A093916 - OEIS

%I #31 Dec 01 2023 03:26:30

%S 2,7,9,24,24,53,47,94,78,147,117,212,164,289,219,378,282,479,353,592,

%T 432,717,519,854,614,1003,717,1164,828,1337,947,1522,1074,1719,1209,

%U 1928,1352,2149,1503,2382,1662,2627,1829,2884,2004,3153,2187,3434,2378,3727,2577,4032,2784,4349,2999,4678,3222,5019,3453,5372

%N a(2*k-1) = (2*k-1)^2 + 2 - k, a(2*k) = 6*k^2 + 2 - k: First column of the triangle A093915.

%C The sequence was initially defined as the first column of the triangle A093915, constructed by trial and error. It is however easy to prove that the sum of the r-th row of A093915, A093917(r), equals twice A006003(r) when r is odd, and three times A006003(r) when r is even. Given the expression of the row sum A093917(r) in terms of the first element a(r), one obtains the explicit formula for a(r). - _M. F. Hasler_, Apr 04 2009

%H Harvey P. Dale, <a href="/A093916/b093916.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F a(n) = ((n^2+1)*b(n) - n + 1)/2 where b(n) = 3 - (n mod 2) = 2 if n odd, = 3 if n even. - _M. F. Hasler_, Apr 04 2009

%F From _Colin Barker_, May 01 2012: (Start)

%F a(n) = (n*(5*n-2) + (n^2+1)*(-1)^n + 7)/4.

%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

%F G.f.: x*(2+7*x+3*x^2+3*x^3+3*x^4+2*x^5)/((1-x)^3*(1+x)^3). (End)

%F E.g.f.: (1/4)*( (7 +3*x +5*x^2)*exp(x) - 8 + (1 -x +x^2)*exp(-x) ). - _G. C. Greubel_, Dec 30 2021

%t LinearRecurrence[{0,3,0,-3,0,1},{2,7,9,24,24,53},80] (* _Harvey P. Dale_, Nov 24 2017 *)

%o (PARI)

%o A093916(n)=((n^2+1)*(3-n%2)-n+1)/2

%o /* or the "experimental" version, trying out all allowed values */

%o A093916(n)={ local( s=(n^3+n)/2, d=(n^2-n)/2, k=ceil((2*s-d)/n)); while( (n*k+d)%s, k++ ); k } \\ _M. F. Hasler_, Apr 04 2009

%o (Magma) [(n*(5*n-2) + (-1)^n*(n^2+1) + 7)/4: n in [1..70]]; // _G. C. Greubel_, Dec 30 2021

%o (SageMath) [(5*n^2 -2*n +7 +(-1)^n*(n^2 +1))/4 for n in (1..70)] # _G. C. Greubel_, Dec 30 2021

%Y Cf. A093915, A093917, A093918.

%K nonn,easy

%O 1,1

%A _Amarnath Murthy_, Apr 25 2004

%E Edited and extended by _M. F. Hasler_, Apr 04 2009