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A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

%I A080151 #59 Aug 19 2025 10:16:58
%S A080151 1,4,9,16,25,36,49,64,81,82,85,90,97,106,117,130,145,162,163,166,171,
%T A080151 178,187,198,211,226,243,244,247,252,259,268,279,292,307,324,325,328,
%U A080151 333,340,349,360,373,388,405,406,409,414,421,430,441,454,469,486,487
%N A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.
%C A080151 Record values in A003132. - _Reinhard Zumkeller_, Jul 10 2011
%H A080151 Robert Israel, <a href="/A080151/b080151.txt">Table of n, a(n) for n = 1..10000</a>
%H A080151 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>
%H A080151 <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 A080151 a(n) = A007953(A002477(n)).
%F A080151 a(n) = sqrt( A080150(n) ).
%F A080151 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 A080151 a(n) = A003132(A051885(n)). - _Reinhard Zumkeller_, Jul 10 2011
%F A080151 a(9*n + k) = 81*n + k^2, with k in range 0 to 9. - _Enrique Pérez Herrero_, Nov 05 2022
%F A080151 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 A080151 Empirical g.f. confirmed. - _Robert Israel_, Aug 05 2019
%p A080151 f := n -> 9*n - 81*frac(1/9*n) + 81*frac(1/9*n)^2:
%p A080151 map(f, [$1..100]); # _Robert Israel_, Aug 05 2019
%t A080151 (* by direct counting *)
%t A080151 Repunit[n_] := (-1 + 10^n)/9; A080151[n_]:=Plus @@ IntegerDigits[Repunit[n]^2];
%t A080151 (* by the formula *)
%t A080151 A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)
%t A080151 (* or alternatively *)
%t A080151 A080151[n_] := 81*(Floor[n/9]+ FractionalPart[n/9]^2) (* _Enrique Pérez Herrero_, Nov 22 2009 *)
%o A080151 (Haskell) a n=(div n 9)*81+(mod n 9)^2
%o A080151           A080151=map a [1..] \\ _Chernin Nadav_, Mar 06 2014
%o A080151 (PARI) vector(100, n, (n\9)*81+(n%9)^2) \\ _Colin Barker_, Mar 05 2014
%Y A080151 Cf. A080150, A002477, A080160, A080161, A080162.
%K A080151 nonn,base,easy
%O A080151 1,2
%A A080151 _Eric W. Weisstein_, Jan 31 2003