harvey P. Dale, Feb 07 2015 *)
CoefficientList[Series[(1 + 5 x + 2 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 12 2017 *)
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PROG
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(PARI) a(n)=4*n^2+3*n+1
(Python)
[4*n**2 + 3*n + 1 for n in range(46)] # Michael S. Branicky, Jan 08 2021
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CROSSREFS
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Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
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.
Cf. A132774.
Sequence in context: A226600 A178072 A185257 * A209992 A212118 A175346
Adjacent sequences: A033948 A033949 A033950 * A033952 A033953 A033954
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Olivier Gorin (gorin(AT)roazhon.inra.fr)
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EXTENSIONS
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Extended (with formula) by Erich Friedman
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STATUS
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approved
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