A024316 -id:A024316 - cp's OEIS frontend

A024322 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (F(2), F(3), ...).

0, 0, 2, 3, 5, 8, 13, 21, 42, 68, 110, 178, 288, 466, 754, 1220, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154231, 250161, 404769, 654930, 1059699, 1714629, 2774328, 4488957, 7263285

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024323, A024324, A024325, A024326, A024327.

Cf. A000045, A010054, A023531.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*Fibonacci(n-j+2): j in [1..Floor((n+1)/2)]]) : n in [1..40]]; // G. C. Greubel, Jan 20 2022

A010054[n_]:= SquaresR[1, 8n+1]/2;
a[n_]:= Sum[A010054[j+1]*Fibonacci[n-j+2], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 20 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*fibonacci(n-j+2) for j in (1..floor((n+1)/2)) ) for n in (1..40)] # G. C. Greubel, Jan 20 2022

From G. C. Greubel, Jan 20 2022: (Start)
a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000045(n-j+1).
a(n) = Sum_{j=1..floor((n+1)/2)} A010054(j+1)*A000045(n-j+2). (End)

A024323 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (odd natural numbers).

Original entry on oeis.org

0, 0, 3, 5, 7, 9, 11, 13, 24, 28, 32, 36, 40, 44, 48, 52, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405, 415, 425, 488, 500, 512, 524, 536, 548, 560, 572, 584

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024324, A024325, A024326, A024327.

Cf. A023531.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*(2*n-2*j+1): j in [1..Floor((n+1)/2)]]) : n in [1..70]]; // G. C. Greubel, Jan 20 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= Sum[A023531[j]*(2*n-2*j+1), {j, Floor[(n+1)/2]}];
Table[a[n], {n, 70}] (* G. C. Greubel, Jan 20 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*(2*n-2*j+1) for j in (1..floor((n+1)/2)) ) for n in (1..70)] # G. C. Greubel, Jan 20 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*(2*n -2*j + 1). - G. C. Greubel, Jan 20 2022

A024324 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A000201 (lower Wythoff sequence).

Original entry on oeis.org

0, 0, 3, 4, 6, 8, 9, 11, 20, 23, 27, 29, 33, 37, 39, 43, 60, 65, 70, 74, 80, 84, 89, 94, 98, 104, 131, 137, 143, 150, 157, 163, 169, 176, 183, 189, 195, 202, 241, 248, 256, 265, 272, 281, 289, 296, 306, 313, 321, 329, 337, 346, 397, 406, 416, 425, 436, 445, 454, 466, 474, 484

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024325, A024326, A024327.

Cf. A000201, A023531.

b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;
A024324:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;
[A024324(n) : n in [1..80]]; // G. C. Greubel, Jan 28 2022

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Floor[n*GoldenRatio^2]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Jan 28 2022 *)

my(phi=quadgen(5)); a(n) = my(L=n>>1,m=2,ret=0); n--; while(n>L, ret += floor(n*phi); n-=(m++)); ret; \\ Kevin Ryde, Feb 03 2022

def b(n,j): return floor( (n+1-j)*(1+sqrt(5))/2 ) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
def A024324(n): return sum( b(n,k) for k in (1..((n+1)//2)) )
[A024324(n) for n in (1..80)] # G. C. Greubel, Jan 28 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000201(n-j+1).

a(62) corrected by Sean A. Irvine, Jun 27 2019

A024325 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A001950 (upper Wythoff sequence).

Original entry on oeis.org

0, 0, 5, 7, 10, 13, 15, 18, 33, 38, 44, 48, 54, 60, 64, 70, 98, 106, 114, 121, 130, 137, 145, 153, 160, 169, 213, 223, 233, 244, 255, 265, 275, 286, 297, 307, 317, 328, 391, 403, 416, 430, 442, 456, 469, 481, 496, 508, 521, 534, 547, 561, 644, 659, 675, 690, 707, 722, 737, 755

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024326, A024327.

Cf. A001950, A023531.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
A024325:= func< n | (&+[A023531(j)*Floor((n-j+1)*(3+Sqrt(5))/2): j in [1..Floor((n+1)/2)]]) >;
[A024325(n) : n in [1..80]]; // G. C. Greubel, Jan 28 2022

A023531[n_] := SquaresR[1, 8n+9]/2;
a[n_]:= a[n]= Sum[A023531[j]*Floor[(n-j+1)*GoldenRatio^2], {j,Floor[(n+1)/2]}];
Table[a[n], {n, 80}] (* G. C. Greubel, Jan 28 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
def A023325(n): return sum( A023531(j)*floor(((n-j+1)*(3+sqrt(5)))/2) for j in (1..((n+1)//2)) )
[A023325(n) for n in (1..80)] # G. C. Greubel, Jan 28 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A001950(n-j+1).

A024326 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A023533.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024327.

Cf. A023531, A023533.

nmax = 120;
A023533:= A023533 = With[{ms= Table[m(m+1)(m+2)/6, {m, 0, nmax+5}]}, Table[If[MemberQ[ms, n], 1, 0], {n, 0, nmax+5}]];
AbsoluteTiming[Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A023533[[n + 1]]; n -= m++]; t, {n, nmax}]] (* G. C. Greubel, Jan 29 2022  *)

nmax=120
@CachedFunction
def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
@CachedFunction
def B_list(N):
    A = []
    for m in range(ceil((6*N)^(1/3))):
        A.extend([0]*(binomial(m+2, 3) -len(A)) +[1])
    return A
A023533 = B_list(nmax+5)
@CachedFunction
def A023324(n): return sum( A023531(j)*A023533[n-j+1] for j in (1..((n+1)//2)) )
[A023324(n) for n in (1..nmax)] # G. C. Greubel, Jan 29 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A023533(n-j+1).

A024327 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor( (n+1)/2 ), s = A023531, t = A014306.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 7, 8, 7, 8, 7

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024328.

Cf. A014306, A023531.

A014306:= With[{ms= Table[m(m+1)(m+2)/6, {m,0,20}]}, Table[If[MemberQ[ms, n], 0, 1], {n,0,150}]];
Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A014306[[n+1]]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Feb 17 2022 *)

nmax=120
@CachedFunction
def b_list(N):
    A = []
    for m in range(ceil((6*N)^(1/3))):
        A.extend([0]*(binomial(m+2, 3) - len(A)) + [1])
    return A
A023533 = b_list(nmax+5)
def A014306(n): return 1 - A023533[n]
def b(n, j): return A014306(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
@CachedFunction
def A024327(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )
[A024327(n) for n in (1..nmax)] # G. C. Greubel, Feb 17 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A014306(n-k+1). - G. C. Greubel, Feb 17 2022

Title corrected by Sean A. Irvine, Jun 30 2019

A178244 Number of distinct permutations of binary digits (0's and 1's) in n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 3, 1, 4, 6, 6, 4, 6, 4, 4, 1, 5, 10, 10, 10, 10, 10, 10, 5, 10, 10, 10, 5, 10, 5, 5, 1, 6, 15, 15, 20, 15, 20, 20, 15, 15, 20, 20, 15, 20, 15, 15, 6, 15, 20, 20, 15, 20, 15, 15, 6, 20, 15, 15, 6, 15, 6, 6, 1, 7, 21, 21, 35, 21, 35, 35, 35, 21, 35, 35, 35, 35, 35, 35, 21

Offset: 0

Juri-Stepan Gerasimov, May 23 2010

			a(0) = 1 because the only permutation is 0 (or 0 written in base 2),
a(1) = 1 because the only permutation is 1 (or 1 written in base 2),
a(3) = 2 because there are two distinct permutations of the binary digits in 2 = 10_2: {01, 10} (identity permutation and transposition tau_12).
a(12) = 6, because the binary digits of 12 = 1100_2 have 6 distinct permutations {0011, 0101, 0110, 1001, 1010, 1100}, of which only 0101, 0110, 1001, 1010 are transpositions (= permutations changing exactly 2 elements).

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Transposition.

Cf. A000120, A001477, A024316, A070939.

f:= proc(n) local L;
  L:= convert(n,base,2);
  binomial(nops(L),convert(L,`+`))
end proc:
map(f, [$0..100]); # Robert Israel, Oct 25 2023

A178244[n_]:=Binomial[BitLength[n],DigitCount[n,2,1]];Array[A178244,100,0] (* Paolo Xausa, Nov 03 2023 *)

A178244(n)=binomial(exponent(n*2+1),hammingweight(n));
apply(A178244, [0..99]) \\ M. F. Hasler, Feb 23 2023

from math import comb
def A178244(n): return comb(n.bit_length(),n.bit_count()) # Chai Wah Wu, Mar 13 2023

a(n) = binomial((A000120(n) + A023416(n)), A000120(n)).

a(n) = binomial(A070939(n), A000120(n)). - Paolo Xausa, Nov 03 2023

Corrected (a 3 in the first group removed) by R. J. Mathar, May 28 2010

Definition corrected by M. F. Hasler, Feb 23 2023

A024328 a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*prime(n-j+1).

Original entry on oeis.org

0, 0, 3, 5, 7, 11, 13, 17, 30, 36, 46, 50, 60, 70, 74, 84, 117, 131, 139, 157, 171, 177, 193, 207, 221, 237, 294, 310, 330, 348, 360, 390, 408, 424, 448, 470, 486, 506, 611, 625, 653, 673, 699, 739, 761, 781, 803, 835, 863, 891, 925, 953, 1078, 1104, 1136, 1180, 1214, 1244, 1270

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A023531 (characteristic function of {n(n+3)/2}).

b:= func< n, j | IsIntegral((Sqrt(8*j+9) -3)/2) select NthPrime(n-j+1) else 0 >;
A024328:= func< n | (&+[b(n, j): j in [1..Floor((n+1)/2)]]) >;
[A024328(n) : n in [1..120]]; // G. C. Greubel, Feb 17 2022

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Prime[n]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Feb 17 2022 *)

A024328(n)=sum(j=1, (n+1)\2, A023531(j)*prime(n-j+1)) \\ M. F. Hasler, Apr 12 2018

def b(n, j): return nth_prime(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
@CachedFunction
def A024327(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )
[A024327(n) for n in (1..120)] # G. C. Greubel, Feb 17 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*prime(n-j+1).

Name edited by M. F. Hasler, Apr 12 2018

cp's OEIS Frontend

A024322 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (F(2), F(3), ...).

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A024323 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (odd natural numbers).

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A024324 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A000201 (lower Wythoff sequence).

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A024325 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A001950 (upper Wythoff sequence).

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A024326 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A023533.

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A024327 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor( (n+1)/2 ), s = A023531, t = A014306.

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A178244 Number of distinct permutations of binary digits (0's and 1's) in n.

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A024322 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (F(2), F(3), ...).

A024323 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (odd natural numbers).

A024324 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A000201 (lower Wythoff sequence).

A024325 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A001950 (upper Wythoff sequence).

A024326 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A023533.

A024327 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor( (n+1)/2 ), s = A023531, t = A014306.