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A056108
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Fourth spoke of a hexagonal spiral.
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12
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1, 5, 15, 31, 53, 81, 115, 155, 201, 253, 311, 375, 445, 521, 603, 691, 785, 885, 991, 1103, 1221, 1345, 1475, 1611, 1753, 1901, 2055, 2215, 2381, 2553, 2731, 2915, 3105, 3301, 3503, 3711, 3925, 4145, 4371, 4603, 4841, 5085, 5335, 5591, 5853, 6121, 6395
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| If Y is a 4-subset of an n-set X then, for n>=4, a(n-4) is the number of 4-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 08 2007
a(n) = A096777(3*n+1) . - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 29 2007
Comment from A. K. Devaraj (dkandadai(AT)gmail.com), Sep 18 2009: (Start)
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod(f(x)); here n belongs to N.
There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z.
However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x.
The present sequence is the integer part when the polynomial is x^2 + x + 1 and x = sqrt(2),
f(x+n*f(x))/f(x) = a(n) + A005563(n)*sqrt(2).
Equals triangle A128229 as an infinite lower triangular matrix * A016777 as a vector, where A016777 = (3n+1).
(End)
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LINKS
| H. Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n) = 3n^2+n+1 = a(n-1)+6n-2 = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+3n = A056106(n)+2n = A056107(n)+n = A056109(n)-n = A003215(n)-2n
a(n) = sum of (n+1)-th row terms of triangle A134234.= - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 14 2007
Equals binomial transform of [1, 4, 6, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 30 2008
a(n)=6*n+a(n-1)-2 (with a(0)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 07 2010]
G.f.: (1+2*x+3*x^2)/(1-3*x+3*x^2-x^3). [Colin Barker, Jan 04 2012]
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MATHEMATICA
| s = 1; lst = {s}; Do[s += n + 3; AppendTo[lst, s], {n, 1, 300, 6}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
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CROSSREFS
| Cf. A054552 for example of square (or octagonal) spiral spoke.
Cf. A134234.
A000217 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
Sequence in context: A048065 A048021 A133268 * A055831 A037984 A073361
Adjacent sequences: A056105 A056106 A056107 * A056109 A056110 A056111
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KEYWORD
| easy,nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jun 09 2000
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