A226199 a(n) = 7^n + n.

1, 8, 51, 346, 2405, 16812, 117655, 823550, 5764809, 40353616, 282475259, 1977326754, 13841287213, 96889010420, 678223072863, 4747561509958, 33232930569617, 232630513987224, 1628413597910467, 11398895185373162, 79792266297612021, 558545864083284028, 3909821048582988071

Offset: 0

Author

Vincenzo Librandi, Jun 16 2013

Keywords

• nonn
• easy

Comments

Smallest prime of this form is a(34) = 54116956037952111668959660883.

In general, the g.f. of a sequence of numbers of the form k^n + n is (1-x-(k-1)*x^2)/((1-k*x)*(x-1)^2) with main linear recurrence (k+2)*a(n-1) - (2*k+1)*a(n-2) + k*a(n-3). - Bruno Berselli, Jun 16 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Crossrefs

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), this sequence (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).

Cf. A199483 (first differences), A370657.

Programs

Magma

[7^n+n: n in [0..20]];

Magma

I:=[1, 8, 51]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+7*Self(n-3): n in [1..30]];

Mathematica

Table[7^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - x - 6 x^2) / ((1 - 7 x) (1 - x)^2), {x, 0, 20}], x]
LinearRecurrence[{9,-15,7},{1,8,51},30] (* Harvey P. Dale, Jun 16 2025 *)

PARI

a(n)=7^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1-x-6*x^2)/((1-7*x)*(1-x)^2).

a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).

E.g.f.: exp(x)*(exp(6*x) + x). - Elmo R. Oliveira, Mar 05 2025