A055270 a(n) = 7*a(n-1) + (-1)^n * binomial(2,2-n) with a(-1)=0.

1, 5, 36, 252, 1764, 12348, 86436, 605052, 4235364, 29647548, 207532836, 1452729852, 10169108964, 71183762748, 498286339236, 3488004374652, 24416030622564, 170912214357948, 1196385500505636, 8374698503539452, 58622889524776164, 410360226673433148, 2872521586714032036

Offset: 0

Author

Barry E. Williams, May 10 2000

Keywords

• easy
• nonn

Comments

For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 6*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (7).

Crossrefs

Cf. A055272 (first differences of 7^n (A000420)).

Programs

Magma

[1,5] cat [36*7^(n-2): n in [2..30]]; // G. C. Greubel, Mar 16 2020

Maple

A055270:= n-> `if`(n<2, 4*n+1, 36*7^(n-2)); seq(A055270(n), n=0..30); # G. C. Greubel, Mar 16 2020

Mathematica

Join[{1,5},NestList[7#&,36,20]] (* Harvey P. Dale, Sep 04 2017 *)

Sage

[1,5]+[36*7^(n-2) for n in (2..30)] # G. C. Greubel, Mar 16 2020

Formula

a(n) = 6^2 * 7^(n-2), n >= 2 with a(0)=1, a(1)=5.

G.f.: (1-x)^2/(1-7*x).

a(n) = Sum_{k=0..n} A201780(n,k)*5^k. - Philippe Deléham, Dec 05 2011

E.g.f.: (13 - 7*x + 36*exp(7*x))/49. - G. C. Greubel, Mar 16 2020

Extensions

Terms a(20) onward added by G. C. Greubel, Mar 16 2020