A022308 -id:A022308 - cp's OEIS frontend

A122195 Numbers that are the sum of exactly 3 sets of Fibonacci numbers.

8, 11, 13, 14, 18, 19, 22, 23, 30, 31, 36, 38, 49, 51, 59, 62, 80, 83, 96, 101, 130, 135, 156, 164, 211, 219, 253, 266, 342, 355, 410, 431, 554, 575, 664, 698, 897, 931, 1075, 1130, 1452, 1507, 1740, 1829, 2350, 2439, 2816, 2960, 3803, 3947, 4557, 4790, 6154

Offset: 1

Ron Knott, Aug 25 2006, corrected Aug 29 2006

nonn

			8 is the sum of only 3 sets of Fibonacci numbers: {8}, {3,5} and {1,2,5};
11 is the sum of only {3,8}, {1,2,8}, {1,2,3,5}.

G. C. Greubel, Table of n, a(n) for n = 1..1000
M. Bicknell-Johnson & D. C. Fielder, The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, Fibonacci Quart. 37.1 (1999) pp. 47 ff.
Ron Knott, Sumthing about Fibonacci Numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,0,0,1,-1).

Cf. A000045, A000071, A013583, A000119, A122194.

a:=[11,13,14,18,19,22,23,30];; for n in [9..60] do a[n]:=a[n-4]+a[n-8]+1; od; Concatenation([8], a); # G. C. Greubel, Jul 13 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9) )); // G. C. Greubel, Jul 13 2019

# first N terms:
series((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(x^9-x^8+x^5-x^4-x+1),x,N+1);

CoefficientList[Series[(8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9), {x, 0, 60}], x] (* G. C. Greubel, Jul 13 2019 *)

my(x='x+O('x^60)); Vec((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8 -3*x^9)/(1-x-x^4+x^5-x^8+x^9)) \\ G. C. Greubel, Jul 13 2019

((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Jul 13 2019

G.f.: (8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1-x-x^4+x^5-x^8+x^9).
a(n) = a(n-4) + a(n-8) + 1.
a(0)=8, a(1)=11, a(2)=13, a(3)=18, then: a(4n) = A022318(n+3) = 2*A000045(n+5) + A000045(n+3) - 1, a(4n+1) = A022406(n+2) = 4*A000045(n+4) - 1, a(4n+2) = A022308(n+4) = 2*A000045(n+4) + A000045(n+6) - 1, a(4n+3) = 3*A000045(n+4) - 1, for n>=1.
a(n) = a(n-1) +a(n-4) -a(n-5) +a(n-8) -a(n-9). - G. C. Greubel, Jul 13 2019

A023554 Convolution of natural numbers >= 3 and (Fib(2), Fib(3), Fib(4), ...).

Original entry on oeis.org

3, 10, 22, 43, 78, 136, 231, 386, 638, 1047, 1710, 2784, 4523, 7338, 11894, 19267, 31198, 50504, 81743, 132290, 214078, 346415, 560542, 907008, 1467603, 2374666, 3842326, 6217051, 10059438, 16276552, 26336055, 42612674, 68948798, 111561543, 180510414

Offset: 1

Clark Kimberling

a(n) is the sum of row n in the triangle T(n,k) defined by: T(n,1) = T(n,n) = 2*n+1 for n>=1 and T(n,k) = 3*T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) for n>2, 2<=k<=n-1. - Lechoslaw Ratajczak, Nov 07 2020

Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 4jesleftforcycseq[ - .25'i + .5'k - .25i' - .5j' + .5k' - .75'ii' + .75'jj' - .25'kk' + .25'jk' - .5'ki' + .25'kj' + .25e ], apart from initial terms. 4jesrightforcycseq = A022308; 2jesforcycseq(n+2) = n+2; identity: jesleft + jesright = jes; vesforcycseq was set to the constant sequence = (-1,-1,-1,-1,-1...). (Dement)

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A213584.

F:=Fibonacci; List([1..40], n-> F(n+5)+2*F(n+3)-(2*n+9)); # G. C. Greubel, Jul 08 2019

F:=Fibonacci; [F(n+5)+2*F(n+3)-(2*n+9): n in [1..40]]; // G. C. Greubel, Jul 08 2019

Table[Fibonacci[n+5] + 2*Fibonacci[n+3] -2*n-9, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)

Vec(x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)) + O(x^60)) \\ Colin Barker, Feb 20 2017

vector(40, n, f=fibonacci; f(n+5)+2*f(n+3)-(2*n+9)) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+5)+2*f(n+3)-(2*n+9) for n in (1..40)] # G. C. Greubel, Jul 08 2019

G.f.: x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)).
2*(n+5) = A022308(n+4) - a(n+1) (conjectured). Note offset of A022308 is 0. - Creighton Dement, Feb 02 2005
From Colin Barker, Feb 20 2017: (Start)
a(n) = -7 + (2^(-1-n)*((1-t)^n*(-19+9*t) + (1+t)^n*(19+9*t)))/t - 2*(1+n) where t=sqrt(5).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4. (End)
a(n) = Fibonacci(n+5) + 2*Fibonacci(n+3) - (2*n + 9). - G. C. Greubel, Jul 08 2019
a(n) = a(n-1) + a(n-2) + 2*n + 3 for n>2. - Lechoslaw Ratajczak, Nov 07 2020

A022309 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=4.

Original entry on oeis.org

0, 4, 5, 10, 16, 27, 44, 72, 117, 190, 308, 499, 808, 1308, 2117, 3426, 5544, 8971, 14516, 23488, 38005, 61494, 99500, 160995, 260496, 421492, 681989, 1103482, 1785472, 2888955, 4674428, 7563384, 12237813, 19801198, 32039012, 51840211, 83879224, 135719436

Offset: 0

N. J. A. Sloane

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

Cf. A000045, A022095, A022308.

RecurrenceTable[{a[0]==0,a[1]==4,a[n]==a[n-1]+a[n-2]+1},a,{n,40}] (* or *) CoefficientList[Series[-x(-4+3x)/((x-1)(x^2+x-1)),{x,0,40}],x]  (* Harvey P. Dale, Apr 24 2011 *)

concat(0, Vec(x*(4-3*x) / ((1-x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Feb 20 2017

From R. J. Mathar, Apr 07 2011: (Start)
G.f. -x*(-4+3*x) / ( (x-1)*(x^2+x-1) ).
a(n) = A022095(n) - 1. (End)
From Colin Barker, Feb 20 2017: (Start)
a(n) = -1 + (2^(-1-n)*((1-t)^n*(-9+t) + (1+t)^n*(9+t)))/t, where t=sqrt(5).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 5*F(n) + F(n-1) - 1, where F = A000045. - Bruno Berselli, Feb 20 2017

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)

cp's OEIS Frontend

A122195 Numbers that are the sum of exactly 3 sets of Fibonacci numbers.

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A023554 Convolution of natural numbers >= 3 and (Fib(2), Fib(3), Fib(4), ...).

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A022309 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=4.

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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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