A034115 - OEIS

%I #30 May 29 2026 08:09:47

%S 35,48,63,80,99,119,142,167,194,223,253,286,321,358,397,437,480,525,

%T 572,621,671,724,779,836,895,955,1018,1083,1150,1219,1289,1362,1437,

%U 1514,1593,1673,1756,1841,1928,2017,2107,2200,2295,2392,2491,2591,2694

%N Fractional part of square root of a(n) starts with 9: first term of runs.

%C How is this different from A034105? - _N. J. A. Sloane_, Mar 30 2007

%C Answer: A034115 has the starts of runs of consecutive values of A034105.  That is, frac{sqrt[a(n)]} >= 0.9, but frac{sqrt[a(n)-1]} < 0.9. - _Don Reble_, Jul 17 2020

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).

%F a(n) = n^2 + 9*n + 25 + floor(4*n / 5) = A027690(n+4)+A090223(n). - _Don Reble_, Jul 17 2020

%F a(n) ~ n^2. - _Charles R Greathouse IV_, May 29 2026

%e 358, 359 and 360 are a run of 3 numbers in A034105, so 358 is in this sequence, but 359 and 360 are not. - _R. J. Mathar_, Jul 21 2020

%t Join[{35},Select[Partition[Select[Range[3000],NumberDigit[Sqrt[#],-1] == 9&],2,1],(#[[2]]-#[[1]]!=1&)][[All,2]]] (* or *) LinearRecurrence[{2,-1,0,0,1,-2,1},{35,48,63,80,99,119,142},50] (* _Harvey P. Dale_, Aug 14 2021 *)

%o (PARI) a(n)=(5*n^2+49*n+[125, 121, 122, 123, 124][n%5+1])/5 \\ _Charles R Greathouse IV_, May 29 2026

%Y Cf. A027690, A034105, A090223, A034115.

%K nonn,base,easy

%O 1,1

%A _Patrick De Geest_, Sep 15 1998