A001610 -id:A001610 - cp's OEIS frontend

A100224 Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.

1, 1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 4, 6, 1, 0, 5, 5, 10, 10, 1, 0, 6, 6, 15, 18, 17, 1, 0, 7, 7, 21, 28, 35, 28, 1, 0, 8, 8, 28, 40, 60, 64, 46, 1, 0, 9, 9, 36, 54, 93, 117, 117, 75, 1, 0, 10, 10, 45, 70, 135, 190, 230, 210, 122

Offset: 0

Paul D. Hanna, Nov 28 2004

Diagonals are: T(n,n)=A001610(n-1) for n>0, with T(0,0)=1, T(n+1,n)=A006490(n), T(n+2,n)=A006491(n), T(n+3,n)=A006492(n), T(n+4,n)=A006493(n). The ratio of the generating functions of any two adjacent diagonals gives: (1-x)/(1-x-x^2) = 1+ x^2+ x^3+ 2*x^4+ 3*x^5+ 5*x^6+ 8*x^7+ 13*x^8+...

			Rows begin:
  [1],
  [1,0],
  [1,0,2],
  [1,0,3,3],
  [1,0,4,4,6],
  [1,0,5,5,10,10],
  [1,0,6,6,15,18,17],
  [1,0,7,7,21,28,35,28],
  [1,0,8,8,28,40,60,64,46],...
where row sums form 2^n-1 for n>0:
2^1-1 = 1+0 = 1
2^2-1 = 1+0+2 = 3
2^3-1 = 1+0+3+3 = 7
2^4-1 = 1+0+4+4+6 = 15
2^5-1 = 1+0+5+5+10+10 = 31.
The main diagonal forms A001610 = [0,2,3,6,10,17,...], where Sum_{n>=1} (A001610(n-1)/n)*x^n = log((1-x)/(1-x-x^2)).

Cf. A100223, A001610, A006490, A006491, A006492, A006493.

PARI
```
T(n,k)=if(n
				
```

G.f.: A(x, y)=(1-2*x*y+2*x^2*y^2)/((1-x*y)*(1-x*y-x^2*y^2-x*(1-x*y))).

A122194 Numbers that are the sum of exactly two sets of Fibonacci numbers.

Original entry on oeis.org

3, 5, 6, 9, 10, 15, 17, 25, 28, 41, 46, 67, 75, 109, 122, 177, 198, 287, 321, 465, 520, 753, 842, 1219, 1363, 1973, 2206, 3193, 3570, 5167, 5777, 8361, 9348, 13529, 15126, 21891, 24475, 35421, 39602, 57313, 64078, 92735, 103681, 150049, 167760, 242785

Offset: 1

Ron Knott, Aug 25 2006

			a(1)=3 as 3 is the sum of just 2 Fibonacci sets {3=Fibonacci(4)} and {1=Fibonacci(2), 2=Fibonacci(3)};
a(2)=5 as 5 is sum of Fibonacci sets {5} and {2,3} only.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.
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 (1,1,-1,1,-1).

Cf. A000032, A000045, A000071, A000119, A013583, A122195.

a:= function(n)
    if n mod 2=0 then return 2*Fibonacci(Int((n+6)/2)) -1;
    else return Lucas(1,-1, Int((n+5)/2))[2] -1;
    fi;
  end;
List([1..50], n-> a(n) ); # G. C. Greubel, Jul 13 2019

f:=Floor; [(n mod 2) eq 0 select 2*Fibonacci(f((n+6)/2))-1 else Lucas(f((n+5)/2))-1: n in [1..50]]; // G. C. Greubel, Jul 13 2019

fib:= combinat[fibonacci]:
lucas:=n->fib(n-1)+fib(n+1):
a:=n -> if n mod 2 = 0 then 2 *fib(n/2+3) -1 else lucas((n+1)/2+2)-1 fi:
seq(a(n), n=1..50);

LinearRecurrence[{1, 1, -1, 1, -1}, {3, 5, 6, 9, 10, 15}, 40] (* Vincenzo Librandi, Jul 25 2017 *)
Table[If[Mod[n,2]==0, 2*Fibonacci[(n+6)/2]-1, LucasL[(n+5)/2]-1], {n,50}] (* G. C. Greubel, Jul 13 2019 *)

vector(50, n, f=fibonacci; if(n%2==0, 2*f((n+6)/2)-1, f((n+7)/2) + f((n+3)/2)-1)) \\ G. C. Greubel, Jul 13 2019

def a(n):
    if (mod(n,2)==0): return 2*fibonacci((n+6)/2) - 1
    else: return lucas_number2((n+5)/2, 1,-1) -1
[a(n) for n in (1..50)] # G. C. Greubel, Jul 13 2019

a(2n-1) = A000032(n+2) - 1,
a(2n) = 2*A000045(n+3) - 1.
a(2n-1) = A001610(n+2), a(2n) = A001595(n+2).
a(1)=3, a(2)=5, a(3)=6, a(4)=9, a(n) = a(n-2) + a(n-4) + 1, n > 4.
G.f.: (3 + 2*x - 2*x^2 + x^3 - 3*x^4)/(1-x-x^2+x^3-x^4+x^5).
a(n) = A272632(n)-1. - R. J. Mathar, Jan 13 2023

A142586 Binomial transform of A014217.

Original entry on oeis.org

1, 2, 5, 14, 39, 107, 290, 779, 2079, 5522, 14615, 38579, 101634, 267347, 702455, 1844114, 4838079, 12686507, 33254210, 87141659, 228301839, 598026002, 1566300455, 4101923939, 10741568514, 28126975907, 73647747815, 192833044754, 504884940879, 1321888886747

Offset: 0

Paul Curtz, Sep 21 2008

The second term in the k-th iterated differences is 2, 3, 6, 10, 17, 28, 46, ... = A001610(k+1).

Colin Barker, Table of n, a(n) for n = 0..1000
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

Cf. A001610, A005248.

[1] cat [Lucas(2*n) - 2^(n-1): n in [1..30]]; // G. C. Greubel, Apr 13 2021

1,seq(combinat[fibonacci](2*n+1) +combinat[fibonacci](2*n-1) -2^(n-1), n = 1..30); # G. C. Greubel, Apr 13 2021

CoefficientList[Series[(1-3x+2x^2+x^3)/((1-3x+x^2)(1-2x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{5,-7,2},{2,5,14},30]] (* Harvey P. Dale, Aug 08 2011 *)

Vec((1-3*x+2*x^2+x^3)/((1-3*x+x^2)*(1-2*x)) + O(x^30)) \\ Colin Barker, Jun 05 2017

[1]+[lucas_number2(2*n,1,-1) -2^(n-1) for n in (1..30)] # G. C. Greubel, Apr 13 2021

From R. J. Mathar, Sep 22 2008: (Start)
G.f.: (1 - 3*x + 2*x^2 + x^3)/((1-3*x+x^2)*(1-2*x)).
a(n) = A005248(n) - 2^(n-1), n>0. (End)
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3); a(0)=1, a(1)=2, a(2)=5, a(3)=14. - Harvey P. Dale, Aug 08 2011
a(n) = (-2^(-1+n) + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n) for n > 0. - Colin Barker, Jun 05 2017

Edited and extended by R. J. Mathar, Sep 22 2008

A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

Original entry on oeis.org

1, 3, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 406, 432, 480, 576, 600, 648, 672, 720, 768, 864, 936, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2208, 2304

Offset: 1

Ctibor O. Zizka, May 22 2009

nonn

Numbers k such that Sum_{i=0..k} A000032(i)/(k+1) is an integer. - Robert G. Wilson v, May 25 2009

Why do the terms in A141767 so closely correspond to A160665? Except for k = 1, 3, 406, 44758, 341446, 1413286, 3170242, 4861698, 7912534, ..., k == 0 (mod 24). - Robert G. Wilson v, May 25 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A001610, A111058, A140826, A140971, A140972.

A000032 := proc(n) option remember ; if n <= 1 then 2-n; else procname(n-1)+procname(n-2) ; fi; end: A001610 := proc(n) add(A000032(i),i=0..n-1) ; end: for n from 1 to 3000 do if A001610(n) mod n = 0 then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

lst = {}; a = 2; b = 1; s = 3; n = 3; While[n < 2447, c = a + b; s = s + c; If[Mod[c, n] == 0, AppendTo[lst, n]]; a = b; b = c; n++ ]; lst (* Robert G. Wilson v, May 25 2009 *)

{k: k | A001610(k)}. - R. J. Mathar, May 25 2009

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

A203009 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(0)=2.

Original entry on oeis.org

1, 3, 11, 50, 374, 4282, 78924, 2322060, 110101476, 8413051008, 1038251025216, 207035781419520, 66749863269991104, 34803836775900988992, 29353783726459293724224, 40050488883338399323186560, 88407698594458813846355350656

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the A000032(j), j=0..n-1, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 2
1 3 2
1 6 11 6
1 10 35 50 24
1 17 105 295 374 168
1 28 292 1450 3619 4282 1848
1 46 796 6706 29719 69424 78924 33264
1 75 2130 29790 224193 931275 2092220 2322060 964656
This here is the first subdiagonal. The diagonal is A135407. The 2nd column is A001610, the 3rd A242300, the 4th A213807. (End)

Cf. A203010.

f[k_] := LucasL[k - 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203009 *)

A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 5, 1, 4, 1, 7, 15, 8, 1, 18, 1, 10, 21, 11, 1, 24, 5, 13, 9, 14, 1, 30, 1, 16, 11, 17, 35, 12, 1, 19, 39, 20, 1, 42, 1, 22, 9, 23, 1, 48, 7, 25, 17, 26, 1, 54, 55, 28, 19, 29, 1, 20, 1

Offset: 0

Paul Curtz, May 26 2014

nonn

The numerators are A189731(n).
B(0,n) = 0, 1, 1, 3/2, 2, 17/6, 4, 23/4, 25/3, 61/5, 18, 107/4, 40, 421/7, ...
is a super autosequence as defined in A242563.
The positive integers in B(0,n) give A064723(n). Corresponding rank: A006093(n+1). B(0,n) is linked to the primes A000040.
Divisor of B(0,n), n > 0: 1, 1, 1, 2, 2, 4, 5, ... = A172128(n+1).
Common (LCM) denominators for the antidiagonals: 1, 1, 1, 2, 2, 6, 6, 12, 12, ... = A139550(n+1)?.
1 = 1
1/2 +  3/2 = 2
1/3 +  5/6 + 17/6 = 4
1/4 + 7/12 +  7/4 + 23/4 = 25/3
etc.
The positive terms of the first bisection are the sum of the corresponding antidiagonal terms upon the 0's.
0 followed by A001610(n), i.e., 0, 0, 2, 3, 6, 10, 17, ... is an autosequence of the second kind.

Cf. A000204, A175386, A001610, A189731, A139550, A242563.

Table[Denominator[(LucasL[n+1]-1)/(n+1)], {n, 0, 100}] (* Artur Jasinski, Nov 06 2022 *)

a(2n+1) = A175386(n).
a(n) = denominator(A001610(n)/(n+1)). [edited by Michel Marcus, Nov 14 2022]
a(n) = denominator((A000204(n+1) - 1)/(n+1)). - Artur Jasinski, Nov 06 2022

a(24)-a(60) from Jean-François Alcover, May 26 2014

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 26, 84, 950, 6000, 62522, 556116, 6259598, 69319848, 874356338, 11384093196, 161462123894, 2397736692144, 37994808171962, 631767062124564, 11088109048500158, 203828700127054008, 3928762035148317314, 79079452776283889820, 1661265965479375937030, 36332908076071038467520, 826376466514358722894154

Offset: 0

Gus Wiseman, Feb 14 2019

nonn

			The a(4) = 2 ordered set partitions are: {{1,3},{2,4}}, {{2,4},{1,3}}.

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A000670, A001610, A032032 (adjacencies allowed), A052841 (singletons allowed), A124323, A169985, A306417, A324011 (orderless case), A324012, A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[stn]!,{stn,Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]}],{n,0,10}]

a(12)-a(26) from Alois P. Heinz, Feb 14 2019

A306418 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} requiring k steps of removing singletons and cyclical adjacency initiators until reaching a fixed point, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 3, 0, 1, 2, 12, 0, 0, 0, 12, 35, 5, 0, 0, 5, 56, 100, 42, 0, 0, 0, 14, 282, 343, 231, 7, 0, 0, 0, 66, 1406, 1476, 1088, 104, 0, 0, 0, 0, 307, 7592, 7383, 4929, 909, 27, 0, 0, 0, 0, 1554, 44227, 40514, 22950, 6240, 470, 20, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 14 2019

See Callan's article for details on this transformation (SeparateIS).

			Triangle begins:
    1
    0    1
    0    2    0
    0    2    3    0
    1    2   12    0    0
    0   12   35    5    0    0
    5   56  100   42    0    0    0
   14  282  343  231    7    0    0    0
   66 1406 1476 1088  104    0    0    0    0
  307 7592 7383 4929  909   27    0    0    0    0

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Row sums are A000110. First column is A324011.

Cf. A000126, A000296, A001610, A306416, A306417, A324012.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
qbj[stn_]:=With[{ini=Join@@Table[Select[s,If[#==Max@@Max@@@stn,MemberQ[s,First[Union@@stn]],MemberQ[s,(Union@@stn)[[Position[Union@@stn,#][[1,1]]+1]]]]&],{s,stn}],sng=Join@@Select[stn,Length[#]==1&]},DeleteCases[Table[Complement[s,Union[sng,ini]],{s,stn}],{}]];
Table[Length[Select[sps[Range[n]],Length[FixedPointList[qbj,#]]-2==k&]],{n,0,8},{k,0,n}]

A146481 Decimal expansion of Product_{n>=2} (1 - 1/(n*(n-1))).

Original entry on oeis.org

2, 9, 6, 6, 7, 5, 1, 3, 4, 7, 4, 3, 5, 9, 1, 0, 3, 4, 5, 7, 0, 1, 5, 5, 0, 2, 0, 2, 1, 9, 1, 4, 2, 8, 6, 4, 8, 6, 4, 8, 3, 1, 5, 1, 9, 1, 7, 8, 9, 4, 7, 8, 9, 0, 8, 1, 6, 7, 3, 5, 7, 3, 3, 1, 6, 5, 9, 0, 6, 1, 6, 2, 9, 1, 5, 1, 9, 6, 0, 8, 8, 8, 3, 6, 6, 7, 4, 8, 1, 6, 4, 0, 2, 1, 2, 6, 2, 2, 1, 4, 5, 4, 1, 7, 7

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant A005596 and the equivalent almost-prime products.

			0.2966751347435910345... = (1 - 1/2)*(1 - 1/6)*(1 - 1/12)*(1 - 1/20)*...

M. Chamberland, A. Straub, On gamma constants and infinite products, arXiv:1309.3455
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], first line Table 3.

Cf. A005596.

phi := (1+sqrt(5))/2; evalf(-sin(Pi*phi)/Pi) ; # R. J. Mathar, Feb 20 2009

RealDigits[-Cos[Pi*Sqrt[5]/2]/Pi, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r=1.
s*Sum_{j=1..floor(s/2)} binomial(s-j-1, j-1)/j = A001610(s-1).
Equals 1/Product_{k=1..2} Gamma(1-x_k) = -sin(A094886)/A000796, where x_k are the 2 roots of the polynomial x*(x+1)-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 11 2013

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.

cp's OEIS Frontend

A100224 Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.

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A122194 Numbers that are the sum of exactly two sets of Fibonacci numbers.

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GAP

Magma

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A142586 Binomial transform of A014217.

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Magma

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A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

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Maple

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A203009 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(0)=2.

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Mathematica

A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

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Mathematica

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A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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Mathematica