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A056109
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Fifth spoke of a hexagonal spiral.
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25
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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; internal format)
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OFFSET
| 0,2
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
H. Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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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
a(n)=6*n+a(n-1)-1 (with a(0)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 07 2010]
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.
A008810(3*n + 1) = A056105(-n) = a(n). - Michael Somos, Aug 03 2006.
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EXAMPLE
| a(1)=6*1+1-1=6; a(2)=6*2+6-1=17; a(3)=6*3+17-1=34 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 07 2010]
1 + 6*x + 17*x^2 + 34*x^3 + 57*x^4 + 86*x^5 + 121*x^6 + 162*x^7 + 209*x^8 + ...
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MATHEMATICA
| s=1; lst={s}; Do[s+=n+5; AppendTo[lst, s], {n, 0, 6!, 6}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 04 2008]
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PROG
| (PARI) {a(n) = 3*n^2 + 2*n + 1} /* Michael Somos, Aug 03 2006 */
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CROSSREFS
| Cf. A054552 for example of square (or octagonal) spiral spoke.
Cf. A003215, A008810, A056105.
Sequence in context: A130051 A038795 A066486 * A023545 A038633 A083045
Adjacent sequences: A056106 A056107 A056108 * A056110 A056111 A056112
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jun 09 2000
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