A093953 a(n) = rightmost term in M^n * [1,1,1], where M = a 3 X 3 matrix composed of the first 3 rows of A050166 (fill in the matrix with zeros): = [1 0 0 / 1 2 0 / 1 4 5].

1, 10, 63, 344, 1781, 9030, 45403, 227524, 1138641, 5695250, 28480343, 142409904, 712065901, 3560362270, 17801876883, 89009515484, 445047839561, 2225239722090, 11126199659023, 55631000392264, 278155006155621, 1390775039166710, 6953875212610763, 34769376096608244, 173846880550150081

Offset

0,2

Comments

A sequence relating to Catalan numbers.

a(n)/a(n-1) tends to 5, a Catalan number. E.g. a(6)/a(5) = 45403/9030 = 4.9948...

Generally, with M = an N X N matrix composed of rows of A050166 (along with zeros), M^n * [1,1,1...] generates terms [a, b, c, d...] such that sequences of which a,b,c,d...are members converge upon the Catalan numbers: 1, 2, 5, 14, 42, 132...

Companion (M^n)[3,2] = 4*A016127(n), (M^n)[3,3] = 5^n = A000351(n), so a(n) = a(n-1) + 4*A016127(n-1) + 5^(n-1) for n>0. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Formula

Or simply with M=[1, 0, 0;1, 2, 0;1, 4, 5], a(n)=(M^n)[3, 1], (adds a leading 0 to sequence). - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005

G.f.: (1+2*x)/(1-8*x+17*x^2-10*x^3). - Colin Barker, Jan 31 2012

Example

a(4) = 1781 since M^4 * [1,1,1] = [1, 31, 1781].

Prog

(PARI) M=[1, 0, 0; 1, 2, 0; 1, 4, 5]; for(i=0, 16, print1((M^i)[3, 1]", ")) \\ Klasen
(PARI) Vec((1+2*x)/(1-8*x+17*x^2-10*x^3) + O(x^25)) \\ Andrew Howroyd, Sep 24 2025

Crossrefs

Cf. A050166, A016127, A000351.

Sequence in context: A077616 A392579 A145885 * A298067 A298716 A271754

Adjacent sequences: A093950 A093951 A093952 * A093954 A093955 A093956

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 18 2004

Extensions

a(16) onwards from Andrew Howroyd, Sep 24 2025

Status

approved