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Partial sums of round(n^2/29).
1

%I #34 Feb 01 2023 02:30:25

%S 0,0,0,0,1,2,3,5,7,10,13,17,22,28,35,43,52,62,73,85,99,114,131,149,

%T 169,191,214,239,266,295,326,359,394,432,472,514,559,606,656,708,763,

%U 821,882,946,1013,1083,1156,1232,1311,1394,1480

%N Partial sums of round(n^2/29).

%C The round function is defined here by round(x) = floor(x + 1/2).

%C There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).

%H Vincenzo Librandi, <a href="/A177332/b177332.txt">Table of n, a(n) for n = 0..905</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

%F a(n) = round(n*(n+1)*(2*n+1)/174).

%F a(n) = floor((n+4)*(2*n^2 - 5*n + 21)/174).

%F a(n) = ceiling((n-3)*(2*n^2 + 9*n + 28)/174).

%F a(n) = a(n-29) + (n+1)*(n-29) + 266, n > 28.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-29) - 3*a(n-30) + 3*a(n-31) - a(n-32). - _R. J. Mathar_, Dec 13 2010

%F G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 - x^6 + x^5 - x^4 + 1)/((x-1)^4*(x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Apr 06 2012

%e a(17) = 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 62.

%p seq(round(n*(n+1)*(2*n+1)/174),n=0..50)

%t Accumulate[Table[Round[n^2/29],{n,0,60}]] (* _Harvey P. Dale_, Dec 18 2010 *)

%o (Magma) [Floor((n+4)*(2*n^2-5*n+21)/174): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011

%o (PARI) a(n)=(2*n^3+3*n^2+n+84)\174 \\ _Charles R Greathouse IV_, Apr 06 2012

%o (Python)

%o def A177332(n): return (n*(n*(2*n + 3) + 1) + 84)//174 # _Chai Wah Wu_, Jan 31 2023

%Y Cf. A177100, A177116.

%K nonn,easy

%O 0,6

%A _Mircea Merca_, Dec 10 2010