A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688

Offset

0,3

Comments

Binomial transform of A081199.

Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010

a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.

Index entries for linear recurrences with constant coefficients, signature (12,-35).

Formula

a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.

G.f.: x/((1-5*x)*(1-7*x)).

a(n) = 7^n/2 - 5^n/2.

a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - Reinhard Zumkeller, Aug 01 2010

a(n) = A121213(n)/2. - Reinhard Zumkeller, Aug 01 2010

E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jun 19 2021

Example

The a(2) = 12 words of length 2 over {A ,B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - Wolfdieter Lang, Jul 17 2017

Mathematica

CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12, -35}, {0, 1}, 30] (* Harvey P. Dale, Feb 07 2014 *)

Prog

(Sage) [lucas_number1(n, 12, 35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) [7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Crossrefs

Cf. A000035, A003462, A005059, A006516, A016161, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.

Sequence in context: A128877 A085797 A016161 * A351161 A016214 A037581

Adjacent sequences: A081197 A081198 A081199 * A081201 A081202 A081203

Keyword

nonn,easy

Author

Paul Barry, Mar 11 2003

Status

approved