A080151 - OEIS

%I #47 Nov 06 2022 07:48:57

%S 1,4,9,16,25,36,49,64,81,82,85,90,97,106,117,130,145,162,163,166,171,

%T 178,187,198,211,226,243,244,247,252,259,268,279,292,307,324,325,328,

%U 333,340,349,360,373,388,405,406,409,414,421,430,441,454,469,486,487

%N Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

%C Record values in A003132: a(n) = A003132(A051885(n)). - _Reinhard Zumkeller_, Jul 10 2011

%H Robert Israel, <a href="/A080151/b080151.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,1,-1).

%F a(n) = A007953(A002477(n)).

%F a(n) = sqrt( A080150(n) ).

%F a(n) = (9^2)*(n/9 - {n/9} + {n/9}^2) = 81*(floor(n/9) + {n/9}^2), where the symbol {n} means fractional part of n. - _Enrique Pérez Herrero_, Nov 22 2009

%F a(9*n + k) = 81*n + k^2, with k in range 0 to 9. - _Enrique Pérez Herrero_, Nov 05 2022

%F Empirical g.f.: x*(17*x^8 + 15*x^7 + 13*x^6 + 11*x^5 + 9*x^4 + 7*x^3 + 5*x^2 + 3*x + 1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). - _Colin Barker_, Mar 05 2014

%F Empirical g.f. confirmed. - _Robert Israel_, Aug 05 2019

%p f := n -> 9*n - 81*frac(1/9*n) + 81*frac(1/9*n)^2:

%p map(f, [$1..100]); # _Robert Israel_, Aug 05 2019

%t (* by direct counting *)

%t Repunit[n_] := (-1 + 10^n)/9; A080151[n_]:=Plus @@ IntegerDigits[Repunit[n]^2];

%t (* by the formula * )

%t A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)

%t (* or alternatively *)

%t A080151[n_] := 81*(Floor[n/9]+ FractionalPart[n/9]^2) (* _Enrique Pérez Herrero_, Nov 22 2009 *)

%o (Haskell) a n=(div n 9)*81+(mod n 9)^2

%o           A080151=map a [1..] \\ _Chernin Nadav_, Mar 06 2014

%o (PARI) vector(100, n, (n\9)*81+(n%9)^2) \\ _Colin Barker_, Mar 05 2014

%Y Cf. A080150, A002477, A080160, A080161, A080162.

%K nonn,base,easy

%O 1,2

%A _Eric W. Weisstein_, Jan 31 2003