A004001 -id:A004001 - cp's OEIS frontend

A282894 Remainder when sum of first n terms of A004001 is divided by A004001(n).

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 6, 6, 6, 6, 6, 7, 9, 0, 0, 2, 5, 5, 8, 8, 8, 10, 10, 10, 10, 10, 9, 9, 10, 12, 15, 15, 18, 22, 3, 3, 7, 12, 12, 17, 17, 17, 21, 26, 26, 1, 1, 1, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 2, 0, 34, 35, 0, 2, 2, 4, 7, 11, 16, 16, 21, 27, 34, 34, 41, 2, 2, 9, 9, 9, 15, 22, 30, 30, 38, 47, 47, 2, 2, 2

Offset: 1

Altug Alkan, Feb 24 2017

			a(6) = 1 since Sum_{k=1..6} A004001(k) = 1 + 1 + 2 + 2 + 3 + 4 = 13 and remainder when 13 is divided by A004001(6) = 4 is 1.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Graph of A282894

Cf. A004001, A282891.

A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
A004001(1):= 1: A004001(2):= 1:
L:= ListTools[PartialSums](map(A004001, [$1..1000])):
seq(L[i] mod A004001(i), i=1..1000); # Robert Israel, Feb 26 2017

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; MapIndexed[Last@ QuotientRemainder[#1, a@ First@ #2] &, Accumulate@ Table[a@ n, {n, 96}]] (* Michael De Vlieger, Feb 24 2017, after Robert G. Wilson v at A004001 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); vector(#a, n, sum(k=1, n, a[k]) % a[n])

first(n)=my(v=vector(n),s); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); for(k=1,n, s+=v[k]; v[k]=s%v[k]); v \\ Charles R Greathouse IV, Feb 26 2017

a(n) = (Sum_{k=1..n} A004001(k)) mod A004001(n).

A283482 Positions of even terms in A004001.

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 13, 14, 15, 16, 18, 20, 21, 23, 24, 28, 29, 30, 31, 32, 34, 36, 39, 41, 42, 44, 45, 49, 52, 53, 54, 59, 60, 61, 62, 63, 64, 66, 68, 70, 71, 73, 75, 76, 78, 81, 84, 85, 86, 88, 91, 94, 95, 96, 98, 99, 103, 104, 105, 106, 108, 109, 113, 114, 115, 116, 122, 123, 124, 125, 126, 127, 128, 130, 132, 134, 137, 139, 141, 142, 144

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8264

Cf. A283481 (complement), A283471 (a subsequence).
Positions of zeros in A095901.
Cf. A004001.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Select[Range@ 144, EvenQ@ a[#] &] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

A283677 a(n) = lcm(b(b(n)), b(n-b(n)+1)) where b(n) = A004001(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 12, 12, 4, 4, 4, 4, 20, 5, 30, 35, 35, 42, 24, 24, 56, 56, 56, 8, 8, 8, 8, 8, 72, 9, 90, 99, 36, 36, 60, 130, 70, 70, 154, 165, 165, 60, 60, 60, 195, 208, 208, 112, 112, 112, 240, 240, 240, 240, 16, 16, 16, 16, 16, 16, 272, 17, 306, 323, 340, 357, 357, 126, 198, 414, 72, 72, 456, 475, 494

Offset: 1

Altug Alkan, Mar 14 2017

See the order of certain subsequences in scatterplot link.

			a(4) = lcm(A004001(A004001(4)), A004001(4-A004001(4)+1)) = lcm(1, 2) = 2.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A283677
Index entries for sequences related to lcm's

Cf. A004001, A080677.

Cf. also A283470, A283673.

b[1] = b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; Table[LCM[b[b[n]], b[n + 1 - b[n]]], {n, 1, 78}] (* Indranil Ghosh, Mar 14 2017 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); va = vector(1000, n, lcm(a[a[n]], a[n+1-a[n]]))

(define (A283677 n) (lcm (A004001 (A004001 n)) (A004001 (+ 1 (- n (A004001 n)))))) ;; (Code for A004001 given under that entry). - Antti Karttunen, Mar 18 2017

a(n) = lcm(A004001(A004001(n)), A004001(A080677(n))). - Antti Karttunen, Mar 18 2017

A088491 a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

Original entry on oeis.org

2, 3, 4, 5, 3, 7, 4, 9, 3, 11, 3, 13, 3, 15, 4, 17, 3, 19, 3, 21, 3, 23, 3, 25, 3, 27, 3, 29, 3, 31, 4, 33, 3, 35, 3, 37, 3, 39, 3, 41, 3, 43, 3, 45, 3, 47, 3, 49, 3, 51, 3, 53, 3, 55, 3, 57, 3, 59, 3, 61, 3, 63, 4, 65, 3, 67, 3, 69, 3, 71, 3, 73, 3, 75, 3, 77, 3, 79, 3, 81, 3, 83, 3, 85, 3, 87

Offset: 2

Roger L. Bagula, Nov 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A004001, A088493.

Conway[n_]:= Conway[n]= If[n<3, 1, Conway[Conway[n-1]] +Conway[n-Conway[n-1]]];
f[n_]:= f[n]= Product[Conway[i], {i,Floor[n/2]}];
a[n_]:= a[n]= Floor[n*f[n-1]/f[n]];
Table[a[n], {n, 2, 100}] (* modified by G. C. Greubel, Mar 27 2022 *)

@CachedFunction
def b(n): # A004001
    if (n<3): return 1
    else: return b(b(n-1)) + b(n-b(n-1))
def f(n): return product( b(j) for j in (1..(n//2)) )
def A088491(n): return (n*f(n-1)//f(n))
[A088491(n) for n in (2..100)] # G. C. Greubel, Mar 27 2022

a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

Edited by G. C. Greubel, Mar 27 2022

A120475 a(n) = Sum_{m=1..n} A000045(m)(A004001(m+1) - 2A004001(m) + A004001(m-1)).

Original entry on oeis.org

-1, 0, -2, 1, 1, -7, -7, 14, 14, 14, -75, 69, -164, -164, -164, 823, 823, 823, 823, -5942, 5004, 5004, -23653, 22715, -52310, -52310, 144108, -173703, -173703, -173703, -173703, 2004606, 2004606, 2004606, 2004606, 2004606, -22153211, 16934958, 16934958, 16934958, -148645183, 119269113

Offset: 1

Roger L. Bagula, Jul 07 2006

sign

Cf. A000045, A004001, A120474.

A000045 := proc(n) option remember ; combinat[fibonacci](n) ; end: A004001 := proc(n) option remember ; if n <= 0 then 0 ; elif n <= 2 then 1; else A004001(A004001(n-1))+A004001(n-A004001(n-1)) ; fi ; end: A120475 := proc(n) add( A000045(m)*(A004001(m+1)-2*A004001(m)+A004001(m-1)),m=1..n) ; end: seq(A120475(n),n=1..80) ; # R. J. Mathar, Sep 18 2007

Conway[0] = 0; Conway[1] = Conway[2] = 1; Conway[n_Integer?Positive] := Conway[n] = Conway[Conway[n - 1]] + Conway[n - Conway[n - 1]]; a[n_] := Fibonacci[n]*(Conway[n + 1] - 2*Conway[n] + Conway[n - 1]); Table[Sum[a[m], {m, 1, n}], {n, 1, 30}]

Edited by N. J. A. Sloane, Aug 14 2006

More terms from R. J. Mathar, Sep 18 2007

A135688 a(n) = A004001(n)*a(n-1) + a(n-2), for n > 2, with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 3, 7, 24, 103, 436, 1847, 9671, 59873, 428782, 3061347, 24919558, 202417811, 1644262046, 13356514179, 121852889657, 1231885410749, 13672592407896, 165302994305501, 1997308524073908, 26130313807266305, 367821701825802178

Offset: 1

Roger L. Bagula, Feb 19 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A004001, A174232.

HC[n_]:= HC[n]= If[n<3, Fibonacci[n], HC[HC[n-1]] +HC[n -HC[n-1]]]; (*A004001*)
a[n_] := a[n] = If[n<3, 1, HC[n]*a[n-1] + a[n-2]];
Table[a[n], {n, 40}]

@cached_function
def HC(n): # HC = A004001
    if (n<3): return fibonacci(n)
    else: return HC(HC(n-1)) +HC(n -HC(n-1))
@CachedFunction
def a(n): # A135688
    if (n<3): return 1
    else: return HC(n)*a(n-1) + a(n-2)
[a(n) for n in (1..40)] # G. C. Greubel, Nov 25 2021

a(n) = A004001(n)*a(n-1) + a(n-2), for n > 2, with a(1) = a(2) = 1.

Edited and corrected by Eric M. Schmidt, Dec 21 2014

A174232 a(n) = a(n-1) - (-1)^nn if (A004001(n) mod 3) = 1, otherwise a(n-1) + (-1)^nn.

Original entry on oeis.org

1, 0, -2, -5, -1, -6, -12, -5, -13, -22, -12, -1, -13, -26, -12, -27, -11, -28, -46, -65, -45, -66, -88, -111, -87, -112, -86, -113, -141, -112, -142, -111, -143, -176, -142, -107, -71, -108, -70, -31, 9, -32, 10, 53, 97, 52, 98, 51, 99, 148, 198, 147, 199, 146

Offset: 0

Roger L. Bagula, Mar 13 2010

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004001, A174218.

HC[n_]:= HC[n]= If[n<3, Fibonacci[n], HC[HC[n-1]] +HC[n -HC[n-1]]]; (*A004001*)
a[n_]:= a[n]= If[n<2, 1-n, If[Mod[HC[n], 3]==1, a[n-1] -(-1)^n*n, a[n-1] + (-1)^n*n]];
Table[a[n], {n,0,80}]

@CachedFunction
def HC(n): # HC = A004001
    if (n<3): return fibonacci(n)
    else: return HC(HC(n-1)) +HC(n -HC(n-1))
def A174232(n):
    if (n<2): return 1-n
    elif (HC(n)%3==1): return A174232(n-1) - (-1)^n*n
    else: return A174232(n-1) + (-1)^n*n
[A174232(n) for n in (0..80)] # G. C. Greubel, Nov 24 2021

a(n) = a(n-1) - (-1)^n*n if (A004001(n) mod 3) = 1, otherwise a(n-1) + (-1)^n*n, with a(0) = 1 and a(1) = 0.

Edited by G. C. Greubel, Nov 24 2021

A266349 a(n) = 1 + A053644(n) - A004001(n+1) = 1 + A072376(n) - A266348(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 8, 7, 6, 5, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 16, 15, 14, 13, 12, 12, 11, 10, 9, 9, 8, 7, 7, 6, 6, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 32, 31, 30, 29, 28, 27, 27, 26, 25, 24, 23, 23, 22, 21, 20, 20, 19, 18, 18, 17, 17, 17, 16, 15, 14, 14, 13, 12, 12, 11, 11, 11, 10

Offset: 1

Antti Karttunen, Jan 22 2016

nonn

Used in a recursive formula of A265754.

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A004001, A053644, A072376, A266348, A265754, A266640.

b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; Table[1 + 2^(Ceiling@ Log2[n + 1] - 1) - b[n + 1], {n, 96}] (* Michael De Vlieger, Jan 26 2016, after Robert G. Wilson v at A004001 *)

a(n) = 1 + A053644(n) - A004001(n+1).

a(n) = 1 + A072376(n) - A266348(n).

A283501 Remainder when sum of first n terms of A004001 is divided by 2*n.

Original entry on oeis.org

1, 2, 4, 6, 9, 1, 3, 5, 8, 12, 17, 22, 2, 6, 10, 14, 19, 25, 32, 0, 6, 13, 21, 29, 38, 47, 2, 10, 18, 26, 34, 42, 51, 61, 2, 12, 23, 34, 46, 59, 73, 3, 16, 30, 44, 59, 74, 89, 7, 22, 37, 53, 69, 85, 102, 7, 22, 37, 53, 69, 85, 101, 117, 5, 20, 36, 53, 71, 90, 110, 130, 7, 27

Offset: 1

Altug Alkan, Mar 09 2017

nonn

Sequence represents b(n, 2) where b(n, i) = (Sum_{k=1..n} A004001(k)) mod (n*i). See also A282891 and corresponding illustration in Links section.

			a(6) = 1 since Sum_{k=1..6} A004001(k) = 1 + 1 + 2 + 2 + 3 + 4 = 13 and remainder when 13 is divided by 12 is 1.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Illustration of Residue Classes Modulo 8

Cf. A004001, A282891.

A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
A004001(1):= 1: A004001(2):= 1:
L:= ListTools[PartialSums](map(A004001, [$1..1000])):
seq(L[i] mod (2*i), i=1..1000); # after Robert Israel at A282891

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[Mod[Total@ Array[a, n], 2 n], {n, 73}] (* Michael De Vlieger, Mar 13 2017, after Robert G. Wilson v at A004001 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); vector(#a, n, sum(k=1, n, a[k]) % (2*n))

a(n) = (Sum_{k=1..n} A004001(k)) mod (2*n).

A283655 a(n) = b(b(n+1)) - b(n-b(n)+1) where b(n) = A004001(n).

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Mar 13 2017

nonn

			a(1) = 0 since a(1) = A004001(A004001(2)) - A004001(1-A004001(1)+1) = 1 - 1 = 0.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Illustration of A283655

Cf. A004001, A004074, A233270, A249071.

b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; a[n_] := b[b[n + 1]] - b[n - b[n] + 1]; Array[a, 100] (* Robert G. Wilson v, Mar 13 2017 *)

a=vector(1001); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); va = vector(1000, n, a[a[n+1]]-a[n+1-a[n]])

cp's OEIS Frontend

A282894 Remainder when sum of first n terms of A004001 is divided by A004001(n).

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A283482 Positions of even terms in A004001.

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A283677 a(n) = lcm(b(b(n)), b(n-b(n)+1)) where b(n) = A004001(n).

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A088491 a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

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A120475 a(n) = Sum_{m=1..n} A000045(m)*(A004001(m+1) - 2*A004001(m) + A004001(m-1)).

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A135688 a(n) = A004001(n)*a(n-1) + a(n-2), for n > 2, with a(1) = a(2) = 1.

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A174232 a(n) = a(n-1) - (-1)^n*n if (A004001(n) mod 3) = 1, otherwise a(n-1) + (-1)^n*n.

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A266349 a(n) = 1 + A053644(n) - A004001(n+1) = 1 + A072376(n) - A266348(n).

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A120475 a(n) = Sum_{m=1..n} A000045(m)(A004001(m+1) - 2A004001(m) + A004001(m-1)).

A174232 a(n) = a(n-1) - (-1)^nn if (A004001(n) mod 3) = 1, otherwise a(n-1) + (-1)^nn.