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%I #24 Jun 13 2015 00:54:42
%S 1,2,5,10,20,36,65,112,193,324,544,900,1489,2442,4005,6534,10660,
%T 17336,28193,45760,74273,120408,195200,316216,512257,829458,1343077,
%U 2174130,3519412,5696124,9219105,14919408,24144289
%N Partial sums of A129361.
%C Sum of labeled numbers of boxes arranged as Pyramid type-II with base Fibonacci(n).
%C 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 The Pyramid arrangements are related to other sequences as follows:
%C Base Number Type-I Type-II
%C ----------- ------ -------
%C Natural A002623 A034828
%C Odd A000292 A128624
%C Fibonacci A129696 a(n)
%C 1 A002620 A002620
%C 1,0 A008805
%C See illustration in links.
%H Kival Ngaokrajang, <a href="/A227356/a227356.jpg">Illustration for some small n.</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,1,-1,0,1).
%F For n >=2, a(n) = a(n-1) + A129361(n-1).
%F 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 a(n) = 2 + A000045(n+4) - A096748(n+6). - _R. J. Mathar_, Jul 20 2013
%o (Small Basic)
%o a[1] = 1
%o k = 0
%o s5 = Math.SquareRoot(5)
%o For n = 2 To 51
%o If Math.Remainder(n,2)=0 Then
%o i = (n+2)/2
%o Else
%o i = (n+1)/2
%o EndIf
%o For j = i To n
%o k = k + Math.Round(Math.Power((1+s5)/2,j)/s5)
%o EndFor
%o a[n] = a[n-1] + k
%o TextWindow.Write(a[n-1] + ", ")
%o k = 0
%o EndFor
%Y Cf. A002623, A034828, A002620, A000292, A128624, A129696, A008805.
%K nonn
%O 1,2
%A _Kival Ngaokrajang_, Jul 08 2013
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