%I #19 Dec 12 2020 19:03:29

%S 1,5,10,30,55,155,280,780,1405,3905,7030,19530,35155,97655,175780,

%T 488280,878905,2441405,4394530,12207030,21972655,61035155,109863280,

%U 305175780,549316405,1525878905,2746582030,7629394530,13732910155,38146972655,68664550780

%N a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, otherwise a(n) = a(n-1) + p^((n-1)/2), where p=5.

%C Partial sums of A133632.

%H Colin Barker, <a href="/A133629/b133629.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5).

%F a(n) = Sum_{k=1..n} A133632(k).

%F The following formulas are given for a general natural parameter p > 1 (p=5 for this sequence).

%F G.f.: x(1+(p-1)x)/((1-px^2)(1-x)).

%F a(n) = (p/(p-1))*(p^(n/2)-1) if n is even, otherwise a(n)=(p/(p-1))*((2p-1)*p^((n-3)/2)-1).

%F a(n) = (p/(p-1))*(p^floor(n/2) + p^floor((n-1)/2) - p^floor((n-2)/2)-1).

%F a(n) = p^floor(n/2) + (p^floor((n+1)/2)-p)/(p-1).

%F a(n) = A132669(a(n+1)) - 1.

%F a(n) = A132669(a(n-1)+1) for n > 0.

%F A132669(a(n)) = a(n-1)+1 for n > 0.

%F From _Colin Barker_, Nov 25 2016: (Start)

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

%F a(n) = (9*5^(n/2-1/2) - 5)/4 for n odd.

%F a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3.

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

%F (End)

%p a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+5 od: seq(a[n], n=1..29); # _Zerinvary Lajos_, Mar 17 2008

%o (PARI) Vec(x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 25 2016

%Y Sequences with similar recurrence rules: A027383 (p=2), A087503 (p=3), A133628 (p=4).

%Y Related sequences: A132666, A132667, A132668, A132669.

%Y Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).

%K nonn,easy

%O 1,2

%A _Hieronymus Fischer_, Sep 19 2007