A054469 - OEIS

%I #39 Jan 05 2025 19:51:36

%S 1,7,28,85,218,499,1053,2092,3970,7272,12958,22596,38739,65535,109714,

%T 182185,300620,493635,807555,1317360,2144396,3485032,5657028,9174560,

%U 14869613,24088399,39009168,63156437,102233030,165466347,267786673

%N A second-order recursive sequence.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H Vincenzo Librandi, <a href="/A054469/b054469.txt">Table of n, a(n) for n = 0..1000</a>

%H A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special Properties of the Sequence W(n){a,b; p,q}</a>, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.

%H A. K. Whitford, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/whitford-a.pdf">Binet's Formula Generalized</a>, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,15,-5,-4,4,-1).

%F a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2.

%F a(n) = a(n-1) + a(n-2) + (n+1)*(n+2)^2*(n+3)/12.

%F a(-n) = 0.

%F a(n) = (Sum_{i=1..floor((n+2)/2)} binomial(n+5-i, n+2-2*i)) + 2*(Sum_{i=1..floor((n+1)/2)} binomial(n+5-i, n+1-2*i)).

%F G.f.: (1+x) / ((1-x)^5*(1-x-x^2)). - _Colin Barker_, Jun 11 2013

%F From _G. C. Greubel_, Oct 21 2024: (Start)

%F a(n) = Fibonacci(n+12) - Sum_{j=0..4} Fibonacci(11-2*j) * binomial(n+j, j).

%F a(n) = Fibonacci(n+12) - (1/12)*(1716 + 802*n + 173*n^2 + 20*n^3 + n^4). (End)

%t RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[ n+3,3]/2},a,{n,30}] (* _Harvey P. Dale_, Sep 22 2013 *)

%t CoefficientList[Series[(1+x)/((1-x)^5*(1-x-x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 23 2013 *)

%o (PARI) a(n) = sum(i=1,(n+2)\2,binomial(n+5-i,n+2-2*i))+2*sum(i=1,(n+1)\2,binomial(n+5-i,n+1-2*i)) \\ _Jason Yuen_, Aug 13 2024

%o (Magma)

%o A054469:= func< n | Fibonacci(n+12) -(1/12)*(1716 +802*n +173*n^2 +20*n^3 +n^4) >;

%o [A054469(n): n in [0..40]]; // _G. C. Greubel_, Oct 21 2024

%o (SageMath)

%o def A054469(n): return fibonacci(n+12) - (1716 + 802*n + 173*n^2 + 20*n^3 + n^4)//12

%o [A054469(n) for n in range(41)] # _G. C. Greubel_, Oct 21 2024

%Y Cf. A000045, A001891, A001911.

%Y Right-hand column 11 of triangle A011794.

%K easy,nonn,changed

%O 0,2

%A _Barry E. Williams_, Mar 31 2000