A103373 -id:A103373 - cp's OEIS frontend

A103383 Primes in A103373.

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

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

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

Original entry on oeis.org

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

A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 256, 265, 293, 349, 419, 475, 504, 521, 558, 642, 768, 894, 979, 1025, 1079

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=6 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 and k=5 case is A103373.
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=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant 1.1127756842787... (see A230160).
The sequence of prime values in this k=6 case is A103384; the sequence of semiprime values in this k=6 case is A103394.

			a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
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 (0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103384, A103394, A230160.

Cf. A103372, A103373, A103375, A103376, A103377, A103378, A103379, A103380

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
RecurrenceTable[{a[n] == a[n - 6] + a[n - 7], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == a[7] == 1}, a, {n, 70}] (* or *)
Rest@ CoefficientList[Series[-x (1 + x) (1 + x + x^2) (x^2 - x + 1)/(-1 + x^6 + x^7), {x, 0, 70}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0,0,0,0,0,1,1},{1,1,1,1,1,1,1},80] (* Harvey P. Dale, Sep 02 2024 *)

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

x='x+O('x^50); Vec(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^6-x^7)) \\ G. C. Greubel, May 01 2017

G.f.: x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( 1-x^6-x^7 ). - R. J. Mathar, Aug 26 2011

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

A103377 a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 128, 129, 136, 157, 192, 227

Offset: 1

Jonathan Vos Post, Feb 15 2005

k=9 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, k=6 case is A103374, k=7 case is A103375, k=8 case is A103376, k=10 case is A103378 and k=11 case is A103379. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=9 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^10 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0757660660868371580595995241652758206925302476392 = A230163. Note that x = (1 + x)^(1/10) = (1 + (1 + (1 + ...)^(1/10))^(1/10))^(1/10). The sequence of prime values in this k=9 case is A103387; The sequence of semiprime values in this k=9 case is A103397.

In analogy to the Fibonacci sequence, one might prefer to start this sequence with offset 0. - M. F. Hasler, Sep 19 2015

			a(83) = 257 because a(83) = a(83-9) + a(83-10). a(74) + a(73) = 129 + 128. This sequence has as elements 5, 17 and 257, which are all Fermat Primes.

A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (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) 291-306
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61(1988)1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103372-A103376, A103378-A103380, A103387, A103397.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* Charles R Greathouse IV, Jan 11 2013 *)

Vec((1+x+x^2)*(1+x^3+x^6)/(1-x^9-x^10)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 1 and for n>10: a(n) = a(n-9) + a(n-10).

O.g.f.: -x*(x^2+x+1)*(x^6+x^3+1)/(-1+x^9+x^10). - R. J. Mathar, May 02 2008

Edited by R. J. Mathar, May 02 2008

Edited by M. F. Hasler, Sep 19 2015

A152831 Triangle read by rows, A007318 repeated five times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A103373 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828, A152830.

Flatten[Table[#, {5}] & /@ Table[Binomial[n, k], {n,0,10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

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).

Original entry on oeis.org

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

A103376 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n-8) + a(n-9).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 257, 265, 293

Offset: 1

Jonathan Vos Post, Feb 05 2005

k=8 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, k=6 case is A103374 and k=7 case is A103375.
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=8 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^9 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0850702454914508283368958640973142340506536310308 = A230162. Note that x = (1 + x)^(1/9) = (1 + (1 + (1 + ...)^(1/9))^(1/9))^(1/9).
The sequence of prime values in this k=8 case is A103386; The sequence of semiprime values in this k=8 case is A103396.

			a(93) = 1200 because a(93) = a(93-8) + a(93-9) = a(85) + a(84) = 642 + 558.

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 (0,0,0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103372-A103375, A103377-A103380, A103386, A103396.

k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 76]
LinearRecurrence[{0,0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1,1},80] (* Harvey P. Dale, May 07 2015 *)

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0,0]^(n-1)*[1;1;1;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)/(1-x^8-x^9). - R. J. Mathar, Dec 14 2009

a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1, a(9)=1, a(n)=a(n-8)+a(n-9). - Harvey P. Dale, May 07 2015

Edited by Ray Chandler, Feb 10 2005

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 9, 9, 12, 12, 15, 16, 17, 21, 21, 27, 28, 32, 37, 38, 48, 49, 59, 65, 70, 85, 87, 107, 114, 129, 150, 157, 192, 201, 236, 264, 286, 342, 358, 428, 465, 522, 606, 644, 770, 823, 950, 1071, 1166, 1376

Offset: 1

Vicente Jesús Maniega Granado, Oct 03 2016

Generalized Fibonacci growth sequence using i = 2 as maturity period, j = 5 as conception period, and k = 2 as growth factor.

Maturity period is the number of periods that a Fibonacci tree node needs for being able to start developing branches. Conception period is the number of periods in a Fibonacci tree node needed to develop new branches since its maturity. Growth factor is the number of additional branches developed by a Fibonacci tree node, plus 1, and equals the base of the exponential series related to the given tree if maturity factor would be zero. Standard Fibonacci would use 1 as maturity period, 1 as conception period, and 2 as growth factor as the series becomes equal to 2^n with a maturity period of 0. Related to Lucas sequences.

			For n = 13 the a(13) = a(8) + a(6) = 2 + 1 = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Julia Collins, Fibonacci Tree
Fractal Foundation, Fibonacci Fractals
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,1).

Cf. A000079 (i = 0, j = 1, k = 2), A000244 (i = 0, j = 1, k = 3), A000302 (i = 0, j = 1, k = 4), A000351 (i = 0, j = 1, k = 5), A000400 (i = 0, j = 1, k = 6), A000420 (i = 0, j = 1, k = 7), A001018 (i = 0, j = 1, k = 8), A001019 (i = 0, j = 1, k = 9), A011557 (i = 0, j = 1, k = 10), A001020 (i = 0, j = 1, k = 11), A001021 (i = 0, j = 1, k = 12), A016116 (i = 0, j = 2, k = 2), A108411 (i = 0, j = 2, k = 3), A213173 (i = 0, j = 2, k = 4), A074872 (i = 0, j = 2, k = 5), A173862 (i = 0, j = 3, k = 2), A127975 (i = 0, j = 3, k = 3), A200675 (i = 0, j = 4, k = 2), A111575 (i = 0, j = 4, k = 3), A000045 (i = 1, j = 1, k = 2), A001045 (i = 1, j = 1, k = 3), A006130 (i = 1, j = 1, k = 4), A006131 (i = 1, j = 1, k = 5), A015440 (i = 1, j = 1, k = 6), A015441 (i = 1, j = 1, k = 7), A015442 (i = 1, j = 1, k = 8), A015443 (i = 1, j = 1, k = 9), A015445 (i = 1, j = 1, k = 10), A015446 (i = 1, j = 1, k = 11), A015447 (i = 1, j = 1, k = 12), A000931 (i = 1, j = 2, k = 2), A159284 (i = 1, j = 2, k = 3), A238389 (i = 1, j = 2, k = 4), A097041 (i = 1, j = 2, k = 10), A079398 (i = 1, j = 3, k = 2), A103372 (i = 1, j = 4, k = 2), A103373 (i = 1, j = 5, k = 2), A103374 (i = 1, j = 6, k = 2), A000930 (i = 2, j = 1, k = 2), A077949 (i = 2, j = 1, k = 3), A084386 (i = 2, j = 1, k = 4), A089977 (i = 2, j = 1, k = 5), A178205 (i = 2, j = 1, k = 11), A103609 (i = 2, j = 2, k = 2), A077953 (i = 2, j = 2, k = 3), A226503 (i = 2, j = 3, k = 2), A122521 (i = 2, j = 6, k = 2), A003269 (i = 3, j = 1, k = 2), A052942 (i = 3, j = 1, k = 3), A005686 (i = 3, j = 2, k = 2), A237714 (i = 3, j = 2, k = 3), A238391 (i = 3, j = 2, k = 4), A247049 (i = 3, j = 3, k = 2), A077886 (i = 3, j = 3, k = 3), A003520 (i = 4, j = 1, k = 2), A108104 (i = 4, j = 2, k = 2), A005708 (i = 5, j = 1, k = 2), A237716 (i = 5, j = 2, k = 3), A005709 (i = 6, j = 1, k = 2), A122522 (i = 6, j = 2, k = 2), A005710 (i = 7, j = 1, k = 2), A237718 (i = 7, j = 2, k = 3), A017903 (i = 8, j = 1, k = 2).

[n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016

LinearRecurrence[{0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 70] (*  or *)
CoefficientList[ Series[(1+x+x^2+x^3+x^4)/(1-x^5-x^7), {x, 0, 70}], x] (* Robert G. Wilson v, Nov 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,a+c}; NestList[nxt,{1,1,1,1,1,1,1},70][[;;,1]] (* Harvey P. Dale, Oct 22 2024 *)

Vec(x*(1+x+x^2+x^3+x^4)/((1-x+x^2)*(1+x-x^3-x^4-x^5)) + O(x^100)) \\ Colin Barker, Oct 27 2016

@CachedFunction # a = A242763
def a(n): return 1 if n<8 else a(n-5) +a(n-7)
[a(n) for n in range(1,76)] # G. C. Greubel, Oct 23 2024

Generic a(n) = 1 for n <= i+j; a(n) = a(n-j) + (k-1)*a(n-(i+j)) for n>i+j where i = maturity period, j = conception period, k = growth factor.
G.f.: x*(1+x+x^2+x^3+x^4) / ((1-x+x^2)*(1+x-x^3-x^4-x^5)). - Colin Barker, Oct 09 2016
Generic g.f.: x*(Sum_{l=0..j-1} x^l) / (1-x^j-(k-1)*x^(i+j)), with i > 0, j > 0 and k > 1.

A321025 a(n) = sum of a(n-4) and a(n-5), with the lowest possible initial values that will generate a sequence where a(n) is always > a(n-1): 4, 5, 6, 7 and 8.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 20, 24, 28, 32, 37, 44, 52, 60, 69, 81, 96, 112, 129, 150, 177, 208, 241, 279, 327, 385, 449, 520, 606, 712, 834, 969, 1126, 1318, 1546, 1803, 2095, 2444, 2864, 3349, 3898, 4539, 5308, 6213, 7247, 8437, 9847, 11521, 13460, 15684

Offset: 1

Mathew Munro, Oct 30 2018

A sum of prior terms in the sequence, like the Fibonacci and Padovan sequences.

			a(6) = a(6-4) + a(6-5) = a(2) + a(1) = 5 + 4 = 9.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1).

Cf. A000045, A000931, A079398, A164317, A103372, A103373

Rest@ CoefficientList[Series[x (4 + 5 x + 6 x^2 + 7 x^3 + 4 x^4)/(1 - x^4 - x^5), {x, 0, 54}], x] (* Michael De Vlieger, Oct 31 2018 *)

a(n) = if(n<=5, n+3, a(n-4) + a(n-5)); \\ Michel Marcus, Oct 31 2018

Vec((4 + 5*x + 6*x^2 + 7*x^3 + 4*x^4)/(1 - x^4 - x^5) + O(x^50)) \\ Andrew Howroyd, Oct 31 2018

a(n) = a(n-4) + a(n-5) with a(1) = 4, a(2) = 5, a(3) = 6, a(4) = 7 and a(5) = 8.

G.f.: x*(4 + 5*x + 6*x^2 + 7*x^3 + 4*x^4)/(1 - x^4 - x^5). - Andrew Howroyd, Oct 31 2018

a(19), a(20) corrected by Georg Fischer, May 24 2019

cp's OEIS Frontend

A103383 Primes in A103373.

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A103393 Semiprimes in A103373.

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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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A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).

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A103377 a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

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A152831 Triangle read by rows, A007318 repeated five times .

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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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