A007781 - OEIS

%I #51 Sep 08 2022 08:44:35

%S 1,3,23,229,2869,43531,776887,15953673,370643273,9612579511,

%T 275311670611,8630788777645,293959006143997,10809131718965763,

%U 426781883555301359,18008850183328692241,808793517812627212561

%N a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

%C (12n^2 + 6n + 1)^2 divides a(6n+1), where (12n^2 + 6n + 1) = (2n+1)^3 - (2n)^3 = A127854(n) = A003215(2n) are the hex (or centered hexagonal) numbers. The prime numbers of the form 12n^2 + 6n + 1 belong to A002407. - _Alexander Adamchuk_, Apr 09 2007

%D Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

%H Doug Bell, <a href="/A007781/b007781.txt">Table of n, a(n) for n = 0..100</a>

%H Andrew Cusumano, <a href="https://www.fq.math.ca/Problems/advanced45-2.pdf">Problem H-656</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; <a href="https://www.fq.math.ca/Problems/Aug2009advanced.pdf">A Sequence Tending To e</a>, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.

%H Ronald K. Hoeflin, <a href="https://web.archive.org/web/20140220060408/http://www.eskimo.com:80/~miyaguch/mega.html">Mega Test</a>. [Wayback Machine link]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerDifferencePrime.html">Power Difference Prime</a>.

%F a(n) = A000312(n+1) - A000312(n) for n>0, a(0) = 1.

%F a(n) = abs(discriminant(x^(n+1)-x+1)).

%F E.g.f.: W(-x)/(1+W(-x)) - W(-x)/((1+W(-x))^3*x) where W is the Lambert W function. - _Robert Israel_, Aug 19 2015

%F Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - _Amiram Eldar_, Jan 03 2022

%e a(14) = 10809131718965763 = 3 * 61^2 * 968299894201.

%p seq( `if`(n=0,1,(n+1)^(n+1) -n^n), n=0..20); # _G. C. Greubel_, Mar 05 2020

%t Join[{1},Table[(n+1)^(n+1)-n^n,{n,20}]]  (* _Harvey P. Dale_, Feb. 09 2011 *)

%t Differences[Table[n^n,{n,0,20}]] (* _Charles R Greathouse IV_, Feb 09 2011 *)

%o (PARI) first(m)=vector(m,i,i--;(i+1)^(i+1) - i^i) /* _Anders Hellström_, Aug 18 2015 */

%o (Magma) [1] cat [(n+1)^(n+1)-n^n: n in [1..20]]; // _Vincenzo Librandi_, Aug 19 2015

%o (Sage) [1]+[(n+1)^(n+1) -n^n for n in (1..20)] # _G. C. Greubel_, Mar 05 2020

%Y Cf. A000312, A068146, A068954, A068955, A068956, A068957.

%Y Cf. A002407, A003215, A127854.

%K nonn,easy

%O 0,2

%A Peter McCormack (peter.mccormack(AT)its.csiro.au)