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%I #33 Feb 18 2024 02:01:08
%S 1,23,24,47,71,118,189,307,496,803,1299,2102,3401,5503,8904,14407,
%T 23311,37718,61029,98747,159776,258523,418299,676822,1095121,1771943,
%U 2867064,4639007,7506071,12145078,19651149,31796227,51447376,83243603,134690979,217934582,352625561,570560143
%N Fibonacci sequence beginning 1, 23.
%C a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(23;n-1-k,k), n>=1, with a(-1)=22. These are the SW-NE diagonals in P(23;n,k), the (23,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by _Paul Barry_, Apr 29 2004. Proof via recursion relations and comparison of inputs.
%H G. C. Greubel, <a href="/A022393/b022393.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1).
%F a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=23. a(-1):=22.
%F G.f.: (1+22*x)/(1-x-x^2).
%t a[1]=1; a[2]=23; a[n_]:=a[n]=a[n - 1]+a[n - 2] (* _José María Grau Ribas_, Feb 15 2010 *)
%t LinearRecurrence[{1,1},{1,23},30] (* _Harvey P. Dale_, Sep 30 2011 *)
%t Table[Fibonacci[n + 2] + 21*Fibonacci[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 02 2018 *)
%o (PARI) for(n=0, 50, print1(fibonacci(n+2) + 21*fibonacci(n), ", ")) \\ _G. C. Greubel_, Mar 02 2018
%o (Magma) [Fibonacci(n+2) + 21*Fibonacci(n): n in [0..50]]; // _G. C. Greubel_, Mar 02 2018
%o (GAP) List([0..40],n->Fibonacci(n+2)+21*Fibonacci(n)); # _Muniru A Asiru_, Mar 03 2018
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Terms a(30) onward added by _G. C. Greubel_, Mar 02 2018
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