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A105309
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a(n) = |b(n)|^2 = x^2 + 3*y*2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,c,c,c) where c = 1/sqrt(3).
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14
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1, 1, 2, 5, 9, 20, 41, 85, 178, 369, 769, 1600, 3329, 6929, 14418, 30005, 62441, 129940, 270409, 562725, 1171042, 2436961, 5071361, 10553600, 21962241, 45703841, 95110562, 197926885, 411889609, 857150100, 1783745641, 3712008565
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Prepending 0 and keeping the offset at 0, turns this into a divisibility sequence with g.f. x(1-x^2)/(1-x-2x^2-x^3+x^4). [From T. D. Noe (noe(AT)sspectra.com), Dec 22 2008]
Equals INVERT transform of (1, 1, 2, 0, 2, 0, 2,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2009]
Sequence gives the norm of the coefficients in 1/(1 - I*x - I*x^2), where I^2=-1. [From Paul D. Hanna, Dec 06 2011]
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LINKS
| Eric Weisstein. "Quaternion"
Index to sequences with linear recurrences with constant coefficients, signature (1,2,1,-1).
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FORMULA
| a(n) = A092886(n+1) - A092886(n-1), n>0.
a(n) = A201837(n)^2 + A201838(n)^2. [From Paul D. Hanna, Dec 06 2011]
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PROG
| (PARI) {a(n)=local(A); if(n<2, n>=0, n++; A=vector(n, i, 1); for(i=3, n, A[i]=A[i-1]+A[i-2]*I); norm(A[n]))} (Somos)
(PARI) {a(n)=norm(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)), n))} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff((1-x^2)/(1-x-2*x^2-x^3+x^4)+x*O(x^n), n)}
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CROSSREFS
| Cf. A092886, A201837, A201838.
Sequence in context: A062092 A079117 A030137 * A192572 A097163 A117186
Adjacent sequences: A105306 A105307 A105308 * A105310 A105311 A105312
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KEYWORD
| nonn,easy
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AUTHOR
| Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 25 2005
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