A001595 -id:A001595 - cp's OEIS frontend

A131411 Triangle read by rows: T(n,k) = Fibonacci(n) + Fibonacci(k) - 1.

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 9, 8, 8, 9, 10, 12, 15, 13, 13, 14, 15, 17, 20, 25, 21, 21, 22, 23, 25, 28, 33, 41, 34, 34, 35, 36, 38, 41, 46, 54, 67, 55, 55, 56, 57, 59, 62, 67, 75, 88, 109, 89, 89, 90, 91, 93, 96, 101, 109, 122, 143, 177, 144, 144, 145, 146, 148, 151, 156, 164, 177, 198, 232, 287

Offset: 1

Gary W. Adamson, Jul 08 2007

Left column = Fibonacci numbers. Right column = A001595: (1, 1, 3, 5, 9, 15, 25,...).

Row sums = A131412: (1, 2, 7, 15, 32, 62, 117, 214,...).

			First few rows of the triangle are:
   1;
   1,  1;
   2,  2,  3;
   3,  3,  4,  5;
   5,  5,  6,  7,  9;
   8,  8,  9, 10, 12, 15;
  13, 13, 14, 15, 17, 20, 25;
  21, 21, 22, 23, 25, 28, 33, 41;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A131412.

Cf. A000012, A000045, A001595, A104763, A127647, A131410.

F:=Fibonacci;; Flat(List([1..15], n-> List([1..n], k-> F(n) +F(k) -1 ))); # G. C. Greubel, Jul 13 2019

F:=Fibonacci; [F(n)+F(k)-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

With[{F=Fibonacci}, Table[F[n]+F[k]-1, {n,15}, {k,n}]//Flatten] (* G. C. Greubel, Jul 13 2019 *)

T(n,k) = if(k<=n, fibonacci(n) + fibonacci(k) - 1, 0); \\ Andrew Howroyd, Aug 10 2018

f=fibonacci; [[f(n)+f(k)-1 for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

Equals A131410 + A104763 - A000012 as infinite lower triangular matrices.

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A145912 Prime Leonardo numbers.

Original entry on oeis.org

3, 5, 41, 67, 109, 1973, 5167, 2692537, 11405773, 126491971, 331160281, 535828591, 279167724889, 145446920496281, 28944668049352441, 5760134388741632239, 63880869269980199809, 167242286979696845953, 597222253637954133837103

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Oct 24 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..50

Cf. A001595.

A228145 = Select[Range[200], PrimeQ[2 Fibonacci[#] - 1] &]; A001595[n_] := 2*Fibonacci[n + 1] - 1; a = A001595[A228145 - 1] (* Amiram Eldar, Sep 04 2017 *)

a(n) = A001595(A228145(n)-1). - Amiram Eldar, Sep 04 2017

Extended by Klaus Brockhaus, Oct 26 2008

A117666 Expansion of (1-3x+x^2)(1-x-x^2)/((1+x+x^2)(1-x+x^2)(1-x)^2).

Original entry on oeis.org

1, -2, -3, 2, 3, 2, 3, 0, -1, 4, 5, 4, 5, 2, 1, 6, 7, 6, 7, 4, 3, 8, 9, 8, 9, 6, 5, 10, 11, 10, 11, 8, 7, 12, 13, 12, 13, 10, 9, 14, 15, 14, 15, 12, 11, 16, 17, 16, 17, 14, 13, 18, 19, 18, 19, 16, 15, 20, 21, 20, 21, 18, 17, 22, 23, 22, 23, 20, 19, 24, 25, 24, 25, 22, 21, 26, 27, 26, 27, 24, 23, 28, 29, 28, 29, 26, 25

Offset: 0

Creighton Dement, Apr 11 2006

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A104161, A001595.

a:=[1,-2,-3,2,3,2];; for n in [7..100] do a[n]:=2*a[n-1] -2*a[n-2] +2*a[n-3] -2*a[n-4] +2*a[n-5] -a[n-6]; od; a; # G. C. Greubel, Jul 13 2019

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2) )); // G. C. Greubel, Jul 13 2019

CoefficientList[Series[(1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2), {x, 0, 100}], x] (* G. C. Greubel, Jul 13 2019 *)

Vec((1-3*x+x^2)*(1-x-x^2)/((1-x)^2*(1+x^2+x^4)) + O(x^100)) \\ Colin Barker, May 18 2019

((1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2)).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jul 13 2019

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>5. - Colin Barker, May 18 2019

A131400 A046854 + A065941 - I (Identity matrix).

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 3, 6, 3, 1, 2, 4, 7, 7, 4, 1, 2, 4, 11, 8, 11, 4, 1, 2, 5, 12, 15, 15, 12, 5, 1, 2, 5, 17, 16, 30, 16, 17, 5, 1, 2, 6, 18, 27, 36, 36, 27, 18, 6, 1, 2, 6, 24, 28, 63, 42, 63, 28, 24, 6, 1, 2, 7, 25, 44, 71, 84, 84, 71, 44, 25, 7, 1

Offset: 0

Gary W. Adamson, Jul 06 2007

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67,...).

			First few rows of the triangle are:
  1;
  2, 1;
  2, 2,  1;
  2, 3,  3, 1;
  2, 3,  6, 3,  1;
  2, 4,  7, 7,  4, 1;
  2, 4, 11, 8, 11, 4, 1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A046854, A065941, A001595.

B:=Binomial;;
T:= function(n,k)
    if k=n then return 1;
    else return B(Int((n+k)/2), k) + B(n - Int((k+1)/2), Int(k/2));
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 13 2019

B:=Binomial; [k eq n select 1 else B(Floor((n+k)/2), k) + B(n - Floor((k+1)/2), Floor(k/2)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 13 2019

With[{B = Binomial}, Table[If[k==n, 1, B[Floor[(n+k)/2], k] + B[n - Floor[(k+1)/2], Floor[k/2]]], {n,0,12}, {k,0,n}]]//Flatten (* G. C. Greubel, Jul 13 2019 *)

b=binomial; T(n,k) = if(k==n, 1, b((n+k)\2, k) + b(n - (k+1)\2, k\2));
for(n=0,12, for(k=0,n, print1(T(n,k), ", ", ))) \\ G. C. Greubel, Jul 13 2019

def T(n, k):
    b=binomial;
    if (k==n): return 1
    else: return b(floor((n+k)/2), k) + b(n - floor((k+1)/2), floor(k/2))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 13 2019

More terms added by G. C. Greubel, Jul 13 2019

A175004 Interspersion related to the Wythoff Array.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 10, 9, 8, 12, 17, 15, 14, 11, 20, 28, 25, 23, 19, 13, 33, 46, 41, 38, 31, 22, 16, 54, 75, 67, 62, 51, 36, 27, 18, 88, 122, 109, 101, 83, 59, 44, 30, 21, 143, 198, 177, 164, 135, 96, 72, 49, 35, 24, 232, 321, 287, 266, 219, 156, 117, 80, 57, 40, 26, 376, 520, 465, 431, 355, 253, 190, 130, 93, 65, 43, 29

Offset: 1

Clark Kimberling, Apr 03 2010

The rows satisfy the recurrence r(n)=r(n-1)+r(n-2)+1.

Every positive integer occurs exactly once, so that as a sequence, A175004 is a permutation of the natural numbers. As an array, it is an interspersion, hence also a dispersion. Specifically, it is the dispersion of the sequence floor(n*x+2/x), where x=(golden ratio). For a discussion of dispersions, see A191426.

			Corner of the array:
1....2....4....7....12...20... (cf. A000071)
3....6....10...17...28...46... (cf. A001610)
5....9....15...25...41...67... (cf. A001595)
8....14...23...38...62...101..

Cf. A035513.

(* program generates the dispersion array T of the complement of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12; (* c= # cols of T, c1=# cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 2/x]
(* f(n) is complement of column 1 *)
mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A175004 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011, added here Jun 03 2011 by Clark Kimberling *)

Let W'=W-1, where W is the Wythoff array, given by A035513.
Row 1 of W' is (0,1,2,4,7,12,...); replace this by (1,2,4,7,12,...).
The resulting array is A175004.

A258316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 or 0011.

Original entry on oeis.org

5, 7, 7, 10, 9, 10, 15, 12, 12, 15, 23, 17, 15, 17, 23, 36, 25, 20, 20, 25, 36, 57, 38, 28, 25, 28, 38, 57, 91, 59, 41, 33, 33, 41, 59, 91, 146, 93, 62, 46, 41, 46, 62, 93, 146, 235, 148, 96, 67, 54, 54, 67, 96, 148, 235, 379, 237, 151, 101, 75, 67, 75, 101, 151, 237, 379, 612

Offset: 1

R. H. Hardin, Jun 29 2015

Table starts
...5...7..10..15..23..36..57..91.146.235.379.612..989.1599.2586.4183.6767.10948
...7...9..12..17..25..38..59..93.148.237.381.614..991.1601.2588.4185.6769.10950
..10..12..15..20..28..41..62..96.151.240.384.617..994.1604.2591.4188.6772.10953
..15..17..20..25..33..46..67.101.156.245.389.622..999.1609.2596.4193.6777.10958
..23..25..28..33..41..54..75.109.164.253.397.630.1007.1617.2604.4201.6785.10966
..36..38..41..46..54..67..88.122.177.266.410.643.1020.1630.2617.4214.6798.10979
..57..59..62..67..75..88.109.143.198.287.431.664.1041.1651.2638.4235.6819.11000
..91..93..96.101.109.122.143.177.232.321.465.698.1075.1685.2672.4269.6853.11034
.146.148.151.156.164.177.198.232.287.376.520.753.1130.1740.2727.4324.6908.11089
.235.237.240.245.253.266.287.321.376.465.609.842.1219.1829.2816.4413.6997.11178
Apparently: put 1s in some number of nonadjacent columns or put 1s in some number of nonadjacent rows

			Some solutions for n=4 k=4
..0..0..0..0..0....1..1..1..1..1....0..0..0..0..0....1..1..1..1..1
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..0..0..0....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0
..0..0..0..0..0....0..0..0..0..0....1..1..1..1..1....0..0..0..0..0
..1..1..1..1..1....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1104

Column 1 is A018910
Column 2 is A157727(n+3)
Column 3 is A187107(n+3)
Diagonal is A001595(n+2)
Superdiagonal 1 is A000071(n+5)
Superdiagonal 2 is A001610(n+3)
Superdiagonal 3 is A001595(n+4)
Superdiagonal 5 is A022308(n+5)
Superdiagonal 6 is A022319(n+5)
Superdiagonal 7 is A022407(n+5)
Superdiagonal 9 is A022323(n+7)

Empirical: T(n,k) = Fibonacci(n+3) +Fibonacci(k+3) -1

Empirical for rows, columns and nw-se diagonals: a(n) = 2*a(n-1) -a(n-3)

A355288 a(0)=1, a(1)=3, a(2)=7; thereafter a(n) = a(n-1) + a(n-2) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 31, 51, 83, 135, 219, 355, 575, 931, 1507, 2439, 3947, 6387, 10335, 16723, 27059, 43783, 70843, 114627, 185471, 300099, 485571, 785671, 1271243, 2056915, 3328159, 5385075, 8713235, 14098311, 22811547, 36909859, 59721407, 96631267, 156352675, 252983943, 409336619, 662320563

Offset: 0

Sumukh Patel, Jun 27 2022

a(n) is the minimum number of nodes required for a full binary tree of height n with every node height-balanced, and the root node has a balance factor of 0.
Full binary tree: A binary tree is called a full binary tree if each node has exactly two or no children.
Essentially the same as A022403. - R. J. Mathar, Sep 23 2022

			The diagrams below illustrate the terms a(3)=11 and a(4)=19.
           R                         R
          / \                       / \
         /   \                     /   \
        /     \                   /     \
       o       o                 /       \
      / \     / \               /         \
     o   N   N   o             /           \
    / \         / \           /             \
   N   N       N   N         o               o
                            / \             / \
                           /   \           /   \
                          /     \         /     \
                         o       o       o       o
                        / \     / \     / \     / \
                       o   N   N   N   N   o   N   N
                      / \                 / \
                     N   N               N   N

Sumukh Patel, Table of n, a(n) for n = 0..1000
Lecture Notes for Computer Science 2530, Height-balanced trees
NIST, Root node
Wikipedia, Full binary tree
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A354902.

[n eq 0 select 1 else 4*Fibonacci(n+1) - 1: n in [0..40]];

Join[{1},Table[4*Fibonacci[n + 1] - 1, {n, 1, 40}]]

a(0)=1, a(1)=3, a(2)=7; thereafter a(n) = a(n-1) + a(n-2) + 1.
From Stefano Spezia, Jun 27 2022: (Start)
G.f.: (1 + x + x^2 - 2*x^3)/((1 - x)*(1 - x - x^2)).
a(n) = 2*a(n-1) - a(n-3) for n > 3.
a(n) = 2^(1-n)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))/sqrt(5) - 1 for n > 0.
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x) - 2. (End)
a(n) = 4*A000045(n+1) - 1, for n >= 1.
a(n) = 2*A001595(n) + 1, for n >= 1.

A362255 a(0) = a(1) = a(2) = 1, for n > 2, a(n) = a(n-1) + a(n-k) + k with k = 2.

Original entry on oeis.org

1, 1, 1, 4, 7, 10, 16, 25, 37, 55, 82, 121, 178, 262, 385, 565, 829, 1216, 1783, 2614, 3832, 5617, 8233, 12067, 17686, 25921, 37990, 55678, 81601, 119593, 175273, 256876, 376471, 551746, 808624, 1185097, 1736845, 2545471, 3730570, 5467417, 8012890, 11743462, 17210881

Offset: 0

Michael De Vlieger, Apr 13 2023

Called Leonardo 2-numbers in the Tan-Leung paper.

Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers (2023) Vol. 23, #A7. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Cf. A001595, A111314, A362256.

LinearRecurrence[{2, -1, 1, -1}, {1, 1, 1, 4}, 40] (* or *)
With[{k = 2}, Nest[Append[#, #[[-1]] + #[[-k - 1]] + k] &, {1, 1, 1}, 40] ]

A362256 a(0) = a(1) = a(2) = 1, for n > 2, a(n) = a(n-1) + a(n-k) + k with k = 3.

Original entry on oeis.org

1, 1, 1, 5, 9, 13, 17, 25, 37, 53, 73, 101, 141, 197, 273, 377, 521, 721, 997, 1377, 1901, 2625, 3625, 5005, 6909, 9537, 13165, 18173, 25085, 34625, 47793, 65969, 91057, 125685, 173481, 239453, 330513, 456201, 629685, 869141, 1199657, 1655861, 2285549, 3154693

Offset: 0

Michael De Vlieger, Apr 13 2023

Called Leonardo 3-numbers in the Tan-Leung paper.

Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers (2023) Vol. 23, #A7. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-1).

Cf. A001595, A111314, A362255.

LinearRecurrence[{2, -1, 0, 1, -1}, {1, 1, 1, 5, 9}, 44] (* or *)
With[{p = 3}, Nest[Append[#, #[[-1]] + #[[-p - 1]] + p] &, {1, 1, 1, 5}, 40] ]

A376108 Non-Leonardo numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Chai Wah Wu, Sep 10 2024

A299174 UNION A005408(A001690 - 1).

Cf. A001595, A299174, A001690.

def A376108(n):
    def f(x):
        a, b, c = 1, 1, n
        while True:
            if b > x: return c
            a, b = b, a+b+1
            c +=1
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m

cp's OEIS Frontend

A131411 Triangle read by rows: T(n,k) = Fibonacci(n) + Fibonacci(k) - 1.

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A145912 Prime Leonardo numbers.

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A117666 Expansion of (1-3*x+x^2)*(1-x-x^2)/((1+x+x^2)*(1-x+x^2)*(1-x)^2).

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A131400 A046854 + A065941 - I (Identity matrix).

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A175004 Interspersion related to the Wythoff Array.

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A258316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 or 0011.

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A355288 a(0)=1, a(1)=3, a(2)=7; thereafter a(n) = a(n-1) + a(n-2) + 1.

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A117666 Expansion of (1-3x+x^2)(1-x-x^2)/((1+x+x^2)(1-x+x^2)(1-x)^2).