A103382 -id:A103382 - cp's OEIS frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613, 4163, 4824, 5684, 6697, 7776

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=4 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) and k=3 case is A079398 (offset so as to begin 1,1,1,1).
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=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0, A160155.
The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.

			a(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.

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

Indranil Ghosh, Table of n, a(n) for n = 1..14857
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 (0,0,0,1,1).

Cf. A000931, A079398, A103373-A103380, A103382, A103392.

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]
LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},70] (* Harvey P. Dale, Apr 22 2015 *)

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

G.f. -x*(1+x)*(1+x^2) / ( -1+x^4+x^5 ). - R. J. Mathar, Aug 26 2011

a(n) = A124789(n-2)+A124798(n-1). - R. J. Mathar, Jun 30 2020

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

A103392 Semiprimes in A103372.

Original entry on oeis.org

4, 9, 15, 21, 33, 38, 58, 86, 106, 122, 485, 905, 1939, 4163, 6697, 12381, 14473, 22889, 107833, 170769, 370729, 687634, 804106, 1093762, 1276259, 2767999, 11140379, 15186449, 210634829, 286937249, 391048473, 532793518, 725995178, 847539571

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103372 with A001358.

			276799 is an element of this sequence because A103372(99) = 276799 and 276799 is semiprime because 276799 = 31 * 8929 where both 31 and 8929 are primes.

Cf. A001358, A103372, A103382, A103393-A103401.

SemiprimeQ[n_] := Plus @@ FactorInteger[n][[All, 2]] == 2; k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 150], SemiprimeQ]]
Union[Select[LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},150],PrimeOmega[#] == 2&]] (* Harvey P. Dale, Apr 22 2015 *)

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

A103383 Primes in A103373.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 157, 227, 293, 349, 419, 557, 1663, 2269, 2609, 3547, 3943, 15761, 17477, 37243, 70481, 105557, 23913779, 84394837, 7057254647, 3915885721591, 4641244746324673, 5266511621347511, 565552908731370799

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103373 with A000040.

			105557 is an element of this sequence because A103373(95) = 105557.

Cf. A000040, A103373, A103382-A103391, A103393.

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 350], PrimeQ]]

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

A103384 Primes in A103374.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 293, 349, 419, 521, 1873, 3877, 4889, 6607, 12289, 19843, 31873, 42703, 80309, 137957, 884987, 976091, 9979037, 15614983, 29738231, 45024979, 812280583, 6882259301, 8479913153, 88930242859, 356874733421

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103374 with A000040.

			1873 is an element of this sequence because A103374(74) = 1873.

Cf. A000040, A103374, A103382-A103391, A103394.

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 260], PrimeQ]]

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

A103385 Primes in A103375.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 1993, 7907, 26293, 29311, 34603, 67477, 3147311, 9159547, 973669469, 6797534657, 9627183689, 297222052181, 4530692779838851, 41748646469705167, 266359428042546661

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103375 with A000040.

			17 is an element of this sequence because A103375(36) = 17.

Cf. A000040, A103375, A103382-A103391, A103395.

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 475], PrimeQ]]

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

A103386 Primes in A103376.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 65657, 152833, 172297, 11229341, 12584983, 26532901, 31220807, 1164893671, 1217349652999, 2346608054761, 8116583338373, 53091879496979, 9758833144565411

Offset: 1

Jonathan Vos Post, Feb 05 2005

Intersection of A103376 with A000040.

			7 is in this sequence because A103376(27) = 7, which is prime.

Cf. A000040, A103376, A103382-A103391, A103396.

k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 500], PrimeQ]]

Edited and extended by Ray Chandler, Feb 10 2005

cp's OEIS Frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

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A103392 Semiprimes in A103372.

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A103383 Primes in A103373.

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A103384 Primes in A103374.

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A103385 Primes in A103375.

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A103386 Primes in A103376.

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