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 A056109 Fifth spoke of a hexagonal spiral. 44
 1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321, 386, 457, 534, 617, 706, 801, 902, 1009, 1122, 1241, 1366, 1497, 1634, 1777, 1926, 2081, 2242, 2409, 2582, 2761, 2946, 3137, 3334, 3537, 3746, 3961, 4182, 4409, 4642, 4881, 5126, 5377, 5634, 5897, 6166, 6441 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Squared distance from (0,0,-1) to (n,n,n) in R^3. - James R. Buddenhagen, Jun 15 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 H. Bottomley, Illustration of initial terms Tanya Khovanova, Recursive Sequences G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 3n^2+2n+1 = a(n-1)+6n-1 = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+4n = A056106(n)+3n = A056107(n)+2n = A056108(n)+n = A003215(n)-n. G.f.: (1+3*x+2*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012 G.f.: (1 + x) * (1 + 2*x) / (1 - x)^3. - Michael Somos, Feb 04 2012 a(n) = A008810(3*n + 1) = A056105(-n). - Michael Somos, Aug 03 2006 E.g.f.: exp(x)*(1 + 5*x + 3*x^2). - Stefano Spezia, Oct 06 2018 MAPLE seq(coeff(series(factorial(n)*(exp(x)*(3*x^2+5*x+1)), x, n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 07 2018 MATHEMATICA s=1; lst={s}; Do[s+=n+5; AppendTo[lst, s], {n, 0, 6!, 6}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 04 2008 *) Table[3 n^2 + 2 n + 1, {n, 0, 100}] (* Vincenzo Librandi, Mar 15 2013 *) CoefficientList[Series[E^x (1 + 5 x + 3 x^2) , {x, 0, 20}], x]*Table[k!, {k, 0, 100}] (* Stefano Spezia, Oct 06 2018 *) LinearRecurrence[{3, -3, 1}, {1, 6, 17}, 60] (* Harvey P. Dale, Mar 28 2019 *) PROG (PARI) {a(n) = 3*n^2 + 2*n + 1}; /* Michael Somos, Aug 03 2006 */ (PARI) Vec((1+3*x+2*x^2)/(1-3*x+3*x^2-x^3)+O(x^100)) \\ Stefano Spezia, Oct 17 2018 (MAGMA) [3*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Mar 15 2013 (GAP) List([0..50], n->3*n^2+2*n+1); # Muniru A Asiru, Oct 07 2018 (Python) for n in range(0, 100): print(int(3*n**2 + 2*n + 1), end=' ') # Stefano Spezia, Oct 16 2018 CROSSREFS Cf. A054552 for example of square (or octagonal) spiral spoke. Cf. A003215, A008810, A056105. Cf. A122430 (prime terms of A056109). - Zak Seidov, Mar 13 2013 Sequence in context: A301711 A066486 A301719 * A023545 A038633 A083045 Adjacent sequences:  A056106 A056107 A056108 * A056110 A056111 A056112 KEYWORD easy,nonn AUTHOR Henry Bottomley, Jun 09 2000 STATUS approved

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Last modified October 20 08:05 EDT 2019. Contains 328252 sequences. (Running on oeis4.)