A103392 -id:A103392 - 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

A103382 Primes in A103372.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 71, 157, 257, 349, 641, 769, 3613, 16763, 233417, 9540317, 7391145211, 139697883473, 163069191377, 562142600387, 108169189705333, 1285424246556809, 1050111983810669984101, 17940124369540827523058309

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103372 with A000040.

			233417 is an element of this sequence because A103372(83) = 233417.

Harvey P. Dale, Table of n, a(n) for n = 1..50

Cf. A000040, A103372, A103383-A103391, A103392.

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 400], PrimeQ]]
Select[LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},500],PrimeQ] // Union (* Harvey P. Dale, May 30 2020 *)

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

A103393 Semiprimes in A103373.

Original entry on oeis.org

4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 249, 895, 989, 1199, 2059, 3073, 9206, 12302, 19766, 33238, 63109, 197459, 252982, 283942, 477931, 691357, 4598261, 8671301, 9819097, 11176233, 51113687, 205150267, 232797043, 496450043, 562358905

Offset: 1

Jonathan Vos Post, Feb 03 2005

Intersection of A103373 with A001358.

			63109 is an element of this sequence because A103373(91) = 63109 and 63109 is semiprime because 63109 = 223 * 283 where both 223 and 283 are primes.

Cf. A001358, A103373, A103383, A103392-A103401.

SemiprimeQ[n_] := Plus @@ FactorInteger[n][[All, 2]] == 2; k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 170], SemiprimeQ]]
Select[LinearRecurrence[{0,0,0,0,1,1},{1,1,1,1,1,1},200],PrimeOmega[ #] == 2&]//Union (* Harvey P. Dale, Jul 20 2019 *)

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

A103394 Semiprimes in A103374.

Original entry on oeis.org

4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 129, 265, 979, 1079, 2279, 7985, 8491, 14019, 15397, 37606, 61289, 71845, 117013, 127401, 196763, 221905, 244414, 265358, 290111, 466319, 555469, 1065241, 1672598, 4276487, 4712791, 5266246, 8178897

Offset: 1

Jonathan Vos Post, Feb 03 2005

			61289 is an element of this sequence because A103374(107) = 61289 and 61289 is semiprime because 61289 = 167 * 367 where both 167 and 367 are primes.

Intersection of A103374 with A001358.

Cf. A001358, A103374, A103384, A103392-A103401.

SemiprimeQ[n_] := Plus @@ FactorInteger[n][[All, 2]] == 2; k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 160], SemiprimeQ]]
Select[LinearRecurrence[{0,0,0,0,0,1,1},{1,1,1,1,1,1,1},200],PrimeOmega[#]==2&]//Union (* Harvey P. Dale, Sep 02 2024 *)

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

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

A103396 Semiprimes in A103376.

Original entry on oeis.org

4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 129, 265, 511, 2038, 2059, 4097, 4174, 7894, 16021, 19857, 31313, 32419, 33238, 37711, 116197, 196609, 220937, 262978, 273926, 955743, 34826059, 64229819, 67835071, 77834009, 497049562, 4370946037

Offset: 1

Jonathan Vos Post, Feb 05 2005

Intersection of A103376 with A001358.

Cf. A001358, A103376, A103386, A103392-A103401.

SemiprimeQ[n_] := Plus @@ FactorInteger[n][[All, 2]] == 2; k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Union[Select[Array[a, 280], SemiprimeQ]]
Union[Select[LinearRecurrence[{0,0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1,1},300],PrimeOmega[#]==2&]] (* Harvey P. Dale, May 07 2015 *)

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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A103382 Primes in A103372.

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A103393 Semiprimes in A103373.

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A103394 Semiprimes in A103374.

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A103395 Semiprimes in A103375.

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A103396 Semiprimes in A103376.

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