A023531 -id:A023531 - cp's OEIS frontend

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).

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

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

A134673 A051731 + A127448 - I where I is the Identity matrix (A023531).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 05 2007

Row sums = A073757: (1, 2, 3, 4, 5, 5, 7, 7, 8, 7, ...).

			First few rows of the triangle:
  1;
  0,  2;
  0,  0,  3;
  1, -1,  0,  4;
  0,  0,  0,  0,  5;
  2, -1, -2,  0,  0,  6;
  0,  0,  0,  0,  0,  0,  7;
  1,  1,  0, -3,  0,  0,  0,  8;
  ... [Typo corrected by _N. J. A. Sloane_, May 22 2010]

Cf. A127448, A073757, A051731.

A134673(n,k) = if(n%k,0,if(k==n,n,k*moebius(n/k)+1)); /* Max Alekseyev, Jan 07 2015 */

a(n) = A051731(n) + A127448(n) - A023531(n).

T(n,k) = k*A008683(n/k) + 1 if k divides n and k < n, T(n,k)=n for k=n, and T(n,k)=0 otherwise. - Max Alekseyev, Jan 07 2015

More terms from Max Alekseyev, Apr 03 2022

A155029 Complement to A051731 with the identity matrix A023531 included.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik, Jan 19 2009

			Table begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 1;
  0, 1, 1, 1, 1;
  0, 0, 0, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1;
  0, 0, 1, 0, 1, 1, 1, 1;
  0, 1, 0, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A023531, A051731, A155031.

[k eq n select 1 else (k eq 0 or n mod k eq 0) select 0 else 1: k in [1..n], n in [1..20]]; // G. C. Greubel, Mar 07 2021

Table[If[k==n, 1, If[k==0, 0, If[Mod[n, k]==0, 0, 1]]], {n, 20}, {k, n}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

flatten([[1 if k==n else 0 if (k==0 or n%k==0) else 1 for k in [1..n]] for n in [1..20]]) # G. C. Greubel, Mar 07 2021

T(n, k) = 0 if n==0 (mod k) otherwise 1 with T(n, n) = 1 and T(n, 1) = 0. - G. C. Greubel, Mar 07 2021

A023565 Convolution of A023531 and A023533.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 1, 3, 0, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2

Offset: 1

Clark Kimberling

nonn

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

Cf. A023531, A023533.

A023531:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 1 else 0 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023531(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022

A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023565[n_]:= A023565[n]= Sum[A023533[k]*A023531[n-k+1], {k,n}];
Table[A023565[n], {n,100}] (* G. C. Greubel, Jul 16 2022 *)

@CachedFunction
def A023531(n): return 1 if ((sqrt(8*n+9) -3)/2).is_integer()  else 0
@CachedFunction
def A023533(n): return 0 if binomial( floor((6*n-1)^(1/3)) +2, 3)!=n else 1
[sum(A023533(k)*A023531(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

a(n) = Sum_{j=1..n} A023533(j) * A023531(n-j+1). - G. C. Greubel, Jul 16 2022

A023567 Convolution of A023531 and primes.

Original entry on oeis.org

0, 2, 3, 5, 9, 14, 18, 24, 32, 39, 51, 57, 71, 85, 94, 108, 124, 142, 152, 176, 193, 205, 229, 249, 271, 295, 315, 336, 364, 386, 408, 444, 468, 490, 526, 561, 583, 617, 663, 681, 717, 745, 781, 831, 862, 894, 924, 968, 1006, 1050, 1100, 1138, 1174

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Primes:= select(isprime, [2,seq(i,i=3..1000,2)]):
seq(add(Primes[n - i*(i+3)/2],i=1..floor((sqrt(1+8*n)-3)/2)), n=2..nops(Primes)); # Robert Israel, Dec 28 2015

vector(20,x,c=0;j=x;t=3;while(j>1,c+=prime(j-1);j-=t;t+=1);c)

a(n) = Sum_{i=0..n-1} A023531(n-i)*p(i+1) where p(i) is the i-th prime.

Corrected (term 781 was missing) by Jeremy Gardiner, Feb 05 2014

A024889 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.

Original entry on oeis.org

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, 1, 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0

Offset: 2

Clark Kimberling

nonn

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

Cf. A023531, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023531:= func< n |  IsSquare(8*n+9) select 1 else 0 >;
A024889:= func< n | (&+[A023531(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A024889(n): n in [2..130]]; // G. C. Greubel, Aug 02 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]]+2,3]!= n,0,1];
A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2],1,0];
A024889[n_]:= A024889[n]= Sum[A023531[j]*A023533[n-j+1], {j, Floor[n/2]}];
Table[A024889[n], {n, 2, 130}] (* G. C. Greubel, Aug 02 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3) != n) else 1
def A023531(n): return 1 if is_square(8*n+9) else 0
def A024889(n): return sum(A023531(k)*A023533(n-k+1) for k in (1..(n//2)))
[A024889(n) for n in (2..130)] # G. C. Greubel, Aug 02 2022

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

A024890 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023531, t = A014306.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 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, 5, 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, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 9, 8, 8, 9, 9, 9

Offset: 2

Clark Kimberling

nonn

a(102) onward corrected by Sean A. Irvine, Jul 27 2019

cp's OEIS Frontend

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

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A134673 A051731 + A127448 - I where I is the Identity matrix (A023531).

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Comments

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PARI

Formula

Extensions

A155029 Complement to A051731 with the identity matrix A023531 included.

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Magma

Mathematica

Sage

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A023565 Convolution of A023531 and A023533.

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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).

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.

A024889 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.