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A133654
a(n) = 2*A000129(n) - 1.
3
1, 3, 9, 23, 57, 139, 337, 815, 1969, 4755, 11481, 27719, 66921, 161563, 390049, 941663, 2273377, 5488419, 13250217, 31988855, 77227929, 186444715, 450117361, 1086679439, 2623476241, 6333631923, 15290740089, 36915112103, 89120964297, 215157040699
OFFSET
1,2
COMMENTS
a(n)/a(n-1) tends to (1 + sqrt(2)).
Define a triangle by T(n,1) = n*(n-1)+1 and T(n,n) = 1, n >= 1. Let interior terms be T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The triangle is 1; 3,1; 7,5,1; 13,15,7,1; etc. The row sums are 1, 4, 13, 36, 93, ... and the differences (sum of terms in row(n) minus those in row(n-1)) are a(n). - J. M. Bergot, Mar 10 2013
FORMULA
a(n) = 2*A000129(n) - 1, where A000129 = the Pell sequence. a(1) = 1, a(2) = 3, then for n>2, a(n) = 2*a(n-1) + a(n-2) + 2.
G.f.: x*(1+x^2)/( (x-1)*(x^2+2*x-1) ). - R. J. Mathar, Nov 14 2007
a(n) = -1+(-(1-sqrt(2))^n+(1+sqrt(2))^n)/sqrt(2). - Colin Barker, Mar 16 2016
EXAMPLE
a(3) = 2*A000129(3) - 1 = 2*5 - 1.
a(5) = 57 = 2*a(4) + a(3) + 2 = 2*23 + 9 + 2.
PROG
(PARI) Vec(x*(1+x^2)/((x-1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, Mar 16 2016
CROSSREFS
Cf. A000129.
Sequence in context: A341029 A045883 A292329 * A193695 A226845 A096574
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Sep 19 2007
EXTENSIONS
More terms from Philippe Deléham, Oct 16 2007, corrected by R. J. Mathar, Mar 12 2013
STATUS
approved