A242876 -id:A242876 - cp's OEIS frontend

A001611 a(n) = Fibonacci(n) + 1.

1, 2, 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156

Offset: 0

N. J. A. Sloane

a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - Amarnath Murthy, Jun 03 2003
a(n+2) = A^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 2=`0`, 3=`10`, 4=`110`, 6=`1110`, ..., in Wythoff code.
The first-difference sequence is the Fibonacci sequence (A000045). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
2 and 3 are the only primes in this sequence.
a(n) is the number of 1 X n nonogram puzzles which can be solved uniquely. See A242876 for puzzle definition. - Lior Manor, Jan 23 2022

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
K.-W. Chen, Greatest Common Divisors in Shifted Fibonacci Sequences, J. Int. Seq. 14 (2011) # 11.4.7.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.1.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 402
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A097280, A097281.
Cf. A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by N. J. A. Sloane, Jun 25 2010 in response to a comment from Aviezri S. Fraenkel]
Cf. A002062, A019863, A160536, A212272, A242876.

a001611 = (+ 1) . a000045
a001611_list = 1 : 2 : map (subtract 1)
                       (zipWith (+) a001611_list $ tail a001611_list)
-- Reinhard Zumkeller, Jul 30 2013

[Fibonacci(n)+1: n in [1..37]]; // Bruno Berselli, Jul 26 2011

A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
with(combinat): seq((fibonacci(n)+1), n=0..35);

a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n-2] + a[n-1] - 1; Table[ a[n], {n, 0, 40} ]
Fibonacci[Range[0,50]]+1  (* Harvey P. Dale, Mar 23 2011 *)

a(n)=fibonacci(n)+1 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: (1-2*x^2)/(1-2*x+x^3).
a(n) = 2*a(n-1) - a(n-3). - Tanya Khovanova, Jul 13 2007
a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
F(4*n) + 1 = F(2*n-1)*L(2*n+1); F(4*n+1) + 1 = F(2*n+1)*L(2*n); F(4*n+2) + 1 = F(2*n+2)*L(2*n); F(4*n+3) + 1 = F(2*n+1)*L(2*n+2) where F(n)=Fibonacci(n) and L(n)=Lucas(n). - R. K. Guy, Feb 27 2003
a(1) = 2; a(n+1)=floor(a(n)*(sqrt(5)+1)/2). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
a(n) = Sum_{k=0..n+1} Fibonacci(k-3). - Ehren Metcalfe, Apr 15 2019
Product_{n>=1} (1 - (-1)^n/a(n)) = sin(3*Pi/10) (A019863). - Amiram Eldar, Nov 28 2024

A384764 Number of uniquely solveable n X m nonograms (hanjie), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 52, 52, 16, 1, 1, 32, 210, 384, 210, 32, 1, 1, 64, 816, 3152, 3152, 816, 64, 1, 1, 128, 3206, 24230, 52362, 24230, 3206, 128, 1, 1, 256, 12536, 189898, 814632, 814632, 189898, 12536, 256, 1, 1, 512, 48962, 1473674, 12819322, 25309575, 12819322, 1473674, 48962, 512, 1

Offset: 0

Bertram Felgenhauer, Jun 09 2025

In this game there is an n X m grid where each square may or may not be filled. Each column and each row is labeled by the length of each successive block of filled squares, but without indication of the number of unfilled squares in between. The object is to determine which squares are filled.

			A(2,2) = 16-2 because out of the possible 2^(2*2) grids, only 10/01 and 01/10 have the same row and column clues.
Top left corner of the array:
  1,  1,    1,      1,        1,         1,           1, ...
  1,  2,    4,      8,       16,        32,          64, ...
  1,  4,   14,     52,      210,       816,        3206, ...
  1,  8,   52,    384,     3152,     24230,      189898, ...
  1, 16,  210,   3152,    52362,    814632,    12819322, ...
  1, 32,  816,  24230,   814632,  25309575,   794378773, ...
  1, 64, 3206, 189898, 12819322, 794378773, 49745060669, ...

Bertram Felgenhauer, Antidiagonals n+m = 0..13, flattened
Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Cf. A242876 (main diagonal), A000012 (column m=0), A000079 (column m=1), A383345 (column m=2).

Cf. A385862 (variant: uniquely solveable n X m yesnograms).

Basic properties include A(n,m) = A(m,n), A(n,m) <= 2^(n*m), A(0,n) = A(n,0) = 1, and A(1,n) = A(n,1) = 2^n.

A385862 Number of n X m yesnograms that can be solved uniquely, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 52, 52, 16, 1, 1, 32, 210, 368, 210, 32, 1, 1, 64, 816, 2992, 2992, 816, 64, 1, 1, 128, 3206, 23058, 49578, 23058, 3206, 128, 1, 1, 256, 12536, 179576, 775204, 775204, 179576, 12536, 256, 1, 1, 512, 48962, 1388978, 12129616, 24177516, 12129616, 1388978, 48962, 512, 1

Offset: 0

Karl W. Heuer, Aug 06 2025

In a nonogram puzzle, there is a hidden bitonal grid (or 0/1 matrix), and each row and each column is labeled by the length of each successive block of foreground pixels, but without indication of the number of background pixels separating them; the object is to determine the grid contents. In this variant, called a "yesnogram", the pixel value that represents foreground for row clues is the complement of the value that represents foreground for column clues.

			For the 3 X 4 grid shown below, the row clues (counting runs of 0s) and the column clues (counting runs of 1s) are sufficient to reconstruct the grid, so this is one of the 2992 solvable grids counted in A(3, 4).
       | 1 3 1 2
   ----+--------
   1   | 0 1 1 1
   1 1 | 0 1 0 1
   2   | 1 1 0 0
Top left corner of the array:
  1,  1,   1,     1,      1,        1,         1, ...
  1,  2,   4,     8,     16,       32,        64, ...
  1,  4,  14,    52,    210,      816,      3206, ...
  1,  8,  52,   368,   2992,    23058,    179576, ...
  1, 16, 210,  2992,  49578,   775204,  12129616, ...
  1, 32, 816, 23058, 775204, 24177516, 754845831, ...

Bertram Felgenhauer, Antidiagonals n+m = 0..13, flattened
Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Cf. A242876 (solvable n X n nonograms), A384764 (solvable n X m nonograms), A383345 (solvable n X 2 nonograms or yesnograms), A385861 (solvable n X n yesnograms).

A(0,n) = 1, and A(1,n) = 2^n. A(n,m) = A(m,n), because a grid is solvable iff its complement-transpose is solvable.

A383345 Number of uniquely solveable n X 2 nonograms (hanjie).

Original entry on oeis.org

1, 4, 14, 52, 210, 816, 3206, 12536, 48962, 191226, 746456, 2913544, 11371040, 44376798, 173181564, 675834086, 2637392942, 10292179494, 40164144690, 156736057740, 611644171812, 2386868430698, 9314465669046

Offset: 0

Bertram Felgenhauer, Jun 11 2025

In this game there is an n X 2 grid where each square may or may not be filled. Each column and each row is labeled by the length of each successive block of filled squares, but without indication of the number of unfilled squares in between. The object is to determine which squares are filled.

The only ambiguous row hint is 1, which has the same solutions regardless of whether black or white squares are counted. So this is also the number of n X 2 "yesnograms".

			a(2) = 16-2 because out of the possible 2^(2*2) grids, only 10/01 and 01/10 have the same row and column clues.

Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Column m=2 of A384764. Also column m=2 of A385862 (n X m yesnograms).

Cf. A242876.

A385861 Number of n X n yesnograms that can be solved uniquely.

Original entry on oeis.org

1, 2, 14, 368, 49578, 24177516, 46985524156

Offset: 0

Karl W. Heuer, Aug 06 2025

A nonogram provides row and column clues indicating runs of black pixels, treating white as blank. In this variant (called a "yesnogram"), the row clues instead indicate runs of white pixels, treating black as blank. Column clues remain unchanged from the standard nonogram.

			a(2) = 14 because, of the 16 2 X 2 grids, 10/01 and 01/10 would have the same set of clues; the other 14 are solvable.

Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Main diagonal of A385862.

Cf. A242876 (solvable n X n nonograms), A384764 (solvable n X m nonograms), A383345 (solvable n X 2 nonograms or yesnograms).

cp's OEIS Frontend

A001611 a(n) = Fibonacci(n) + 1.

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A384764 Number of uniquely solveable n X m nonograms (hanjie), read by antidiagonals.

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A385862 Number of n X m yesnograms that can be solved uniquely, read by antidiagonals.

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A383345 Number of uniquely solveable n X 2 nonograms (hanjie).

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A385861 Number of n X n yesnograms that can be solved uniquely.

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