A103344 -id:A103344 - cp's OEIS frontend

A104449 Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.

3, 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620

Offset: 0

Casey Mongoven, Mar 08 2005

The old name was: The Pibonacci numbers (a Fibonacci-type sequence): each term is the sum of the two previous terms.
The 6th row in the Wythoff array begins with the 6th term of the sequence (14, 23, 37, 60, 97, 157, ...). a(n) = f(n-3) + f(n+2) for the Fibonacci numbers f(n) = f(n-1) + f(n-2); f(0) = 0, f(1) = 1.
(a(2*k), a(2*k+1)) give for k >= 0 the proper positive solutions of one of two families (or classes) of solutions (x, y) of the indefinite binary quadratic form x^2 + x*y - y^2 of discriminant 5 representing 11. The other family of such solutions is given by (x2, y2) = (b(2*k), b(2*k+1)) with b = A013655. See the formula in terms of Chebyshev S polynomials S(n, 3) = A001906(n+1) below, which follows from the fundamental solution (3, 1) by applying positive powers of the automorphic matrix given in a comment in A013655. See also A089270 with the Alfred Brousseau link with D = 11. - Wolfdieter Lang, May 28 2019

V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

G. C. Greubel, Table of n, a(n) for n = 0..1000
John Conway, Alex Ryba, The extra Fibonacci series and the Empire State Building, Math. Intelligencer 38 (2016), no. 1, 41-48. (Uses the name Pibonacci.)
Tanya Khovanova, Recursive Sequences
Ron Knott, Fibonacci Numbers and the Golden Section .
Eric Weisstein's World of Mathematics, Fibonacci Number
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. Other Fibonacci-type sequences: A000045, A000032, A013655. Other related sequences: A001906, A013655, A089270, A103343, A103344.
Wythoff array: A035513.
Essentially the same as A000285.

a:=[3,1];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 29 2019

[Fibonacci(n-1) + Lucas(n): n in [0..40]]; // G. C. Greubel, May 29 2019

a:=n->3*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007

LinearRecurrence[{1,1},{3,1},40] (* Harvey P. Dale, May 23 2014 *)

a(n)=3*fibonacci(n-1)+fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011

((3-2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 29 2019

a(n) = a(n-1) + a(n-2) with a(0) = 3, a(1) = 1.
a(n) = 3*Fibonacci(n-1) + Fibonacci(n). - Zerinvary Lajos, Oct 05 2007
G.f.: (3-2*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = ( (3*sqrt(5)-1)*((1+sqrt(5))/2)^n + (3*sqrt(5)+1)*((1-sqrt(5) )/2)^n )/(2*sqrt(5)). - Bogart B. Strauss, Jul 19 2013
Bisection: a(2*k) = 4*S(k-1, 3) - 3*S(k-2, 3), a(2*k+1) = 2*S(k-1, 3) + S(k, 3) for k >= 0, with the Chebyshev S(n, 3) polynomials from A001906(n+1) for n >= -1. - Wolfdieter Lang, May 28 2019
a(n) = Fibonacci(n-1) + Lucas(n). - G. C. Greubel, May 29 2019
a(3n + 4)/a(3n + 1) = continued fraction 4,4,4,...,4,9 (that's n 4's followed by a single 9). - Greg Dresden and Shaoxiong Yuan, Jul 16 2019
E.g.f.: (exp((1/2)*(1 - sqrt(5))*x)*(1 + 3*sqrt(5) + (- 1 + 3*sqrt(5))*exp(sqrt(5)*x)))/(2*sqrt(5)). - Stefano Spezia, Jul 18 2019

Name changed by Wolfdieter Lang, Jun 17 2019

A103343 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 3, 1, 4, 5, 9, 14, 23, 37, ....

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 3, 2, 1, 3, 4, 3, 1, 2, 3, 4, 3, 0, 3, 5, 4, 2, 2, 4, 5, 3, 0, 3, 4, 4, 3, 1, 4, 6, 5, 2, 3, 5, 6, 4, 0, 4, 6, 5, 3, 2, 5, 6, 4, 1, 3, 4, 5, 4, 0, 4, 7, 6, 3, 3, 6, 8, 5, 0, 5, 7, 6, 4, 2, 6, 8, 6, 2, 4, 6, 7, 5, 0, 5, 8, 6, 3, 3, 6, 7, 4, 0, 4, 5, 5, 4, 1, 5

Offset: 0

Casey Mongoven, Feb 01 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..11933
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.
Ron Knott, Ron Knott's Sequence Visualiser.
Casey Mongoven, V(n) Rep Sequence no. 1; electronic music created with this sequence.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 corrected by Alois P. Heinz, Sep 16 2015

A219479 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 6, 7, 13, 20, 33, 53, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 3, 6, 3, 0, 0, 0, 0, 5, 5

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498.

Alois P. Heinz, Table of n, a(n) for n = 0..17101
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.
C. Mongoven, Musical Composition Based on This Sequence.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219480 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 7, 8, 15, 23, 38, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 0, 4, 4, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..12166
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219481 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 8, 9, 17, 26, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..13763
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219482 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 9, 10, 19, 29, 48, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15360
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219483 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 7, 9, 16, 25, 41, ....

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 3, 0, 2, 0, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 1, 0, 4, 0, 3, 0, 0, 0, 0, 3, 0, 5, 0, 2, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 2, 0, 5, 0, 3, 0, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..13153
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219484 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 9, 11, 20, 31, 51, ....

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 3, 0, 0, 0, 0, 0, 0, 3, 0, 5, 0, 2, 0, 0, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16347
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219485 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 7, 10, 17, 27, 44, ....

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 0, 3, 0, 0, 5, 0, 0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 2, 0, 0, 5, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..14140
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219486 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 8, 11, 19, 30, 49, ....

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15737
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Cf. A000121, A000119, A067595, A003263, A103344.

a(0)=1 from Alois P. Heinz, Sep 16 2015

cp's OEIS Frontend

A104449 Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.

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A103343 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 3, 1, 4, 5, 9, 14, 23, 37, ....

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A219479 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 6, 7, 13, 20, 33, 53, ....

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A219480 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 7, 8, 15, 23, 38, ....

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A219481 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 8, 9, 17, 26, ....

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A219482 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 9, 10, 19, 29, 48, ....

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A219483 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 7, 9, 16, 25, 41, ....

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A219484 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 9, 11, 20, 31, 51, ....

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Author

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Crossrefs

Extensions

A219485 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 7, 10, 17, 27, 44, ....

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Author

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Links

Crossrefs

Extensions

A219486 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 8, 11, 19, 30, 49, ....

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