A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.

1, 1, 43, 85, 1891, 5461, 84883, 314245, 3879331, 17077621, 180009523, 897269605, 8457669571, 46142992981, 401365114963, 2339370820165, 19196705648611, 117450280095541, 923711917337203, 5856623681349925, 44652524209512451, 290630718826209301, 2166036735625732243

Offset: 0

Author

Len Smiley, Dec 07 2001

Keywords

• nonn
• easy

Comments

A second-order recurrence of promic type (integer roots).

If the number j = A002378(m) is promic (= i(i+1)), then a(n) = a(n-1) + j*a(n-2), a(0) = a(1) = 1 has a closed-form solution involving only powers of integers. The binomial coefficient sum solves the recurrence regardless of promicity (cf. GKP reference).

Hankel transform is := 1,42,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

References

• R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, p. 204.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..150
• Index entries for linear recurrences with constant coefficients, signature (1,42).

Crossrefs

Cf. A001045 (j=2), A015441 (j=6), A053404 (j=12), A053428 (j=20), A053430 (j=30).

Programs

Maple

n->sum(binomial(n-k, k)*(42)^k, k=0..n)

Mathematica

LinearRecurrence[{1,42},{1,1},30] (* Harvey P. Dale, Apr 30 2017 *)

PARI

a(n) = { (7^(n+1) - (-6)^(n+1))/13 } \\ Harry J. Smith, Nov 02 2009

Formula

a(n) = a(n-1) + 42a(n-2); a(0) = a(1) = 1.

G.f.: -1/((6*x+1)*(7*x-1)). - R. J. Mathar, Nov 16 2007