A100478 Pentanacci pi function: a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n) = pi(Sum_{j=1..5} a(n-j)) where pi = A000720.

1, 1, 1, 1, 1, 3, 4, 4, 6, 7, 9, 10, 11, 14, 15, 17, 19, 21, 23, 24, 27, 30, 30, 32, 34, 36, 37, 39, 40, 42, 44, 46, 47, 47, 48, 50, 51, 53, 53, 54, 55, 56, 58, 58, 60, 61, 62, 62, 62, 63, 63, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 1

Author

Jonathan Vos Post, Nov 22 2004

Keywords

• easy
• nonn

Comments

Starting with other values of a(1), a(2), a(3), a(4), a(5) what behaviors are possible? Does the sequence always stick at a single integer after some point, or can it go into a loop, or is there a third pattern?

a(n) is equal to 66 for 54 <= n <= 10^7. - G. C. Greubel, Apr 06 2023

Examples

			a(6) = pi(a(1)+a(2)+a(3)+a(4)+a(5)) = pi(1+1+1+1+1) = pi(5) = 3.
a(7) = pi(a(2)+a(3)+a(4)+a(5)+a(6)) = pi(1+1+1+1+3) = pi(7) = 4.
a(8) = pi(a(3)+a(4)+a(5)+a(6)+a(7)) = pi(1+1+1+3+4) = pi(10) = 4.
a(9) = pi(a(4)+a(5)+a(6)+a(7)+a(8)) = pi(1+1+3+4+4) = pi(13) = 6.
a(10) = pi(a(5)+a(6)+a(7)+a(8)+a(9)) = pi(1+3+4+4+6) = pi(18) = 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Andrew Booker, The Nth Prime Page.
• I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
• V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
• Eric Weisstein's World of Mathematics, Prime Counting Function

Crossrefs

Cf. A001591, A038607.

Programs

Mathematica

a[n_]:= a[n]= If[n<6,1,PrimePi[Sum[a[n-j], {j,5}]]];
Table[a[n], {n,80}] (* Robert G. Wilson v, Dec 03 2004 *)

SageMath

@CachedFunction
def a(n): # a = A100478
    if (n<6): return 1
    else: return prime_pi(sum(a(n-j) for j in range(1,6)))
[a(n) for n in range(1, 81)] # G. C. Greubel, Apr 06 2023

Formula

a(n) = pi(a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5)) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.

Extensions

Edited and extended by Robert G. Wilson v, Dec 03 2004