A066178 Number of binary bit strings of length n with no block of 8 or more 0's. Nonzero heptanacci numbers, A122189.

1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600

Offset: 0

Author

Len Smiley, Dec 14 2001

Keywords

• nonn
• easy

Comments

Analogous bit string description and o.g.f. (1-x)/(1-2x+x^{k+1}) works for nonzero k-nacci numbers.

Compositions of n into parts <= 7. - Joerg Arndt, Aug 06 2012

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
• Zhao Hui Du, Link giving derivation and proof of the formula
• Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
• László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
• Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
• Eric Weisstein's World of Mathematics, Heptanacci Number
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Crossrefs

Cf. A000045 (k=2, Fibonacci numbers), A000073 (k=3, tribonacci) A000078 (k=4, tetranacci) A001591 (k=5, pentanacci) A001592 (k=6, hexanacci), A122189 (k=7, heptanacci).

Row 7 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

Programs

Mathematica

a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 8; a[5] = 16; a[6] = 32; a[7] = 64; a[n_] := 2*a[n - 1] - a[n - 8]; Array[a, 31, 0]
CoefficientList[ Series[(1 - x)/(1 - 2 x + x^8), {x, 0, 30}], x]
LinearRecurrence[{1,1,1,1,1,1,1},{1,1,2,4,8,16,32},40] (* Harvey P. Dale, Nov 16 2014 *)

Formula

O.g.f.: 1/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).

a(n) = Sum_{i=n-7..n-1} a(i).

a(n) = round((r-1)/((t+1)*r - 2*t) * r^(n-1)), where r is the heptanacci constant, the real root of the equation x^(t+1) - 2*x^t + 1 = 0 which is greater than 1. The formula could also be used for a k-step Fibonacci sequence if r is replaced by the k-bonacci constant, as in A000045, A000073, A000078, A001591, A001592. - Zhao Hui Du, Aug 24 2008

a(n) = 2*a(n-1) - a(n-8). - Vincenzo Librandi, Dec 20 2010

Extensions

Definition corrected by Vincenzo Librandi, Dec 20 2010