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A085810
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Number of three-choice paths along a corridor of width 6, starting from one side.
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7
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1, 2, 5, 13, 35, 96, 266, 741, 2070, 5791, 16213, 45409, 127206, 356384, 998509, 2797678, 7838801, 21963661, 61540563, 172432468, 483144522, 1353740121, 3793094450, 10628012915, 29779028189
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Narrower corridors produce A000012, A000079, A000129, A001519, A057960. An infinitely wide corridor would produce A005773.
Diagonal sums of A114164. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
C(n):= a(n)*(-1)^n appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= A181880(n-2)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
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REFERENCES
| P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.3.
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FORMULA
| a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3);
G.f.: (1-2x)/(1-4x+3x^2+x^3); a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, C(n-k, j)C(j+k, 2k)}}; a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, C(n-k, k+j)*C(k, k-j)2^(n-2k-j)}}; a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-j, n-2k-j)C(k, j)(-1)^j*2^(n-2k-j)}}; - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
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CROSSREFS
| Cf. A000012 A000079 A000129 A001519 A057960 A005773.
Sequence in context: A007075 A000107 A063028 * A005773 A022855 A091190
Adjacent sequences: A085807 A085808 A085809 * A085811 A085812 A085813
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KEYWORD
| easy,nonn
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2003
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