A024315 -id:A024315 - cp's OEIS frontend

A024318 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 = (Fibonacci numbers).

0, 0, 1, 2, 3, 5, 8, 13, 26, 42, 68, 110, 178, 288, 466, 754, 1254, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154608, 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, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000045, A023531.

b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;
A024318:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;
[A024318(n) : n in [1..80]]; // G. C. Greubel, Jan 19 2022

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Fibonacci[n+1]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Jan 19 2022 *)

def b(n,j): return fibonacci(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
def A024318(n): return sum( b(n,j) for j in (1..floor((n+1)/2)) )
[A024318(n) for n in (1..120)] # G. C. Greubel, Jan 19 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*Fibonacci(n-j+1). - G. C. Greubel, Jan 19 2022

A024316 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = A023531.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Cf. A023531.

a024316 n = sum $ take (div (n + 1) 2) $ zipWith (*) zs $ reverse zs
            where zs = take n $ tail a023531_list
-- Reinhard Zumkeller, Feb 14 2015

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*A023531(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..110]]; // G. C. Greubel, Jan 17 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= a[n]= Sum[A023531[j]*A023531[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 110}] (* G. C. Greubel, Jan 17 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*A023531(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..110)] # G. C. Greubel, Jan 17 2022

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

A024317 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 = A023532.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 4, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 4, 5, 5, 4, 5, 4, 4, 5, 3, 5, 5, 5, 4, 5, 5, 5, 6, 5, 5, 6, 6, 5, 5, 5, 6, 6, 5, 5, 6, 5, 6, 7, 7, 5, 7, 7, 7, 7, 4, 7, 6, 6, 7, 7, 6, 6, 7, 7, 7, 8, 7, 6, 8, 8, 7, 8, 7, 7, 7, 7, 8, 8, 8, 6, 8, 7, 8, 8, 7, 8, 9, 8, 8

Offset: 1

Clark Kimberling

nonn

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

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

Cf. A023531. A023532.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*(1 - A023531(n-j+1)): j in [1..Floor((n+1)/2)]]) : n in [1..90]]; // G. C. Greubel, Jan 19 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= Sum[A023531[j]*(1 - A023531[n-j+1]), {j, Floor[(n+1)/2]}];
Table[a[n], {n, 90}] (* G. C. Greubel, Jan 19 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*(1-A023531(n-j+1)) for j in (1..floor((n+1)/2)) ) for n in (1..90)] # G. C. Greubel, Jan 19 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A023532(n-k+1). - G. C. Greubel, Jan 19 2022

A024312 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3).

Original entry on oeis.org

9, 12, 31, 38, 70, 82, 130, 148, 215, 240, 329, 362, 476, 518, 660, 712, 885, 948, 1155, 1230, 1474, 1562, 1846, 1948, 2275, 2392, 2765, 2898, 3320, 3470, 3944, 4112, 4641, 4828, 5415, 5622, 6270, 6498, 7210, 7460, 8239, 8512, 9361, 9658, 10580, 10902, 11900, 12248, 13325

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

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

[(&+[j*(n+5-j): j in [3..Floor((n+5)/2)]]) : n in [1..50]]; // G. C. Greubel, Jan 17 2022

Table[Sum[j*(n-j+5), {j, 3, Floor[(n+5)/2]}], {n, 50}] (* G. C. Greubel, Jan 17 2022 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{9,12,31,38,70,82,130},60] (* Harvey P. Dale, Aug 23 2024 *)

a(n)=((75+182*n+63*n^2+4*n^3)-3*(25+10*n+n^2)*(-1)^n)/48 \\ Charles R Greathouse IV, Oct 21 2022

[( (75 +182*n +63*n^2 +4*n^3) - 3*(25 +10*n +n^2)*(-1)^n )/48 for n in (1..50)] # G. C. Greubel, Jan 17 2022

From G. C. Greubel, Jan 17 2022: (Start)
a(n) = ( (75 + 182*n + 63*n^2 + 4*n^3) - 3*(25 + 10*n + n^2)*(-1)^n )/48.
G.f.: x*(9 + 3*x - 8*x^2 - 2*x^3 + 2*x^4)/((1-x)^4 * (1+x)^3).
a(n) = (-60 - 18*n + (14 + 3*n)*f(n) + 3*(4+n)*f(n)^2 - 2*f(n)^3)/6, where f(n) = floor((n+5)/2). (End)

A024313 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

Original entry on oeis.org

3, 3, 10, 17, 37, 59, 114, 185, 334, 540, 938, 1518, 2573, 4163, 6946, 11239, 18559, 30029, 49248, 79685, 130090, 210490, 342596, 554332, 900423, 1456915, 2363370, 3824013, 6197753

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

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

Cf. A000032, A000045, A023531.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+3] + F[n+2] - LucasL[n/2 +3] - (n/2 +1)*F[n/2 +2], LucasL[n+3] + F[n+2] - LucasL[(n+5)/2]-(n+3)/2*Fibonacci[(n+3)/2]]];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

def A024313_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(3 -2*x^2 +4*x^3 -x^4 -3*x^5 -2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024313_list(41); a[1:] # G. C. Greubel, Jan 17 2022

G.f.: x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
From G. C. Greubel, Jan 17 2022: (Start)
a(2*n) = Lucas(2*n+3) + F(2*n+2) - Lucas(n+3) - (n+1)*F(n+2).
a(2*n+1) = Lucas(2*n+4) + F(2*n+3) - Lucas(n+3) - (n+2)*F(n+2), where F(n) = A000045(n). (End)

A024314 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

Original entry on oeis.org

3, 9, 24, 37, 81, 133, 256, 413, 746, 1208, 2098, 3394, 5753, 9309, 15532, 25131, 41499, 67147, 110122, 178181, 290890, 470670, 766068, 1239524, 2013407, 3257761, 5284656, 8550753

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

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

Cf. A000032, A000045, A023532.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

a[n_]:= With[{F=Fibonacci}, 6*F[n+3] +F[n+1] - (1/2)*((1+(-1)^n)*(((n+2)/2 )*LucasL[(n+4)/2] + 5*F[(n+6)/2]) - (1-(-1)^n)*(((n+3)/2)*LucasL[(n+3)/2] +5*F[(n+5)/2] ))];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

def A024314_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024314_list(41); a[1:] # G. C. Greubel, Jan 17 2022

G.f.: x*(3 + 6*x + 6*x^2 - 8*x^3 - 7*x^4 + x^5 - 4*x^6 + 2*x^7)/((1 - x - x^2)*(1 - x^2 - x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
From G. C. Greubel, Jan 17 2022: (Start)
a(2*n) = 6*F(2*n+3) + F(2*n+1) - (n+6)*F(n+3) - (n+1)*F(n+1).
a(2*n+1) = 6*F(2*n+2) + F(2*n) - (n+6)*F(n+2) - (n+1)*F(n), where F(n) = A000045(n). (End)

A024319 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 = (Lucas numbers).

Original entry on oeis.org

0, 0, 3, 4, 7, 11, 18, 29, 58, 94, 152, 246, 398, 644, 1042, 1686, 2804, 4537, 7341, 11878, 19219, 31097, 50316, 81413, 131729, 213142, 345714, 559377, 905091, 1464468, 2369559, 3834027, 6203586

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, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A023531.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*Lucas(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..40]]; // G. C. Greubel, Jan 19 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= Sum[A023531[j]*LucasL[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 19 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*lucas_number2(n-j+1,1,-1) for j in (1..floor((n+1)/2)) ) for n in (1..40)] # G. C. Greubel, Jan 19 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*Lucas(n-j+1). - G. C. Greubel, Jan 19 2022

A024320 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 = (1, p(1), p(2), ... ).

Original entry on oeis.org

0, 0, 2, 3, 5, 7, 11, 13, 24, 30, 36, 46, 50, 60, 70, 74, 103, 117, 131, 139, 157, 171, 177, 193, 207, 221, 278, 294, 310, 330, 348, 360, 390, 408, 424, 448, 470, 486, 573, 611, 625, 653, 673, 699, 739, 761, 781, 803, 835, 863, 891, 925, 1054, 1078, 1104, 1136, 1180, 1214

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, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000040, A023531.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
p:= func< n | n eq 1 select 1 else NthPrime(n-1) >;
[ (&+[A023531(j)*p(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..60]]; // G. C. Greubel, Jan 19 2022

A023531[n_]:= SquaresR[1, 8*n+9]/2;
p[n_]:= If[n==1, 1, Prime[n-1]];
a[n_]:= Sum[A023531[j]*p[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 60}] (* G. C. Greubel, Jan 19 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
def p(n):
    if (n==1): return 1
    else: return nth_prime(n-1)
[sum( A023531(j)*p(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..60)] # G. C. Greubel, Jan 19 2022

a(n) = A023531(1) + Sum_{j=2..floor((n+1)/2)} A023531(j)*Prime(n-j+1). - G. C. Greubel, Jan 19 2022

A024321 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 = (composite numbers).

Original entry on oeis.org

0, 0, 6, 8, 9, 10, 12, 14, 25, 28, 32, 35, 37, 40, 44, 46, 64, 69, 73, 77, 81, 85, 89, 93, 96, 100, 128, 133, 139, 144, 148, 154, 162, 166, 170, 176, 181, 187, 223, 229, 236, 242, 248, 255, 262, 268, 275, 281, 287, 294, 301, 308, 354, 361, 370, 380, 386, 394, 401, 408, 418, 425

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, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A002808, A023531.

A002808:= [n : n in [2..100] | not IsPrime(n) ];
A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*A002808[n-j+1]: j in [1..Floor((n+1)/2)]]) : n in [1..70]]; // G. C. Greubel, Jan 19 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
Composite[n_]:= FixedPoint[n +PrimePi[#] +1 &, n];
a[n_]:= Sum[A023531[j]*Composite[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 70}] (* G. C. Greubel, Jan 19 2022 *)

A002808 = [n for n in (1..250) if sloane.A001222(n) > 1]
def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*A002808[n-j] for j in (1..floor((n+1)/2)) ) for n in (1..70)] # G. C. Greubel, Jan 19 2022

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A002808(n-j+1). - G. C. Greubel, Jan 19 2022

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

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)

cp's OEIS Frontend

A024318 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 = (Fibonacci numbers).

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A024316 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = A023531.

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A024317 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 = A023532.

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A024312 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3).

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A024313 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

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A024314 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

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A024319 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 = (Lucas numbers).

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Formula

A024318 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 = (Fibonacci numbers).

A024316 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = A023531.

A024317 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 = A023532.

A024312 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3).

A024313 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

A024314 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

A024319 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 = (Lucas numbers).

A024320 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 = (1, p(1), p(2), ... ).

A024321 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 = (composite numbers).

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), ...).