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Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.
2

%I #49 Mar 03 2023 12:07:32

%S 1,3,7,14,25,43,69,110,167,255,375,558,805,1179,1681,2438,3451,4975,

%T 7011,10070,14153,20283,28461,40734,57103,81663,114415,163550,229069,

%U 327355,458409,654998,917123,1310319,1834587,2620998,3669553,5242395,7339525,10485230

%N Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

%H Vincenzo Librandi, <a href="/A193911/b193911.txt">Table of n, a(n) for n = 1..2000</a>

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,1,4,-2).

%F All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - _Henry Bottomley_, Feb 04 2000

%F So we get the sums from the piecewise function:

%F for odd n>=1, a(n)=2^((n+1)/2)-n+((n+1)/2)-2+Sum_{i=0..((n+1)/2)-1}(2*i+1)*(2^(((n+1)/2)-i) -1);

%F for even n>=2, a(n)=2^((n/2)+2)-n-4+Sum_{i=0..(n/2)-1}(2*i+1)*(2^((n/2)-i) -1). - _Jeffrey R. Goodwin_, Aug 09 2011

%F Let odd m>=3, then a(n)=a(m)-A000295(((m+1)/2)+1), where n>=2 is even. - _Jeffrey R. Goodwin_, Aug 09 2011

%F Let even m>=2, then a(n)=a(m)-A077802(m/2)=a(m)-A095151(m/2), where n>=1 is odd. - _Jeffrey R. Goodwin_, Aug 09 2011

%F G.f.:(1 + x - x^2)/((-1 + x)^3*(-1 - x + 2*x^2 + 2*x^3)). - _Alexander R. Povolotsky_, Aug 09 2011

%F a(n+4) = -2*a(n)+3*a(n+2)+n+5. - _Alexander R. Povolotsky_, Aug 09 2011

%F a(n) = 1/8*(2^(n/2+2)*((10-7*sqrt(2))*(-1)^n+10+7*sqrt(2))-(-1)^n-2*n*(n+12)-79). - _Alexander R. Povolotsky, Aug 09 2011

%e Portion of the first three rows:

%e A033627, 2-Stohr 1 2 4 7

%e A026474, 3-Stohr 1 2 4 8

%e A051039, 4-Stohr 1 2 4 8

%e Thus a(1)=1, a(2)=2+1=3, and a(3)=4+2+1=7.

%t A193911=t={0,1}; Do[AppendTo[t,t[[-2]]+t[[-1]]]; AppendTo[t,2*t[[-2]]],{n,41}]; Drop[Nest[Accumulate,t,2],1] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2012 *)

%t LinearRecurrence[{2,2,-6,1,4,-2},{1,3,7,14,25,43},40] (* _Harvey P. Dale_, Jun 20 2015 *)

%K nonn,easy

%O 1,2

%A _Jeffrey R. Goodwin_, Aug 08 2011