A002487 -id:A002487 - cp's OEIS frontend

A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)x^n/n ), where xexp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

1, 1, 4, 3, 15, 12, 37, 25, 100, 75, 219, 144, 501, 357, 972, 615, 1995, 1380, 3665, 2285, 7052, 4767, 12255, 7488, 22305, 14817, 37524, 22707, 65775, 43068, 106837, 63769, 180436, 116667, 286251, 169584, 471173, 301589, 729404, 427815, 1169211, 741396, 1778545, 1037149

Offset: 0

Paul D. Hanna, Sep 20 2011

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...
where
log(A(x)) = x + 7*x^2/2 - 2*x^3/3 + 31*x^4/4 + x^5/5 - 14*x^6/6 + x^7/7 + 127*x^8/8 +...+ A195587(n)*x^n/n +...
Let C(x) be the odd bisection of g.f. A(x):
C(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 + 615*x^7 + 1380*x^8 + 2285*x^9 + 4767*x^10 + 7488*x^11 + 14817*x^12 +...+ A237650(n)*x^n +...
then C(x) equals the cube of an integer series:
C(x)^(1/3) = 1 + x + 3*x^2 + 2*x^3 + 9*x^4 + 7*x^5 + 17*x^6 + 10*x^7 + 41*x^8 + 31*x^9 + 75*x^10 + 44*x^11 + 150*x^12 +...+ A237651(n)*x^n +...
which equals A(x)/C(x^2)^(1/3).
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * (1+x^4+x^8)^6 * (1+x^8+x^16)^12 * (1+x^16+x^32)^24 *...* (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n) *...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A195587, A163658, A163659, A237650, A002487; variant: A237646.

{A163659(n)=if(n<1,0,if(n%3,1,-2)*sigma(2^valuation(n,2)))}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0,50,print1(a(n),", "))

/* G.f.: A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2: */
{a(n)=local(A=1+x); for(i=1, #binary(n), A=(1+x+x^2)*(1+x^2+x^4)*subst(A^2, x, x^2) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* G.f.: (1+x+x^2) * Product_{n>=0} (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n): */
{a(n)=local(A=1+x); A=(1+x+x^2)*prod(k=0, #binary(n), (1+x^(2*2^k)+x^(4*2^k)+x*O(x^n))^(3*2^k)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} A195587(n)*x^n/n ), where A195587(n) = A163659(n^2).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(3*2^n).
(3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
Bisections: let A(x) = B(x^2) + x*C(x^2), then
(4) B(x) = (1+x) * C(x).
(5) C(x) = (1+x+x^2)^3 * C(x^2)^2.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^3 = C(x) * C(x^2).
(8) A(x)^2 = C(x) / (1+x+x^2).
(9) A(x) = ( C(x)/A(x) - C(x^2)^2/A(x^2)^2 ) / (2*x).

Entry and formulas revised by Paul D. Hanna, May 04 2014

A244472 2nd-largest term in n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, 16114, 26073, 42187, 68260, 110447, 178707, 289154, 467861, 757015, 1224876, 1981891, 3206767, 5188658, 8395425, 13584083, 21979508, 35563591, 57543099

Offset: 1

N. J. A. Sloane, Jul 01 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A002487, A013655, A100545 (bisection).

Cf. A244473, A244474, A244475, A244476.

I:=[1, 2, 4, 7, 12]; [n le 5 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Wesley Ivan Hurt, Jul 10 2015

A244472 := proc(n)
    if n < 4 then
        op(n,[1,2,4]) ;
    else
        combinat[fibonacci](n+2)-combinat[fibonacci](n-3) ;
    end if;
end proc:
seq(A244472(n),n=1..50) ; # R. J. Mathar, Jul 05 2014

CoefficientList[Series[-(x^4 + x^3 + x^2 + x + 1)/(x^2 + x - 1), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 10 2015 *)
Join[{1, 2, 4}, LinearRecurrence[{1, 1}, {7, 12}, 50]] (* Vincenzo Librandi, Jul 11 2015 *)

Vec(-x*(x^4+x^3+x^2+x+1)/(x^2+x-1) + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = A013655(n-1), n>3.
a(n) = a(n-1)+a(n-2), n>5. - Colin Barker, Jul 10 2015
G.f.: -x*(x^4+x^3+x^2+x+1) / (x^2+x-1). - Colin Barker, Jul 10 2015

A244473 3rd-largest term in n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

1, 3, 5, 11, 18, 30, 49, 80, 129, 209, 338, 547, 885, 1432, 2317, 3749, 6066, 9815, 15881, 25696, 41577, 67273, 108850, 176123, 284973, 461096, 746069, 1207165, 1953234, 3160399, 5113633, 8274032, 13387665, 21661697, 35049362, 56711059, 91760421

Offset: 2

N. J. A. Sloane, Jul 01 2014

Colin Barker, Table of n, a(n) for n = 2..1000
Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A002487, A244472-A244476.

A244473 := proc(n)
    if n < 8 then
        op(n,[-1,1,3,5,11,18,30]) ;
    else
        combinat[fibonacci](n+1)+5*combinat[fibonacci](n-4) ;
    end if;
end proc:
seq(A244473(n),n=2..50) ; # R. J. Mathar, Jul 05 2014

Join[{1,3,5,11,18,30},LinearRecurrence[{1,1},{49,80},40]] (* Harvey P. Dale, Jan 13 2015 *)

Vec(-x^2*(x^7+x^6+x^5+2*x^4+3*x^3+x^2+2*x+1)/(x^2+x-1) + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = a(n-1)+a(n-2) for n>9. - Colin Barker, Jul 10 2015

G.f.: -x^2*(x^7+x^6+x^5+2*x^4+3*x^3+x^2+2*x+1) / (x^2+x-1). - Colin Barker, Jul 10 2015

A244476 6th-largest term in n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

2, 8, 15, 26, 45, 75, 121, 199, 322, 542, 877, 1427, 2309, 3739, 6050, 9790, 15841, 25632, 41473, 67105, 108578, 175683, 284261, 459944, 744205, 1204149, 1948354, 3152503, 5100857, 8253360

Offset: 4

N. J. A. Sloane, Jul 01 2014

Jennifer Lansing, Largest Values for the Stern Sequence, Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.5.

Cf. A002487, A244472, A244473, A244474, A244475.

from functools import reduce
from itertools import product
def A244476(n): return sorted(set(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if y else (x[0]+x[1],x[1]),k,(1,0))) for k in product((False,True),repeat=n)),reverse=True)[5] # Chai Wah Wu, Jun 19 2022

a(24)-a(33) from Chai Wah Wu, Jun 19 2022

A266405 Start with a(1) = 1, then always choose for a(n) the least unused number such that A002487(a(n)a(n-1)) = A002487(a(n)) A002487(a(n-1)), where A002487 is Stern-Brocot sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 14, 16, 11, 17, 13, 12, 18, 15, 26, 24, 23, 19, 32, 20, 28, 34, 22, 27, 33, 29, 31, 25, 45, 49, 36, 30, 52, 48, 35, 64, 21, 69, 42, 128, 37, 256, 38, 46, 66, 54, 41, 83, 82, 108, 44, 39, 88, 68, 56, 40, 55, 65, 47, 130, 59, 96, 51, 192, 70, 72, 60, 104, 71, 80, 57, 63, 61, 126, 98, 90, 50, 62, 58, 124, 100, 121, 127

Offset: 1

Antti Karttunen, Dec 29 2015

nonn

This is a permutation of natural numbers for the same reason that A266195 and A266351 are. If nothing else works for the value of next a(n), then at least the next unused power of 2 will save the sequence from dying, and will also immediately pick up as its succeeding pair the least term not used so far. This follows because A002487(2^m) = 1 and A002487(2^m * n) = A002487(n) for all n and m.

Still, it would be nice to know when 149 will appear in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..2016
Index entries for sequences that are permutations of the natural numbers

Inverse: A266406.
Cf. A002487.
Cf. A266195, A266351, A265405 (for sequences with similar definitions).

A284007 a(n) = a(a(n-A002487(n))) + a(n-a(n-A002487(n))) with a(1) = a(2) = 1, where A002487 = Stern-Brocot sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 10, 11, 11, 11, 12, 12, 14, 13, 14, 14, 15, 16, 15, 16, 16, 18, 18, 18, 19, 19, 20, 20, 22, 22, 24, 23, 25, 24, 25, 25, 25, 26, 26, 27, 27, 28, 29, 28, 30, 29, 30, 31, 30, 32, 34, 34, 33, 33, 34, 34, 36, 37, 36, 37, 38, 38, 40, 41, 39, 40, 43, 42, 44, 43, 43, 44, 44, 45, 45, 45, 50

Offset: 1

Antti Karttunen, Mar 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A002487, A004001, A283474, A284000, A284007, A284013.

a(1) = a(2) = 1, for n > 2, a(n) = a(a(A284013(n))) + a(n-a(A284013(n))).

A317838 a(n) = Sum_{d|n} A002487(d).

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 4, 7, 8, 6, 9, 6, 8, 10, 5, 6, 14, 8, 12, 14, 12, 8, 12, 11, 12, 15, 12, 8, 20, 6, 6, 14, 12, 16, 21, 12, 16, 18, 16, 12, 28, 14, 18, 26, 16, 10, 15, 13, 22, 20, 18, 14, 30, 20, 16, 20, 16, 12, 30, 10, 12, 24, 7, 16, 28, 12, 18, 24, 32, 14, 28, 16, 24, 35, 24, 26, 36, 14, 20, 29, 24, 20, 42, 30, 28, 28, 24

Offset: 1

Antti Karttunen, Aug 09 2018

nonn

Inverse Möbius transform of A002487, Stern's Diatomic sequence.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A317837, A317839, A317840, A317841, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317838(n) = sumdiv(n,d,A002487(d));

a(n) = Sum_{d|n} A002487(d).

a(n) = A317837(n) + A002487(n).

A331600 a(n) = A002487(A331595(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 4, 3, 1, 4, 1, 3, 4, 2, 1, 3, 2, 2, 3, 3, 1, 4, 1, 5, 4, 2, 4, 3, 1, 2, 4, 3, 1, 4, 1, 3, 7, 2, 1, 5, 2, 12, 4, 3, 1, 3, 8, 3, 4, 2, 1, 3, 1, 2, 7, 5, 8, 4, 1, 3, 4, 12, 1, 5, 1, 2, 4, 3, 4, 4, 1, 5, 3, 2, 1, 3, 8, 2, 4, 3, 1, 3, 8, 3, 4, 2, 8, 5, 1, 16, 7, 3, 1, 4, 1, 3, 18

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Stern's sequences

Cf. A002487, A122111, A241909, A331595, A331731.

Cf. also A323901, A323902, A323903, A331601.

Array[If[# == 1, 1, NestWhile[If[OddQ[#3], {#1, #1 + #2, #4}, {#1 + #2, #2, #4}] & @@ Append[#, Floor[#[[-1]]/2]] &, {1, 0, #}, #[[-1]] > 0 &][[2]] &@ Apply[GCD, {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]}] &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 25 2020, after JungHwan Min at A122111 *)

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));
A331600(n) = A002487(A331595(n));

a(n) = A002487(A331595(n)) = A002487(gcd(A122111(n), A241909(n))).

a(n) = A002487(A331731(n)).

A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8, 54

Offset: 0

Antti Karttunen, Feb 05 2020

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A002487(A163511(n))].
For all i, j:
  a(i) = a(j) => A331748(i) = A331748(j),
  a(i) = a(j) => A331749(i) = A331749(j).

Cf. A002487, A163511, A323901, A324400, A331742, A331744, A331748, A331749.
Differs from A331745 for the first time at n=77, where a(77) = 40, while A331745(77) = 24.
Differs from A103391(1+n) for the first time at n=191, where a(191) = 23, while A103391(192) = 97.

\\ Needs also code from A323901.
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux331743(n) = [A002487(n), A323901(n)];
v331743 = rgs_transform(vector(1+up_to, n, Aux331743(n-1)));
A331743(n) = v331743[1+n];

a(2^n) = 2 for all n >= 0.

A135510 Least positive number missing from row n of Stern's diatomic array (see A049456 or A002487).

Original entry on oeis.org

2, 3, 4, 6, 6, 14, 20, 28, 38, 54, 90, 150, 216, 350, 506, 876, 1230, 2034, 3160, 4470, 7764, 12190, 18816, 29952, 43800, 73968, 112602, 182210, 285780, 474558, 729432, 1194078, 1843110, 2990880, 4662450, 7608720, 11801580, 18489120, 29790300

Offset: 1

mc (da-da(AT)lycos.de), Feb 09 2008

nonn

The old definition was "Least numbers not generated by Eisenstein's algorithm: m=1 n=1, then insert between them m+n, at stage p=1. (E.g. next stage (p=2) of Eisenstein's algorithm would be m, m+m+n, m+n, m+n+n, n). The maximum of these symmetric row elements at stage p is fibonacci(p+2); but how to determine the first number not generated at stage p?"

Don Reble, C++ program for A135510 and A293160
G. Eisenstein, Über ein einfaches Mittel zur Auffindung der höheren Reciprocitätsgesetze und der mit ihnen zu verbindenden Ergänzungssätze, Journal für die reine und angewandte Mathematik, Volume 39 (1850), page 351ff.
M. A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

Cf. A049456, A002487, A293160.

A049456 := proc(n, k)
    option remember;
    if n =1 then
        if k >= 0 and k <=1 then
            1;
        else
            0 ;
        end if;
    elif type(k, 'even') then
        procname(n-1, k/2) ;
    else
        procname(n-1, (k+1)/2)+procname(n-1, (k-1)/2) ;
    end if;
end proc: # R. J. Mathar, Dec 12 2014
mex := proc(L)
        local k;
        for k from 1 do
                if not k in L then
                        return k;
                end if;
        end do:
end proc:
rho:=n->[seq(A049456(n,k),k=0..2^(n-1))];
[seq(mex(rho(n)),n=1..16)]; # N. J. A. Sloane, Oct 14 2017

(* T is A049456 *)
T[n_, k_] := T[n, k] = If[n == 1, If[k >= 0 && k <= 1, 1, 0], If[EvenQ[k], T[n-1, k/2], T[n-1, (k+1)/2] + T[n-1, (k-1)/2]]];
mex[L_] := Module[{k}, For[k = 1, True, k++, If[FreeQ[L, k], Return[k]]]];
rho[n_] := Table[T[n, k], {k, 0, 2^(n-1)}];
a[n_] := a[n] = mex[rho[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Aug 03 2023, after Maple code *)

Entry revised by N. J. A. Sloane, Oct 14 2017

a(29)-a(39) from Don Reble, Oct 16 2016

cp's OEIS Frontend

A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

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A244472 2nd-largest term in n-th row of Stern's diatomic triangle A002487.

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A244473 3rd-largest term in n-th row of Stern's diatomic triangle A002487.

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A244476 6th-largest term in n-th row of Stern's diatomic triangle A002487.

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A266405 Start with a(1) = 1, then always choose for a(n) the least unused number such that A002487(a(n)*a(n-1)) = A002487(a(n)) * A002487(a(n-1)), where A002487 is Stern-Brocot sequence.

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A284007 a(n) = a(a(n-A002487(n))) + a(n-a(n-A002487(n))) with a(1) = a(2) = 1, where A002487 = Stern-Brocot sequence.

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A317838 a(n) = Sum_{d|n} A002487(d).

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A331600 a(n) = A002487(A331595(n)).

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A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)x^n/n ), where xexp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

A266405 Start with a(1) = 1, then always choose for a(n) the least unused number such that A002487(a(n)a(n-1)) = A002487(a(n)) A002487(a(n-1)), where A002487 is Stern-Brocot sequence.