A002262 -id:A002262 - cp's OEIS frontend

A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 2, 3, 1, 0, 5, 6, 2, 3, 1, 0, 6, 8, 9, 2, 3, 1, 0, 7, 3, 8, 9, 2, 3, 1, 0, 8, 7, 5, 5, 6, 2, 3, 1, 0, 9, 10, 4, 4, 7, 8, 2, 3, 1, 0, 10, 12, 13, 6, 4, 6, 7, 2, 3, 1, 0, 11, 15, 12, 13, 5, 4, 6, 7, 2, 3, 1, 0, 12, 11, 17, 17, 18, 5, 4, 6, 7, 2, 3, 1, 0, 13, 5, 16, 16, 19, 20, 5, 4, 6, 7, 2, 3, 1, 0, 14, 13, 7, 18, 16, 18, 19, 5, 4, 6, 7, 2, 3, 1, 0

Offset: 0

Antti Karttunen, Feb 14 2016

			The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11,  5, 13, 16, 14, 18, 20, 23, 19
0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16,  7,  6, 15, 14, 21, 20, 25, 24
0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10,  8, 15,  7, 21, 25, 24, 26
0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11,  8,  9, 26, 27, 24, 25
0, 1, 3, 2, 8, 6, 4, 5, 20, 18,  9, 17,  7, 11, 10, 12, 28, 26, 33, 25
0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10,  9,  8, 13, 12, 27, 26, 35, 34
0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12,  8, 10, 13,  9, 27, 35, 38, 36
0, 1, 3, 2, 7, 6, 4, 5, 12, 13, 14, 15,  8,  9, 10, 11, 36, 37, 38, 39
0, 1, 3, 2, 7, 6, 4, 5, 14, 16, 11, 15,  8,  9, 12, 10, 38, 40, 35, 39
0, 1, 3, 2, 7, 6, 4, 5, 17, 16, 13, 12,  8,  9, 11, 10, 41, 40, 37, 36
0, 1, 3, 2, 7, 6, 4, 5, 17, 13, 12, 14,  8,  9, 11, 10, 41, 37, 36, 38
0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 12, 13,  8,  9, 11, 10, 38, 39, 36, 37
0, 1, 3, 2, 7, 6, 4, 5, 16, 14, 12, 13,  8,  9, 11, 10, 40, 38, 21, 37
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 38, 23, 22
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 23, 26, 24
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 24, 25, 26, 27

Antti Karttunen, Table of n, a(n) for n = 0..32895; the first 256 antidiagonals of array

Cf. A003188, A006068.
Inverses of these permutations can be found in table A268820.
Row 0: A001477, Row 1: A268718, Row 2: A268822, Row 3: A268824, Row 4: A268826, Row 5: A268828, Row 6: A268832, Row 7: A268934.
Rows converge towards A006068.

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def A(r, c): return c if r==0 else 0 if c==0 else 1 + A(r - 1, a278618(c) - 1)
for r in range(21): print([A(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268830 n) (A268830bi (A002262 n) (A025581 n))) ;; o=0: Square array of shifted powers of A268718.
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (- (A268718 col) 1))))))
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (A003188 (+ -1 (A006068 col))))))))

A032531 An inventory sequence: triangle read by rows, where T(n, k), 0 <= k <= n, records the number of k's thus far in the flattened sequence.

Original entry on oeis.org

0, 1, 1, 1, 3, 0, 2, 3, 1, 2, 2, 4, 3, 3, 1, 2, 5, 4, 4, 3, 1, 2, 6, 5, 5, 3, 3, 1, 2, 7, 6, 7, 3, 3, 2, 2, 2, 7, 9, 9, 3, 3, 2, 3, 0, 3, 7, 10, 13, 3, 3, 2, 4, 0, 2, 4, 7, 12, 15, 5, 4, 2, 5, 0, 2, 1, 5, 8, 14, 15, 6, 6, 4, 5, 1, 2, 1, 0, 6, 10, 15, 15, 7, 7, 5, 7, 1, 2, 2, 0, 1, 7, 12, 17

Offset: 0

Dmitri Papichev (Dmitri.Papichev(AT)iname.com)

Old name: a(n) = number of a(i) for 0<=iA002262(n).
This sequence is a variation of the Inventory sequence A342585. The same rules apply except that in this variation each row ends after k terms, where k is the current row count which starts at 1. The behavior up to the first 1 million terms is similar to A342585 but beyond that the most common terms do not increase, likely due to the rows being cut off after k terms thus numbers such as 1 and 2 no longer make regular appearances. Larger number terms do increase and overtake the leading early terms, and it appears this pattern repeats as n increases. See the linked images. - Scott R. Shannon, Sep 13 2021
The complexity of this sequence derives from the totals being updated during the calculation of each row. If each row recorded an inventory of only the earlier rows, we would get the much simpler A025581. - Peter Munn, May 06 2023

			Triangle begins:
  0;
  1, 1;
  1, 3, 0;
  2, 3, 1, 2;
  2, 4, 3, 3, 1;
  2, 5, 4, 4, 3, 1;
  2, 6, 5, 5, 3, 3, 1;
  2, 7, 6, 7, 3, 3, 2, 2;
  2, 7, 9, 9, 3, 3, 2, 3, 0;
  ...

Scott R. Shannon, Table of n, a(n) for n = 0..20000.
Scott R. Shannon, Image of the first 10^6 terms.
Scott R. Shannon, Image of the first 10^7 terms.
Scott R. Shannon, Image of the first 10^8 terms.
Index entries for sequences related to inventory sequences

Cf. A002262, A025581, A342585.

A002262 := proc(n)
    n - binomial(floor(1/2+sqrt(2*(1+n))), 2);
end proc:
A032531 := proc(n)
    option remember;
    local a,piv,i ;
    a := 0 ;
    piv := A002262(n) ;
    for i from 0 to n-1  do
        if procname(i) = piv then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A032531(n),n=0..100) ; # R. J. Mathar, May 08 2020

A002262[n_] :=  n - Binomial[Floor[1/2 + Sqrt[2*(1 + n)]], 2];
A032531[n_] := A032531[n] = Module[{a, piv, i}, a = 0; piv = A002262[n]; For[i = 0, i <= n-1, i++, If[A032531[i] == piv, a++]]; a];
Table[A032531[n], {n, 0, 100}] (* Jean-François Alcover, Mar 25 2024, after R. J. Mathar *)

from math import comb, isqrt
from collections import Counter
def idx(n): return n - comb((1+isqrt(8+8*n))//2, 2)
def aupton(nn):
    num, alst, inventory = 0, [0], Counter([0])
    for n in range(1, nn+1):
        c = inventory[idx(n)]
        alst.append(c)
        inventory[c] += 1
    return alst
print(aupton(93)) # Michael S. Branicky, May 07 2023

New name from Peter Munn, May 06 2023

A059389 Sums of two nonzero Fibonacci numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, 22, 23, 24, 26, 29, 34, 35, 36, 37, 39, 42, 47, 55, 56, 57, 58, 60, 63, 68, 76, 89, 90, 91, 92, 94, 97, 102, 110, 123, 144, 145, 146, 147, 149, 152, 157, 165, 178, 199, 233, 234, 235, 236, 238, 241, 246, 254, 267

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001

The sums of two distinct nonzero Fibonacci numbers is essentially the same sequence: 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, ... (only 2 is missing), since F(i) + F(i) = F(i-2) + F(i+1). - Colm Mulcahy, Mar 02 2008

To elaborate on Mulcahy's comment above: all terms of A078642 are in this sequence; those are numbers with two distinct representations as the sum of two Fibonacci numbers, which are, as Alekseyev proved, numbers of the form 2*F(i) greater than 2. - Alonso del Arte, Jul 07 2013

			10 is in the sequence because 10 = 2 + 8.
11 is in the sequence because 11 = 3 + 8.
12 is not in the sequence because no pair of Fibonacci numbers adds up to 12.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000045, A059390 (complement). Similar in nature to A048645. Essentially the same as A084176. Intersection with A049997 is A226857.

N:= 1000: # to get all terms <= N
R:= NULL:
for j from 1 do
  r:= combinat:-fibonacci(j);
  if r > N then break fi;
  R:= R, r;
end:
R:= {R}:
select(`<=`, {seq(seq(r+s, s=R),r=R)},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Feb 15 2015

max = 13; Select[Union[Total/@Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] &] (* Harvey P. Dale, Mar 13 2011 *)

list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([2])); if(fibonacci(t)>lim,t--); for(i=3,upper, t=fibonacci(i); for(j=2,i-1,tt=t+fibonacci(j); if(tt>lim, break, listput(v,tt)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2012

a(1) = 2 and for n >= 2 a(n) = F_(trinv(n-2)+2) + F_(n-((trinv(n-2)*(trinv(n-2)-1))/2)) where F_n is the n-th Fibonacci number, F_1 = 1 F_2 = 1 F_3 = 2 ... and the definition of trinv(n) is in A002262. - Noam Katz (noamkj(AT)hotmail.com), Feb 04 2001

log a(n) ~ sqrt(n log phi) where phi is the golden ratio A001622. There are (log x/log phi)^2 + O(log x) members of this sequence up to x. - Charles R Greathouse IV, Jul 24 2012

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

A061885 n + largest triangular number less than or equal to n.

Original entry on oeis.org

0, 2, 3, 6, 7, 8, 12, 13, 14, 15, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 42, 43, 44, 45, 46, 47, 48, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 74, 75, 76, 77, 78, 79, 80, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 132

Offset: 0

Henry Bottomley, May 12 2001

nonn

A253607(a(n)) > 0. - Reinhard Zumkeller, Jan 05 2015

Also possible values of floor(x*floor(x)) for real x < 1. - Jianing Song, Feb 16 2021

			a(9) = 9+6 = 15;
a(10) = 10+10 = 20;
a(11) = 11+10 = 21.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A060985.

Cf. A064801 (complement), A253607.

a061885 n = n + a057944 n  -- Reinhard Zumkeller, Feb 03 2012

from math import comb, isqrt
def A061885(n): return n+comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 09 2024

a(n) = n+A057944(n) = 2n-A002262(n) = n+[(sqrt(1+8*n)-1)/2]*[(sqrt(1+8*n)+1)/2]/2.

a(n) = A004201(n+1) - 1. - Franklin T. Adams-Watters, Jul 05 2009

A071654 Inverse permutation to A071653.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 5, 7, 19, 15, 4, 22, 16, 52, 14, 13, 20, 60, 43, 51, 41, 11, 18, 53, 178, 42, 153, 39, 10, 21, 47, 155, 177, 125, 151, 38, 12, 61, 56, 136, 154, 555, 123, 150, 40, 33, 55, 179, 164, 135, 479, 553, 122, 152, 117, 29, 17, 159, 557, 163, 417, 477, 552, 124

Offset: 0

Antti Karttunen, May 30 2002

A014137(n-1) = A071654(A072638(n)) for n>0 - Antti Karttunen, Jul 30 2012, based on Paul D. Hanna's similar observation in A071653.

Antti Karttunen, Rows n=0..66 of triangle, flattened
A. Karttunen, Alternative Catalan Orderings
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071653. A071672 gives the corresponding parenthesizations (from the term 1 onward) encoded as binary numbers, i.e. A071672(n) = A063171(A071654(n)) for n >= 1.

a(n) = A057163(A071652(n)). Cf. also A014486, A002262, A025581, A071651, A071652.

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

Original entry on oeis.org

0, 1, 2, 3, 12, 4, 7, 17, 18, 5, 6, 23, 40, 24, 8, 11, 31, 49, 50, 25, 9, 10, 30, 59, 84, 60, 32, 13, 16, 39, 71, 97, 98, 61, 33, 14, 15, 38, 70, 111, 144, 112, 72, 41, 19, 22, 48, 83, 127, 161, 162, 113, 73, 42, 20, 21, 47, 82, 126, 179, 220, 180, 128, 85, 51, 26, 29, 58

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072735, projections: A072740 & A072741, variant of the same theme: A072732. Used to construct the global arithmetic ranking scheme of plane binary trees presented in A072787/A072788. Cf. also A001477 and its projections A025581 & A002262.

(define (A072734 n) (packA072734 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072734 x y) (let ((x-y (- x y))) (cond ((negative? x-y) (packA001477 (+ (* 2 x) (modulo (1+ x-y) 2)) (+ (* 2 x) (floor->exact (/ (+ (- x-y) (modulo x-y 2)) 2))))) ((< x-y 3) (packA001477 (+ (* 2 y) x-y) (* 2 y))) (else (packA001477 (+ (* 2 y) (floor->exact (/ (1+ x-y) 2)) (modulo (1+ x-y) 2)) (+ (* 2 y) (modulo x-y 2)))))))

A079217 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 6, 2, 1, 0, 0, 1, 10, 0, 0, 0, 0, 0, 1, 11, 5, 0, 1, 0, 0, 0, 1, 18, 0, 2, 0, 0, 0, 0, 0, 1, 21, 11, 0, 0, 1, 0, 0, 0, 0, 1, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35, 26, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 68, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 69, 66, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Antti Karttunen Jan 03 2002

Index entries for sequences related to parenthesizing

The row sums equal to the left edge shifted left once = A057546 = first row of A079216 (the latter gives the Maple procedure PFixedByA057511).

Cf. A079218, A079219, A079220, A079221, A079222 and A003056 and A002262.

[seq(A079217(n),n=0..119)]; A079217 := n -> PFixedByA057511(A003056(n)+1,1, A002262(n)+1);

A079222 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the six-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 14, 9, 0, 1, 38, 42, 28, 2, 0, 1, 111, 124, 90, 0, 0, 6, 1, 332, 379, 285, 5, 0, 27, 0, 1, 1029, 1178, 914, 0, 0, 110, 0, 0, 1, 3232, 3742, 2955, 14, 1, 429, 0, 0, 0, 1, 10374, 12024, 9666, 0, 0, 1614, 0, 0, 0, 0, 1, 33679, 39200, 31853, 42, 0

Offset: 0

Antti Karttunen Jan 03 2002

Note: the counts given here are inclusive, i.e. T(n,d) includes also the counts A079218(n,d) and A079219(n,d).

The row sums equal to the left edge shifted left once = A079227 = sixth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079221 and A003056 & A002262.

[seq(A079222(n),n=0..119)]; A079222 := n -> PFixedByA057511(A003056(n)+1,6, A002262(n)+1);

A163233 Two-dimensional Binary Reflected Gray Code: a(i,j) = bits of binary expansion of A003188(i) interleaved with that of A003188(j).

Original entry on oeis.org

0, 1, 2, 5, 3, 10, 4, 7, 11, 8, 20, 6, 15, 9, 40, 21, 22, 14, 13, 41, 42, 17, 23, 30, 12, 45, 43, 34, 16, 19, 31, 28, 44, 47, 35, 32, 80, 18, 27, 29, 60, 46, 39, 33, 160, 81, 82, 26, 25, 61, 62, 38, 37, 161, 162, 85, 83, 90, 24, 57, 63, 54, 36, 165, 163, 170, 84, 87, 91

Offset: 0

Antti Karttunen, Jul 29 2009

The top left 8 X 8 corner of the array is
+0 +1 +5 +4 20 21 17 16
+2 +3 +7 +6 22 23 19 18
10 11 15 14 30 31 27 26
+8 +9 13 12 28 29 25 24
40 41 45 44 60 61 57 56
42 43 47 46 62 63 59 58
34 35 39 38 54 55 51 50
32 33 37 36 52 53 49 48
By taking the top left 2 X 2 corner, 2 X 4 rectangle ((0,1,5,4),(2,3,7,6)) or 4 X 4 corner one obtains Karnaugh map templates for 2, 3 or 4 variables respectively (although not the standard ones usually given in the textbooks).

Antti Karttunen, Table of n, a(n) for n = 0..2079
Wikipedia, Karnaugh map
Index entries for sequences that are permutations of the natural numbers

Inverse: A163234. a(n) = A057300(A163235(n)). Transpose: A163235. Row sums: A163242. Cf. A054238, A147995.

Table[Function[k, FromDigits[#, 2] &@ Apply[Function[{a, b}, Riffle @@ Map[PadLeft[#, Max[Length /@ {a, b}]] &, {a, b}]], Map[IntegerDigits[#, 2] &@ BitXor[#, Floor[#/2]] &, {k, j}]]][i - j], {i, 0, 11}, {j, i, 0, -1}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

def a000695(n):
    n=bin(n)[2:]
    x=len(n)
    return sum([int(n[i])*4**(x - 1 - i) for i in range(x)])
def a003188(n): return n^(n>>1)
def a(n, k): return a000695(a003188(n)) + 2*a000695(a003188(k))
for n in range(21): print([a(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 25 2017

(define (A163233bi x y) (+ (A000695 (A003188 x)) (* 2 (A000695 (A003188 y)))))
(define (A163233 n) (A163233bi (A025581 n) (A002262 n)))

a(x,y) = A000695(A003188(x)) + 2*A000695(A003188(y)).

A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 9, 5, 7, 27, 10, 12, 8, 28, 30, 11, 13, 15, 29, 31, 33, 18, 14, 16, 36, 32, 34, 54, 19, 21, 17, 37, 39, 35, 55, 57, 20, 22, 24, 38, 40, 42, 56, 58, 60, 81, 23, 25, 45, 41, 43, 63, 59, 61, 243, 82, 84, 26, 46, 48, 44, 64, 66, 62, 244, 246, 83, 85, 87, 47, 49

Offset: 0

Antti Karttunen, Jul 29 2009

			From _Kevin Ryde_, Oct 06 2020: (Start)
Array A(y,x) read by downwards antidiagonals, so 0, 1,3, 2,4,6, etc.
        x=0   1   2   3   4   5   6   7   8
      +--------------------------------------
  y=0 |   0,  1,  2,  9, 10, 11, 18, 19, 20,
    1 |   3,  4,  5, 12, 13, 14, 21, 22,
    2 |   6,  7,  8, 15, 16, 17, 24,
    3 |  27, 28, 29, 36, 37, 38,
    4 |  30, 31, 32, 39, 40,
    5 |  33, 34, 35, 42,
    6 |  54, 55, 56,
    7 |  57, 58,
    8 |  60,
(End)

Antti Karttunen, Table of n, a(n) for n = 0..3320
Index entries for sequences that are permutations of the natural numbers

Inverse: A163329. Transpose: A163330. Cf. A037314 (row y=0), A208665 (column x=0)

Cf. A054238 is an analogous sequence for binary. Cf. A007089, A163327, A163332, A163334.

A(y,x) = 3*fromdigits(digits(y,3),9) + fromdigits(digits(x,3),9); \\ Kevin Ryde, Oct 06 2020

(define (A163328 n) (+ (A037314 (A025581 n)) (* 3 (A037314 (A002262 n)))))

a(n) = A037314(A025581(n)) + 3*A037314(A002262(n))

a(n) = A163327(A163330(n)).

Edited by Charles R Greathouse IV, Nov 01 2009

cp's OEIS Frontend

A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Scheme

A032531 An inventory sequence: triangle read by rows, where T(n, k), 0 <= k <= n, records the number of k's thus far in the flattened sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A059389 Sums of two nonzero Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A061885 n + largest triangular number less than or equal to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Python

Formula

A071654 Inverse permutation to A071653.

Views

Author

Keywords

Comments

Links

Crossrefs

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

A079217 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple