A103376 -id:A103376 - cp's OEIS frontend

A103386 Primes in A103376.

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

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

A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 33, 38, 48, 58, 63, 65, 71, 86, 106, 121, 128, 136, 157, 192, 227, 249, 264, 293, 349, 419, 476, 513, 557, 642, 768, 895, 989, 1070, 1199, 1410, 1663, 1884, 2059, 2269

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=5 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) and k=4 case is A103372.
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=5 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^6 - x - 1 = 0. This is the real constant 1.1347241384015194926054460545064728402796672263828014859251495516682....
The sequence of prime values in this k=5 case is A103383; the sequence of semiprime values in this k=5 case is A103393.

			a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.

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

Cf. A000045, A000931, A079398, A103383, A103393.

Cf. A103372, A103374, A103375, A103376, A103377, A103378, A103379, A103380

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]
RecurrenceTable[{a[n] == a[n - 5] + a[n - 6], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1}, a, {n, 65}] (* or *)
Rest@ CoefficientList[Series[-x (1 + x + x^2 + x^3 + x^4)/(-1 + x^5 + x^6), {x, 0, 65}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0,0,0,0,1,1},{1,1,1,1,1,1},70] (* Harvey P. Dale, Jul 20 2019 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,0,0,0,0]^(n-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+x^2+x^3+x^4)/(1-x^5-x^6 )) \\ G. C. Greubel, May 01 2017

G.f.: x*(1+x+x^2+x^3+x^4) / (1-x^5-x^6 ). - R. J. Mathar, Aug 26 2011

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

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

A152846 Triangle read by rows, A007318 rows repeated eight times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 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, 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

Offset: 0

Philippe Deléham, Dec 14 2008

Diagonal sums : A103376 .

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

Cf. A007318, A152198, A152815, A152828, A152830, A152831, A152844, A152845.

Flatten[Table[#, {8}] & /@ Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

cp's OEIS Frontend

A103386 Primes in A103376.

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A103396 Semiprimes in A103376.

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A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).

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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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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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A152846 Triangle read by rows, A007318 rows repeated eight times .

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