Search: id:A022095
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%I A022095
%S A022095 1,5,6,11,17,28,45,73,118,191,309,500,809,1309,2118,3427,
%T A022095 5545,8972,14517,23489,38006,61495,99501,160996,260497,
%U A022095 421493,681990,1103483,1785473,2888956,4674429,7563385
%N A022095 Fibonacci sequence beginning 1 5.
%C A022095 a(n-1)=sum(P(5;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=4. These 
               are the sums of the SW-NE diagonals in P(5;n,k), the (5,1) Pascal 
               triangle A093562. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 
               29 2004. Proof via recursion relations and comparison of inputs. 
               Also sums of the SW-NE diagonals in the (1,4)-Pascal triangle A095666.
%H A022095 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A022095 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%F A022095 a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=5. a(-1):=4.
%F A022095 G.f.: (1+4*x)/(1-x-x^2).
%F A022095 Row sums of triangle A131776 starting (1, 5, 6, 11, 17, 28,...). - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
%F A022095 a(n)=4*fibonacci(n-1)+fibonacci(n), n>=1 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 05 2007
%F A022095 a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 2*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/
               (2^(n-2)*sqrt5). Offset 1. a(3)=6 [From Al Hakanson (hawkuu(AT)gmail.com), 
               Jan 14 2009]
%p A022095 a:=n->4*fibonacci(n-1)+fibonacci(n): seq(a(n), n=1..32); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%t A022095 a={};b=1;c=5;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;
               AppendTo[a,c],{n,1,9,1}];a (Vladimir Orlovsky, Jul 22 2008)
%Y A022095 a(n) = A101220(4, 0, n+1).
%Y A022095 a(n) = A109754(4, n+1).
%Y A022095 Cf. A131776.
%Y A022095 Sequence in context: A136974 A101187 A070373 this_sequence A042531 A042839 
               A041373
%Y A022095 Adjacent sequences: A022092 A022093 A022094 this_sequence A022096 A022097 
               A022098
%K A022095 nonn
%O A022095 0,2
%A A022095 N. J. A. Sloane (njas(AT)research.att.com).

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