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A227356 Partial sums of A129361.

%I A227356 #26 Jun 30 2025 16:47:26
%S A227356 1,2,5,10,20,36,65,112,193,324,544,900,1489,2442,4005,6534,10660,
%T A227356 17336,28193,45760,74273,120408,195200,316216,512257,829458,1343077,
%U A227356 2174130,3519412,5696124,9219105,14919408,24144289
%N A227356 Partial sums of A129361.
%C A227356 Sum of labeled numbers of boxes arranged as Pyramid type-II with base Fibonacci(n).
%C A227356 Let us call a Pyramid "type-I" when each row starts with the same number as the leftmost base number, and "type-II" when each column has the same number as the base.
%C A227356 The Pyramid arrangements are related to other sequences as follows:
%C A227356    Base Number     Type-I     Type-II
%C A227356    -----------     ------     -------
%C A227356    Natural         A002623    A034828
%C A227356    Odd             A000292    A128624
%C A227356    Fibonacci       A129696    a(n)
%C A227356    1               A002620    A002620
%C A227356    1,0             A008805
%C A227356 See illustration in links.
%H A227356 Kival Ngaokrajang, <a href="/A227356/a227356.jpg">Illustration for some small n.</a>
%H A227356 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,1,-1,0,1).
%F A227356 For n >=2, a(n) = a(n-1) + A129361(n-1).
%F A227356 G.f. -x*(1+x)*(x^2-x+1) / ( (x-1)*(x^2+x-1)*(x^4+x^2-1) ). - _Joerg Arndt_, Jul 10 2013
%F A227356 a(n) = 2 + A000045(n+4) - A096748(n+6). - _R. J. Mathar_, Jul 20 2013
%t A227356 LinearRecurrence[{2,1,-3,1,-1,0,1},{1,2,5,10,20,36,65},40] (* _Harvey P. Dale_, Jun 30 2025 *)
%o A227356 (Small Basic)
%o A227356 a[1] = 1
%o A227356 k = 0
%o A227356 s5 = Math.SquareRoot(5)
%o A227356 For n = 2 To 51
%o A227356   If Math.Remainder(n,2)=0 Then
%o A227356     i = (n+2)/2
%o A227356   Else
%o A227356     i = (n+1)/2
%o A227356   EndIf
%o A227356   For j = i To n
%o A227356     k = k + Math.Round(Math.Power((1+s5)/2,j)/s5)
%o A227356   EndFor
%o A227356   a[n] = a[n-1] + k
%o A227356   TextWindow.Write(a[n-1] + ", ")
%o A227356   k = 0
%o A227356 EndFor
%Y A227356 Cf. A002623, A034828, A002620, A000292, A128624, A129696, A008805.
%K A227356 nonn
%O A227356 1,2
%A A227356 _Kival Ngaokrajang_, Jul 08 2013