cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

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), ...).

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024323, A024324, A024325, A024326, A024327.
Cf. A000045, A010054, A023531.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • Sage
    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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024324, A024325, A024326, A024327.
Cf. A023531.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • Sage
    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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024325, A024326, A024327.
Cf. A000201, A023531.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Extensions

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

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024327.
Cf. A023531, A023533.

Programs

  • Mathematica
    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  *)
  • Sage
    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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024328.
Cf. A014306, A023531.

Programs

  • Mathematica
    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 *)
  • Sage
    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

Formula

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

Extensions

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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}).

Programs

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

Formula

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

Extensions

Name edited by M. F. Hasler, Apr 12 2018
Previous Showing 11-16 of 16 results.

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