A103385 -id:A103385 - cp's OEIS frontend

A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 512

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=7 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, k=5 case is A103373 and k=6 case is A103374.
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
For this k=7 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^8 - x - 1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059....
The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395.

			a(30) = 12 because a(30) = a(30-7) + a(30-8) = a(24) + a(23) = 7 + 5 = 12.

Zanten, A. J. van, "The golden ratio in the arts of painting, building and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103372-A103374, A103376-A103380, A103385, A103395.

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 73]
LinearRecurrence[{0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1},80]

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0]^(n-1)*[1;1;1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x^7+x^8). - R. J. Mathar, Dec 14 2009

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

Corrected (one more 8 inserted) by R. J. Mathar, Dec 14 2009

A103395 Semiprimes in A103375.

Original entry on oeis.org

4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 129, 265, 2049, 3865, 4163, 8557, 14005, 80413, 104757, 116333, 152713, 241354, 2273893, 2492909, 16432401, 31701485, 34090613, 263504954, 424792297, 1534443805, 3233454667, 10580401481

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103375 with A001358.

			14005 is an element of this sequence because A103375(106) = 14005 and
14005 is semiprime because 14005 = 5 * 2801 where both 5 and 2801 are primes. It is coincidence here that 106 = 2 * 53 is also semiprime.

Cf. A001358, A103375, A103385, A103392-A103401.

SemiprimeQ[n_] := Plus @@ FactorInteger[n][[All, 2]] == 2; k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 255], SemiprimeQ]]

Edited, corrected and extended by Ray Chandler and Robert G. Wilson v, Feb 06 2005

cp's OEIS Frontend

A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).

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