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 A022393 Fibonacci sequence beginning 1, 23. 1
 1, 23, 24, 47, 71, 118, 189, 307, 496, 803, 1299, 2102, 3401, 5503, 8904, 14407, 23311, 37718, 61029, 98747, 159776, 258523, 418299, 676822, 1095121, 1771943, 2867064, 4639007, 7506071, 12145078, 19651149, 31796227, 51447376, 83243603, 134690979, 217934582, 352625561, 570560143 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (1, 1). FORMULA a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=23. a(-1):=22. G.f.: (1+22*x)/(1-x-x^2). a(n) = 23*A000045(n) + A000045(n-1). - Paolo P. Lava, May 19 2015 MAPLE with(numtheory): with(combinat): P:=proc(q) local n; for n from 0 to q do print(23*fibonacci(n)+fibonacci(n-1)); od; end: P(30); # Paolo P. Lava, May 19 2015 MATHEMATICA a[1]=1; a[2]=23; a[n_]:=a[n]=a[n - 1]+a[n - 2] (*  José María Grau Ribas, Feb 15 2010 *) LinearRecurrence[{1, 1}, {1, 23}, 30] (* Harvey P. Dale, Sep 30 2011 *) Table[Fibonacci[n + 2] + 21*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *) PROG (PARI) for(n=0, 50, print1(fibonacci(n+2) + 21*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018 (MAGMA) [Fibonacci(n+2) + 21*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018 (GAP) List([0..40], n->Fibonacci(n+2)+21*Fibonacci(n)); # Muniru A Asiru, Mar 03 2018 CROSSREFS Sequence in context: A007638 A031332 A122470 * A042068 A042066 A042070 Adjacent sequences:  A022390 A022391 A022392 * A022394 A022395 A022396 KEYWORD nonn AUTHOR EXTENSIONS Terms a(30) onward added by G. C. Greubel, Mar 02 2018 STATUS approved

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Last modified October 19 20:55 EDT 2019. Contains 328224 sequences. (Running on oeis4.)