A005169 -id:A005169 - cp's OEIS frontend

A226999 Inverse Euler transform of A005169 (fountains of coins).

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 35, 55, 93, 149, 248, 403, 671, 1098, 1827, 3013, 5013, 8313, 13859, 23063, 38534, 64341, 107715, 180355, 302565, 507784, 853507, 1435415, 2416941, 4072272, 6868062, 11590807, 19577555, 33088481, 55964327, 94712212

Offset: 1

R. J. Mathar, Jun 26 2013

nonn

If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Alois P. Heinz, Table of n, a(n) for n = 1..4178
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Cf. A005169, A005170.

max = 100;
A005169 = Series[1 - Fold[Function[1 - x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]&;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A005169 // Rest] (* Jean-François Alcover, Jan 06 2020 *)

a(n) ~ 1 / (n * r^n), where r = A347901 = 0.57614876914275660229786857371993878235472466311897446868515653431946822937499... - Vaclav Kotesovec, Oct 09 2019

A289080 a(0) = 1 and a(n) = A005169(n) - A005169(n-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 4, 6, 11, 19, 33, 57, 99, 172, 298, 518, 898, 1559, 2706, 4696, 8151, 14147, 24554, 42617, 73969, 128384, 222831, 386759, 671282, 1165118, 2022251, 3509944, 6092077, 10573790, 18352530, 31853801, 55287454, 95960372, 166554844, 289083043

Offset: 0

Seiichi Manyama, Sep 02 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..4179
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Column 2 of A288267 (except a(0)).

Cf. A005169.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, j), j=1..min(i+1, n)))
    end:
a:= n-> b(n, 0) -b(n-1, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 02 2017

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j], {j, 1, Min[i+1, n]}]];
a[n_] := b[n, 0] - b[n-1, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

G.f.: (1-x)/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(...)))))).

A167749 Riordan array (1,xf(x)) where f(x) is the g.f. of A005169 (fountains of coins).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 3, 3, 1, 0, 3, 6, 6, 4, 1, 0, 5, 11, 13, 10, 5, 1, 0, 9, 20, 27, 24, 15, 6, 1, 0, 15, 38, 54, 55, 40, 21, 7, 1, 0, 26, 70, 109, 120, 100, 62, 28, 8, 1, 0, 45, 129, 216, 258, 236, 168, 91, 36, 9, 1, 0, 78, 238, 423, 544, 540, 426, 266, 128, 45, 10, 1

Offset: 0

Paul Barry, Nov 10 2009

Row sums are A167750. Diagonal sums are A167751.

			Triangle begins
1,
0, 1,
0, 1, 1,
0, 1, 2, 1,
0, 2, 3, 3, 1,
0, 3, 6, 6, 4, 1,
0, 5, 11, 13, 10, 5, 1,
0, 9, 20, 27, 24, 15, 6, 1,
0, 15, 38, 54, 55, 40, 21, 7, 1,
0, 26, 70, 109, 120, 100, 62, 28, 8, 1,
0, 45, 129, 216, 258, 236, 168, 91, 36, 9, 1,
0, 78, 238, 423, 544, 540, 426, 266, 128, 45, 10, 1,
0, 135, 437, 824, 1127, 1205, 1035, 721, 402, 174, 55, 11, 1

G.f.: 1/(1-xy/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-... (continued fraction).

A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 13, 23, 35, 59, 93, 154, 248, 413, 671, 1111, 1827, 3036, 5013, 8348, 13859, 23122, 38534, 64434, 107715, 180509, 302565, 508032, 853507, 1435828, 2416941, 4072943, 6868062, 11591918, 19577555, 33090308, 55964327, 94715248, 160391045

Offset: 1

Ilya Gutkovskiy, Jun 11 2018

nonn

Inverse weigh transform of A005169.

			(1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^4)^2 * (1 + x^5)^2 * ... * (1 + x^n)^a(n) * ... = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).

Alois P. Heinz, Table of n, a(n) for n = 1..3000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A005169, A226999.

nn = 39; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

Product_{n>=1} (1 + x^n)^a(n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).

a(n) ~ 1 / (n * A347901^n). - Vaclav Kotesovec, Sep 18 2021

A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

Original entry on oeis.org

1, 1, 4, 5, 11, 22, 36, 69, 121, 221, 386, 686, 1210, 2122, 3734, 6517, 11408, 19903, 34714, 60485, 105312, 183272, 318758, 554262, 963361, 1674076, 2908426, 5052066, 8774386, 15237482, 26458718, 45939797, 79759442, 138468656, 240382216, 417289619, 724369536, 1257396992

Offset: 1

Paul D. Hanna, Jul 17 2013

nonn

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 11*x^5/5 + 22*x^6/6 +...
such L(x) = log(P(x)) - log(Q(x)) where
P(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...+ A224898(n)*x^n +...
Q(x) = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 - 2*x^15 - x^16 - 3*x^17 - 3*x^18 +...+ A039924(n)*x^n +...
log(P(x)) = -2*x^2/2 - 3*x^3/3 - 6*x^4/4 - 10*x^5/5 - 11*x^6/6 - 21*x^7/7 - 22*x^8/8 - 39*x^9/9 - 42*x^10/10 +...
log(Q(x)) = -x - 3*x^2/2 - 7*x^3/3 - 11*x^4/4 - 21*x^5/5 - 33*x^6/6 - 57*x^7/7 - 91*x^8/8 - 160*x^9/9 - 263*x^10/10 +...

Cf. A227543, A005169, A039924, A224898.

/* As the log of a continued fraction: */
{a(n)=local(A=x, CF=1+x); for(k=0, n, CF=1/(1-x^(n-k+1)*CF+x*O(x^n)); A=log(CF)); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=x, P=1+x, Q=1);
P=sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k));
Q=sum(m=0, sqrtint(n), (-1)^m*x^(m^2)/prod(k=1, m, 1-x^k));
A=log(P/(Q+x*O(x^n))); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: log( 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(1-...)))))) ), the logarithm of a continued fraction.
L.g.f.: log( P(x) / Q(x) ) where
P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity.

A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, 28, 26, 30, 30, 28, 32, 30, 32, 32, 32, 32, 40, 33, 31, 38, 35, 33, 39, 40, 37, 38, 40, 39

Offset: 1

Simon Plouffe and N. J. A. Sloane, May 20 1991

Rate of growth is not known. In fact it is not even known if this sequence is defined for all positive n.
Roman Pearce, Aug 29 2014, has computed that a(n) exists for n <= 10^10. - N. J. A. Sloane
a(n) exists for n <= 3*10^10. - M. Eric Carr, Jul 02 2023

			a(18) = 11 because a(17) is 10 and a(16) is 9, so we take a(18 - 10) + a(18 - 9) = a(8) + a(9) = 5 + 6 = 11.

B. W. Conolly, "Meta-Fibonacci sequences," in S. Vajda, editor, Fibonacci and Lucas Numbers and the Golden Section. Halstead Press, NY, 1989, pp. 127-138.
R. K. Guy, Unsolved Problems in Number Theory, Sect. E31.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics, Vol. 16 (2018), pp. 1490-1500.
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, 2017.
B. Balamohan, A. Kuznetsov, and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Thomas Bloom, Problem 422, Erdős Problems.
Paul Bourke, Hofstadter "Q" Series.
Paul Bourke, Hofstadter "Q" Series. [Cached copy, pdf only, with permission]
Paul Bourke, Listen to first 300 terms of this sequence. [Cached copy from the Hofstadter "Q" web page, with permission]
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See p. 794.
Benoit Cloitre, Plot of a(n)/n - 1/2 for n=1 up to 2^19.
J.-P. Davalan, Douglas Hofstadter's sequences.
Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
Nathan Fox, A New Approach to the Hofstadter Q-Recurrence, arXiv:1807.01365 [math.NT], 2018.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)].
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Nick Hobson, Python program for this sequence.
D. R. Hofstadter, Eta-Lore. [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences. [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991.
D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
D. R. Hofstadter, Plot of first 100 million terms.
Abraham Isgur, Mustazee Rahman, and Stephen Tanny, Solving non-homogeneous nested recursions using trees, Annals of Combinatorics 17.4 (2013): 695-710. See p. 695. - _N. J. A. Sloane_, Apr 16 2014
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505.
Piotr Miska, Bartosz Sobolewski, and Maciej Ulas, Binary sequences meet the Fibonacci sequence, arXiv:2412.11319 [math.NT], 2024. See p. 20.
K. Pinn, Order and chaos in Hofstadter's Q(n) sequence, Complexity, 4:3 (1999), 41-46.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
K. Pinn, A Chaotic Cousin Of Conway's Recursive Sequence, arXiv:cond-mat/9808031 [cond-mat.stat-mech], 1998.
N. J. A. Sloane and Brady Haran, Amazing Graphs III, Numberphile video (2019).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991.
A. M. M. Sharif Ullah, Surface Roughness Modeling Using Q-Sequence, Mathematical and Computational Applications, 2017.
Eric Weisstein's World of Mathematics, Hofstadter's Q-Sequence.
Wikipedia, Hofstadter sequence.
Pedro Zanetti, C++ code snippet for this sequence.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadter-type sequences

Cf. A004001, A005206, A005374, A005375, A005378.
Cf. A005379, A070867, A226222, A239913, A284019.
Cf. A081827 (first differences).
Cf. A226244, A226245 (record values and where they occur).
See A244477 for a different start.

#include 
#define LIM 20
int Qa[LIM];
int Q(int n){if (n==1 || n==2){return 1;} else{return Qa[n-Qa[n-1]]+Qa[n-Qa[n-2]];}}
int main(){int i;printf("n\tQ\n");for(i=1; iGonzalo Ciruelos, Aug 01 2013

a005185 n = a005185_list !! (n-1)
a005185_list = 1 : 1 : zipWith (+)
   (map a005185 $ zipWith (-) [3..] a005185_list)
   (map a005185 $ zipWith (-) [3..] $ tail a005185_list)
-- Reinhard Zumkeller, Jun 02 2013, Sep 15 2011

I:=[1,1]; [n le 2 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-2)): n in [1..90]]; // Vincenzo Librandi, Aug 08 2014

A005185 := proc(n) option remember;
    if n<=2 then 1
    elif n > procname(n-1) and n > procname(n-2) then
        RETURN(procname(n-procname(n-1))+procname(n-procname(n-2)));
    else
        ERROR(" died at n= ", n);
    fi; end proc;
# More generally, the following defines the Hofstadter-Huber sequence Q(r,s) - N. J. A. Sloane, Apr 15 2014
r:=1; s:=2;
a:=proc(n) option remember; global r,s;
if n <= s then  1
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi; fi; end;
[seq(a(n), n=1..100)];

a[1]=a[2]=1; a[n_]:= a[n]= a[n -a[n-1]] + a[n -a[n-2]]; Table[ a[n], {n,70}]

q:=proc(n) option remember; begin if n<=2 then 1 else q(n-q(n-1))+q(n-q(n-2)) end_if; end_proc: q(i)$i=1..100; // Zerinvary Lajos, Apr 03 2007

{a(n)= local(A); if(n<1, 0, A=vector(n,k,1); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A[n])} /* Michael Somos, Jul 16 2007 */

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n):
    if n < 3: return 1
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 26 2021

@CachedFunction
def a(n):
    if (n<3): return 1
    else: return a(n -a(n-1)) + a(n -a(n-2))
[a(n) for n in (1..70)] # G. C. Greubel, Feb 13 2020

(define q (lambda (n) (cond ( (eqv? n 0) 1) ( (eqv? n 1) 1) ( #t (+ (q (- n (q (- n 1)))) (q (- n (q (- n 2)))))))))

;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A005185 n) (if (<= n 2) 1 (+ (A005185 (- n (A005185 (- n 1)))) (A005185 (- n (A005185 (- n 2)))))))
;; Antti Karttunen, Mar 22 2017

A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42

Offset: 1

N. J. A. Sloane

On Jul 15 1988 during a colloquium talk at Bell Labs, John Conway stated that he could prove that a(n)/n -> 1/2 as n approached infinity, but that the proof was extremely difficult. He therefore offered $100 to someone who could find an n_0 such that for all n >= n_0, we have |a(n)/n - 1/2| < 0.05, and he offered $10,000 for the least such n_0. I took notes (a scan of my notebook pages appears below), plus the talk - like all Bell Labs Colloquia at that time - was recorded on video. John said afterwards that he meant to say $1000, but in fact he said $10,000. I was in the front row. The prize was claimed by Colin Mallows, who agreed not to cash the check. - N. J. A. Sloane, Oct 21 2015
a(n) - a(n-1) = 0 or 1 (see the D. Newman reference). - Emeric Deutsch, Jun 06 2005
a(A188163(n)) = n and a(m) < n for m < A188163(n). - Reinhard Zumkeller, Jun 03 2011
From Daniel Forgues, Oct 04 2019: (Start)
Conjectures:
  a(n) = n/2 iff n = 2^k, k >= 1.
  a(n) = 2^(k-1): k times, for n = 2^k - (k-1) to 2^k, k >= 1. (End)

			If n=4, 2^4=16, a(16-i) = 2^(4-1) = 8 for 0 <= i <= 4-1 = 3, hence a(16)=a(15)=a(14)=a(13)=8.

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
B. W. Conolly, Meta-Fibonacci sequences, in S. Vajda, editor, "Fibonacci and Lucas Numbers and the Golden Section", Halstead Press, NY, 1989, pp. 127-138.
R. K. Guy, Unsolved Problems Number Theory, Sect. E31.
D. R. Hofstadter, personal communication.
C. A. Pickover, Wonders of Numbers, "Cards, Frogs and Fractal sequences", Chapter 96, pp. 217-221, Oxford Univ. Press, NY, 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

T. D. Noe, Table of n, a(n) for n = 1..10000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan and O. Ozgur Aybar, On Families of Solutions for Meta-Fibonacci Recursions Related to Hofstadter-Conway $10000 Sequence, Slides of talk at presented in 5th International Interdisciplinary Chaos Symposium on Chaos and Complex Systems, May 9-12, 2019.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, Volume 2017, Article ID 2614163, 8 pages.
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.
Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, Plot of 100000 terms of a(n)-n/2
D. R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, Oct 10 2014; Part 1, Part 2.
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
N. J. A. Sloane, Scan of notebook pages with notes on John Conway's Colloquium talk on Jul 15 1988 [See Comments above]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991
Eric Weisstein's World of Mathematics, Hofstadter-Conway 10000-Dollar Sequence.
Eric Weisstein's World of Mathematics, Newman-Conway Sequence
Wikipedia, Hofstadter sequence
Index entries for sequences related to binary expansion of n
Index entries for Hofstadter-type sequences

Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences), A265332, A266341, A055748 (a chaotic cousin), A188163 (greedy inverse).
Cf. A004074 (A249071), A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Cf. also permutations A267111-A267112 and arrays A265901, A265903.

a004001 n = a004001_list !! (n-1)
a004001_list = 1 : 1 : h 3 1  {- memoization -}
  where h n x = x' : h (n + 1) x'
          where x' = a004001 x + a004001 (n - x)
-- Reinhard Zumkeller, Jun 03 2011

[n le 2 select 1 else Self(Self(n-1))+ Self(n-Self(n-1)):n in [1..75]]; // Marius A. Burtea, Aug 16 2019

A004001 := proc(n) option remember; if n<=2 then 1 else procname(procname(n-1)) +procname(n-procname(n-1)); fi; end;

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[ a[n], {n, 1, 75}] (* Robert G. Wilson v *)

a=vector(100);a[1]=a[2]=1;for(n=3,#a,a[n]=a[a[n-1]]+a[n-a[n-1]]);a \\ Charles R Greathouse IV, Jun 10 2011

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

def a004001(n):
    A = {1: 1, 2: 1}
    c = 1 #counter
    while n not in A.keys():
        if c not in A.keys():
            A[c] = A[A[c-1]] + A[c-A[c-1]]
        c += 1
    return A[n]
# Edward Minnix III, Nov 02 2015

@CachedFunction
def a(n): # a = A004001
    if n<3: return 1
    else: return a(a(n-1)) + a(n-a(n-1))
[a(n) for n in range(1,101)] # G. C. Greubel, Apr 25 2024

;; An implementation of memoization-macro definec can be found for example from: http://oeis.org/wiki/Memoization
(definec (A004001 n) (if (<= n 2) 1 (+ (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; Antti Karttunen, Oct 22 2014

Limit_{n->infinity} a(n)/n = 1/2 and as special cases, if n > 0, a(2^n-i) = 2^(n-1) for 0 <= i < = n-1; a(2^n+1) = 2^(n-1) + 1. - Benoit Cloitre, Aug 04 2002 [Corrected by Altug Alkan, Apr 03 2017]

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785

Offset: 0

N. J. A. Sloane and Herman P. Robinson

Expansion of Rogers-Ramanujan function G(x) in powers of x.
Same as number of partitions into distinct parts where the difference between successive parts is >= 2.
As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
Coefficients in expansion of permanent of infinite tridiagonal matrix:
  1 1 0 0 0 0 0 0 ...
  x 1 1 0 0 0 0 0 ...
  0 x^2 1 1 0 0 0 ...
  0 0 x^3 1 1 0 0 ...
  0 0 0 x^4 1 1 0 ...
  ................... - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006
Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006
a(n) = number of NW partitions of n, for n >= 1; see A237981.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[1](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are
  [ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 3]  [ 4 4 1 1 1 1 1 1 1 1 ]
  [ 4]  [ 4 4 4 1 1 1 1 ]
  [ 5]  [ 4 4 4 4 ]
  [ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ]
  [ 7]  [ 6 4 1 1 1 1 1 1 ]
  [ 8]  [ 6 4 4 1 1 ]
  [ 9]  [ 6 6 1 1 1 1 ]
  [10]  [ 6 6 4 ]
  [11]  [ 9 1 1 1 1 1 1 1 ]
  [12]  [ 9 4 1 1 1 ]
  [13]  [ 9 6 1 ]
  [14]  [ 11 1 1 1 1 1 ]
  [15]  [ 11 4 1 ]
  [16]  [ 14 1 1 ]
  [17]  [ 16 ]
The a(16)=17 partitions of 16 where successive parts differ by at least 2 are
  [ 1]  [ 7 5 3 1 ]
  [ 2]  [ 8 5 3 ]
  [ 3]  [ 8 6 2 ]
  [ 4]  [ 9 5 2 ]
  [ 5]  [ 9 6 1 ]
  [ 6]  [ 9 7 ]
  [ 7]  [ 10 4 2 ]
  [ 8]  [ 10 5 1 ]
  [ 9]  [ 10 6 ]
  [10]  [ 11 4 1 ]
  [11]  [ 11 5 ]
  [12]  [ 12 3 1 ]
  [13]  [ 12 4 ]
  [14]  [ 13 3 ]
  [15]  [ 14 2 ]
  [16]  [ 15 1 ]
  [17]  [ 16 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 14.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities.
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
Cf. A188216 (least part k occurs at least k times).
Cf. A047209, A203776, A237981.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Row sums of A268187.

a003114 = p a047209_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 05 2011

a003114 = p 1 where
   p _ 0 = 1
   p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Feb 19 2013

g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); # Emeric Deutsch, Feb 27 2006

CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-François Alcover, Apr 08 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 0, 24}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 60; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* Michael Somos, Oct 15 2008 */

G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014
G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008
Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, May 17 2015
Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 23 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.

Original entry on oeis.org

0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981

Offset: 0

N. J. A. Sloane

Conjecture: for n > 2, smallest prime divisor of a(n) > n. - Gerald McGarvey, Jun 19 2004
Rebuttal: This is not true; see Zivkovic link (Math. Comp. 68 (1999), pp. 403-409) has demonstrated that 3612703 divides a(n) for all n >= 3612702. - Paul Jobling, Oct 18 2004
Conjecture: For n>1, a(n) is the number of lattice paths from (0,0) to (n+1,0) that do not cross above y=x or below the x-axis using up-steps +(1,a) and down-steps +(1,-b) where a and b are positive integers. For example, a(3) = 5: [(1,1)(1,1)(1,1)(1,-3)], [(1,1)(1,-1)(1,3)(1,-3)], [(1,1)(1,-1)(1,2)(1,-2)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,-1)(1,-1)]. - Nicholas Ham, Aug 23 2015
Ham's claim is true for n=2. We proceed with a proof for n>2 by induction. On the j-th step, from (j-1,y) to (j,y'), there are j options for y': 0, 1, ..., y-1, y+1, ..., j. Thus there are n! possible paths from (0,0) to x=n that stay between y=0 and y=x. (Then the final step is determined.) However, because +(1,0) is not an allowable step, we cannot land on (n,0) on the n-th step. Therefore, the number of acceptable lattice paths is n! - a(n-1). - Danny Rorabaugh, Nov 30 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B10, pp. 152-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Richard K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
Richard K. Guy, Letter to N. J. A. Sloane, 1987
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: 103 and 130.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Eric Wegrzynowski, Séries de factorielles.
Eric Weisstein's World of Mathematics, Alternating Factorial and Factorial.
Miodrag Živković, The number of primes Sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), pp. 403-409.
Index entries for sequences related to factorial numbers.

Cf. A000142, A001272, A003422, A071828, A303697, A343187, A359808.

List([0..30],n->Sum([1..n],i->(-1)^(n-i)*Factorial(i))); # Muniru A Asiru, Jun 01 2018

a005165 n = a005165_list !! n
a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list
-- Reinhard Zumkeller, Jul 21 2013

A005165 := proc(n) local i; add((-1)^(n-i)*i!,i=1..n); end;

nn=25;With[{fctrls=Range[nn]!},Table[Abs[Total[Times@@@Partition[ Riffle[ Take[ fctrls,n],{1,-1}],2]]],{n,nn}]] (* Harvey P. Dale, Dec 10 2011 *)
a[0] = 0; a[n_] := n! - a[n - 1]; Array[a, 26, 0] (* Robert G. Wilson v, Aug 06 2012 *)
RecurrenceTable[{a[n] == n! - a[n - 1], a[0] == 0}, a, {n, 0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)
AlternatingFactorial[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
a[n_] = (-1)^n (Exp[1]((-1)^n Gamma[-1-n,1] Gamma[2+n] - ExpIntegralEi[-1]) - 1)
Table[a[n] // FullSimplify, {n, 0, 20}] (* Gerry Martens, May 22 2018 *)

a(n)=if(n<0,0,sum(k=0,n-1,(-1)^k*(n-k)!))

first(m)=vector(m,j,sum(i=0,j-1,((-1)^i)*(j-i)!)) \\ Anders Hellström, Aug 23 2015

a(n)=round((-1)^n*(exp(1)*(gamma(n+2)*incgam(-1-n,1)*(-1)^n +eint1(1))-1)) \\ Gerry Martens, May 22 2018

a = 0
f = 1
for n in range(1, 33):
    print(a, end=",")
    f *= n
    a = f - a
# Alex Ratushnyak, Aug 05 2012

a(0) = 0, a(n) = n! - a(n-1) for n > 0; also a(n) = n*a(n-2) + (n-1)*a(n-1) for n > 1. Sum_{n>=1} Pi^n/a(n) ~ 30.00005. - Gerald McGarvey, Jun 19 2004
E.g.f.: 1/(1-x) + exp(-x)*(e*(Ei(1,1)-Ei(1,1-x)) - 1). - Robert Israel, Dec 01 2015
a(n) = (-1)^n*(exp(1)*(gamma(n+2)*gamma(-1-n,1)*(-1)^n +Ei(1))-1). - Gerry Martens, May 22 2018
Sum_{n>=1} 1/a(n) = A343187. - Amiram Eldar, Jun 01 2023

cp's OEIS Frontend

A347901 Decimal expansion of a constant related to the asymptotics of A005169.

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A226999 Inverse Euler transform of A005169 (fountains of coins).

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A289080 a(0) = 1 and a(n) = A005169(n) - A005169(n-1).

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A167749 Riordan array (1,xf(x)) where f(x) is the g.f. of A005169 (fountains of coins).

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A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins).

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A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

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A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

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