A002487 -id:A002487 - cp's OEIS frontend

A174868 Partial sums of Stern's diatomic series A002487.

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset: 0

Jonathan Vos Post, Dec 01 2010

After the initial 0, identical to A007729.

			a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1)),initial=0)
A174868_list = list(islice(A174868_gen(),30)) # Chai Wah Wu, May 07 2023

a(n) = Sum_{i=0..n} A002487(i).
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017
a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018
a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)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).

Original entry on oeis.org

1, 1, 8, 7, 63, 56, 329, 273, 1736, 1463, 7511, 6048, 32585, 26537, 124440, 97903, 475287, 377384, 1658881, 1281497, 5783960, 4502463, 18825023, 14322560, 61171649, 46849089, 188181672, 141332583, 577889023, 436556440, 1696298665, 1259742225, 4970284200, 3710541975, 14019036535, 10308494560

Offset: 0

Paul D. Hanna, May 03 2014

nonn

Compare to the g.f. of A195586.

			G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +...
where
log(A(x)) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A237649(n)*x^n/n +...
Bisections: let A(x) = B(x^2) + x*C(x^2), then:
B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +...
C(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 + 377384*x^8 + 1281497*x^9 + 4502463*x^10 +...+ A237647(n)*x^n +...
Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series:
C(x)^(1/7) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 + 379*x^9 + 1451*x^10 + 1072*x^11 + 5210*x^12 +...+ A237648(n)*x^n +...
Also, C(x) / (1+x+x^2)^3 = A(x)^4:
A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +...
Further, C(x)*C(x^2)^3 = A(x)^7:
A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +...
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^7 * (1+x^4+x^8)^28 * (1+x^8+x^16)^112 * (1+x^16+x^32)^448 *...* (1 + x^(2*2^n) + x^(4*2^n))^(7*4^n) *...

Cf. A237647, A237648, A237649, A195586.

{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^3)*x^k/k)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^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)^7 * C(x^2)^4.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^7 = C(x) * C(x^2)^3.
(8) A(x)^4 = C(x) / (1+x+x^2)^3.
(9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5).

A237649 a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 8191, 1, -30, 1, 127, -2, 15, 1, -2046, 1, 15, -2, 127, 1, -30, 1, 65535, -2, 15, 1, -254, 1, 15, -2, 1023, 1, -30, 1, 127, -2, 15, 1, -16382, 1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 524287, 1, -30, 1, 127

Offset: 1

Paul D. Hanna, May 03 2014

Multiplicative because A163659 is. - Andrew Howroyd, Jul 27 2018

			L.g.f.: L(x) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A163659(n^3)*x^n/n +...
where
exp(L(x)) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 +...+ A237646(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A237646, A163659.

{A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=A163659(n^3)}
for(n=1,64,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n),A);A=log(1+X+X^2) + sum(k=0,#binary(n),7*4^k*log(1 + X^(2*2^k) + X^(4*2^k)));n*polcoeff(A,n)}
for(n=1,64,print1(a(n),", "))

L.g.f.: log(1+x+x^2) + Sum_{n>=0} 7*4^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n.

G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 14*8^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)).

A244474 4th-largest term in n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

2, 4, 10, 17, 29, 47, 79, 128, 208, 337, 546, 883, 1429, 2312, 3741, 6053, 9794, 15847, 25641, 41488, 67129, 108617

Offset: 3

N. J. A. Sloane, Jul 01 2014

Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.

Cf. A002487, A244472, A244473, A244475, A244476.

A002487 := proc(n,k)
    option remember;
    if k =0 then
        1;
    elif k = 2^n-1 then
        n+1 ;
    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:
A244474 := proc(n)
    {seq(A002487(n,k),k=0..2^n-1)} ;
    sort(%) ;
    op(-4,%) ;
end proc:
for n from 3 do
    print(A244474(n)) ;
od: # R. J. Mathar, Oct 25 2014

s[n_] := s[n] = Switch[n, 0, 0, 1, 1, _, If[EvenQ[n], s[n/2], s[(n - 1)/2] + s[(n - 1)/2 + 1]]];
T = Table[s[n], {n, 0, 2^25}] // Flatten // SplitBy[#, If[# == 1, 1, 0]&]& // DeleteCases[#, {1}]&;
Union[#][[-4]]& /@ T[[5 ;;]] (* Jean-François Alcover, Mar 12 2023 *)

from itertools import product
from functools import reduce
def A244474(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)[3] # Chai Wah Wu, Jun 20 2022

G.f.: (-2-2*x-4*x^2-3*x^3-2*x^4-x^5-3*x^6-2*x^7-x^8-x^9-x^10)/(-1+x+x^2) (conjectured) - Jean-François Alcover, Mar 12 2023

a(24) from Jean-François Alcover, Mar 12 2023

A244475 5th-largest term in the n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

1, 3, 9, 16, 27, 46, 76, 123, 207, 335, 545, 882, 1428, 2311, 3740, 6051, 9791, 15842, 25633, 41475, 67108, 108583, 175691, 284274, 459965, 744239, 1204204, 1948443, 3152647, 5101090, 8253737

Offset: 3

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, A244476.

A002487 := proc(n,k)
    option remember;
    if k =0 then
        1;
    elif k = 2^n-1 then
        n+1 ;
    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:
A244475 := proc(n)
    {seq(A002487(n,k),k=0..2^n-1)} ;
    sort(%) ;
    op(-5,%) ;
end proc:
for n from 3 do
    print(A244475(n)) ;
od: # R. J. Mathar, Oct 25 2014

s[n_, k_] := s[n, k] = Which[k == 0, 1, k == 2^n-1, n+1, EvenQ[k], s[n-1, k/2], True, s[n-1, (k-1)/2] + s[n-1, (k+1)/2]];
row[n_] := Table[s[n, k], {k, 0, 2^n-1}];
a[n_] := If[n == 3, 1, Union[row[n]][[-5]]];
Table[Print[n, " ", a[n]]; a[n], {n, 3, 23}] (* Jean-François Alcover, Mar 13 2023, after R. J. Mathar *)

from itertools import product
from functools import reduce
def A244475(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)[4] # Chai Wah Wu, Jun 19 2022

Conjectured g.f.: -x^3*(x^14+ x^13+ x^12+ 2*x^11 +3*x^10 +5*x^9 +8*x^8 +x^7 +3*x^6 +3*x^5 +2*x^4 +4*x^3 +5*x^2 +2*x +1) / (x^2+x-1). - Alois P. Heinz, Jun 20 2022

a(24)-a(25) from Alois P. Heinz, Jun 19 2022

a(26)-a(33) from Chai Wah Wu, Jun 20 2022

A283474 a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(n-A002487(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 14, 20, 40, 48, 68, 82, 150, 190, 272, 354, 708, 858, 1130, 1280, 2138, 2328, 3186, 3894, 7080, 8210, 10538, 11818, 20028, 23214, 33752, 44290, 88580, 100398, 123612, 134150, 222730, 233268, 277558, 300772, 534040, 567792, 691404, 725156, 1025928, 1126326, 1427098, 1704656, 3131754, 3665794

Offset: 0

Antti Karttunen, Mar 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000045, A002487, A078510, A284007, A284013.

Cf. A283479 (first differences).

a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(A284013(n)).

A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 3, 1, 3, 3, 3, 2, 3, 3, 3, 1, 5, 4, 7, 3, 7, 5, 7, 2, 7, 5, 7, 3, 7, 4, 5, 1, 5, 5, 5, 4, 7, 7, 7, 3, 11, 8, 13, 5, 7, 7, 7, 2, 7, 7, 7, 5, 13, 8, 11, 3, 7, 7, 7, 4, 5, 5, 5, 1, 7, 6, 7, 5, 13, 9, 13, 4, 15, 11, 15, 7, 15, 10, 11, 3, 11, 11, 11, 8, 13, 13, 13, 5, 13, 12, 15, 7, 15, 9, 11, 2, 11, 9, 15, 7, 15, 12, 13, 5, 13, 13, 13, 8

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A002487, A283986.

Cf. A002487, A003986, A283977, A283978.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, a[n/2], BitOr[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 108}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, n, if(n%2, bitor(A(n\2), A((n + 1)/2)), A(n\2)));
for(n=0, 101, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283976 n) (if (even? n) (A002487 n) (A003986bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A003986bi implements bitwise-OR (A003986).

a(2n) = A002487(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).
a(n) = A283977(n) + A283978(n).
a(n) = A002487(n) - A283978(n).

A283977 a(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 2, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 5, 4, 7, 3, 6, 5, 7, 2, 7, 5, 6, 3, 7, 4, 5, 1, 4, 5, 1, 4, 3, 7, 4, 3, 11, 8, 13, 5, 2, 7, 5, 2, 5, 7, 2, 5, 13, 8, 11, 3, 4, 7, 3, 4, 1, 5, 4, 1, 7, 6, 3, 5, 12, 9, 13, 4, 15, 11, 12, 7, 13, 10, 9, 3, 8, 11, 3, 8, 5, 13, 8, 5, 9, 12, 11, 7, 14, 9, 11, 2, 11, 9, 14, 7, 11, 12, 9, 5, 8, 13, 5, 8, 3, 11, 8, 3

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A002487, A283987.

Cf. A003987, A283976, A283978.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, a[n/2], BitXor[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 112}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, n, if(n%2, bitxor(A(n\2), A((n + 1)/2)), A(n\2)));
for(n=0, 120, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283977 n) (if (even? n) (A002487 n) (A003987bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(2n) = A002487(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).
a(n) = A283976(n) - A283978(n).
a(n) = A002487(n) - 2*A283978(n).

A283978 a(2n) = 0, a(2n+1) = A002487(n) AND A002487(n+1), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 4, 0, 3, 0, 0, 0, 0, 0, 5, 0, 2, 0, 2, 0, 5, 0, 0, 0, 0, 0, 3, 0, 4, 0, 4, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 2, 0, 2, 0, 3, 0, 8, 0, 8, 0, 5, 0, 4, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 4, 0, 5, 0, 8, 0, 8, 0, 3, 0, 2, 0, 2, 0, 3, 0, 0, 0

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A000004, A283988.

Cf. A002487, A004198, A283976, A283977.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, 0, BitAnd[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 120}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, 0, if(n%2, bitand(A(n\2), A((n + 1)/2)), 0));
for(n=0, 120, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283978 n) (if (even? n) 0 (A004198bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(2n) = 0, a(2n+1) = A002487(n) AND A002487(n+1), where AND is bitwise-and (A004198).
a(n) = A283976(n) - A283977(n).
a(n) = A002487(n) - A283976(n) = (A002487(n) - A283977(n))/2.

A284460 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A245327/A245328 into the enumeration system A002487/A002487' (Calkin-Wilf), and A020651/A020650 (Yu-Ting inverted) into A162911/A162912(Drib).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 7, 14, 13, 8, 11, 10, 9, 12, 15, 30, 29, 24, 27, 18, 17, 20, 23, 22, 21, 16, 19, 26, 25, 28, 31, 62, 61, 56, 59, 50, 49, 52, 55, 38, 37, 32, 35, 42, 41, 44, 47, 46, 45, 40, 43, 34, 33, 36, 39, 54, 53, 48, 51, 58, 57, 60, 63

Offset: 1

Yosu Yurramendi, Mar 28 2017

nonn

The inverse permutation is A284459.

Yosu Yurramendi, Table of n, a(n) for n = 1..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A002487, A020650, A020651, A162911, A162912, A245327, A245328, A284459.

maxrow <- 4 # by choice
a <- 1
b01 <- 1
for(m in 0:maxrow){
  b01 <- c(b01,rep(1,2^(m+1))); b01[(2^(m+1)+2^m-2^(m-1)):(2^(m+1)+2^m+2^(m-1)-1)] <- 0
  for(k in 0:(2^m-1)){
    a[2^(m+1) +       k] <- a[2^m + k] + 2^(m + b01[2^(m+1) +       k])
    a[2^(m+1) + 2^m + k] <- a[2^m + k] + 2^(m + b01[2^(m+1) + 2^m + k])
}}
a
# Yosu Yurramendi, Mar 28 2017

a(n) = A231550(A258996(n)) = A092569(A231550(n)), n > 0 . - Yosu Yurramendi, Apr 10 2017

cp's OEIS Frontend

A174868 Partial sums of Stern's diatomic series A002487.

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A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)*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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A237649 a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

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PARI

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A244474 4th-largest term in n-th row of Stern's diatomic triangle A002487.

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A244475 5th-largest term in the n-th row of Stern's diatomic triangle A002487.

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A283474 a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(n-A002487(n)).

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A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

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A283977 a(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).

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A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)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).