A024316 -id:A024316 - cp's OEIS frontend

A023531 a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.

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

Offset: 0

Clark Kimberling, Jun 14 1998

Can be read as table: a(n,m) = 1 if n = m >= 0, else 0 (unit matrix).
a(n) = number of 1's between successive 0's (see also A005614, A003589 and A007538). - Eric Angelini, Jul 06 2005
Triangle T(n,k), 0 <= k <= n, read by rows, given by A000004 DELTA A000007 where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order.
A023531 is reverse reluctant sequence of sequence A000007. - Boris Putievskiy, Jan 11 2013
Also the Bell transform (and the inverse Bell transform) of 0^n (A000007). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
This is the turn sequence of the triangle spiral.  To form the spiral: go a unit step forward, turn left a(n)*120 degrees, and repeat.  The triangle sides are the runs of a(n)=0 (no turn).  The sequence can be generated by a morphism with a special symbol S for the start of the sequence: S -> S,1; 1 -> 0,1; 0->0.  The expansion lengthens each existing side and inserts a new unit side at the start.  See the Fractint L-system in the links to draw the spiral this way. - Kevin Ryde, Dec 06 2019

			As a triangle:
       1
      0 1
     0 0 1
    0 0 0 1
   0 0 0 0 1
  0 0 0 0 0 1
G.f. = 1 + x^2 + x^5 + x^9 + x^14 + x^20 + x^27 + x^35 + x^44 + x^54 + ...
From _Kevin Ryde_, Dec 06 2019: (Start)
.
              1            Triangular spiral: start at S;
             / \             go a unit step forward,
            0   0   .        turn left a(n)*120 degrees,
           /     \   .       repeat.
          0   1   0   .
         /   / \   \   \   Each side's length is 1 greater
        0   0   0   0   0    than that of the previous side.
       /   /     \   \   \
      0   0   S---1   0   0
     /   /             \   \
    0   1---0---0---0---1   0
   /                         \
  1---0---0---0---0---0---0---1
(End)

Antti Karttunen, Table of n, a(n) for n = 0..100127
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Kevin Ryde, Fractint L-System drawing the spiral.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.
Index entries for characteristic functions.

Cf. A000217, A003057, A010054, A000007, A023532.

Cf. A000096, A007701, A024316.

a023531 n = a023531_list !! n
a023531_list = concat $ iterate ([0,1] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

seq(op([0$m,1]),m=0..10); # Robert Israel, Jan 18 2015
# alternative
A023531 := proc(n)
    option remember ;
    local m,t ;
    for m from 0 do
        t := m*(m+3)/2 ;
        if t > n then
            return 0 ;
        elif t = n then
            return 1 ;
        end if;
    end do:
end proc:
seq(A023531(n),n=0..40) ; # R. J. Mathar, May 15 2025

If[IntegerQ[(Sqrt[9+8#]-3)/2],1,0]&/@Range[0,100] (* Harvey P. Dale, Jul 27 2011 *)
a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt[ 8 n + 9]]; (* Michael Somos, May 17 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)) - 1) / x, {x, 0, n}]; (* Michael Somos, May 17 2014 *)

{a(n) = if( n<0, 0, issquare(8*n + 9))}; /* Michael Somos, May 17 2014 */

A023531(n)=issquare(8*n+9) \\ M. F. Hasler, Apr 12 2018

from math import isqrt
def A023531(n): return int((k:=n+1<<1)==(m:=isqrt(k))*(m+1)) # Chai Wah Wu, Nov 09 2024

def A023531_row(n) :
    if n == 0: return [1]
    return [0] + A023531_row(n-1)
for n in (0..9): print(A023531_row(n))  # Peter Luschny, Jul 22 2012

If (floor(sqrt(2*n))-(2*n/(floor(sqrt(2*n)))) = -1, 1, 0). - Gerald Hillier, Sep 11 2005
a(n) = 1 - A023532(n); a(n) = 1 - mod(floor(((10^(n+2) - 10)/9)10^(n+1 - binomial(floor((1+sqrt(9+8n))/2), 2) - (1+floor(log((10^(n+2) - 10)/9, 10))))), 10). - Paul Barry, May 25 2004
a(n) = floor((sqrt(9+8n)-1)/2) - floor((sqrt(1+8n)-1)/2). - Paul Barry, May 25 2004
a(n) = round(sqrt(2n+3)) - round(sqrt(2n+2)). - Hieronymus Fischer, Aug 06 2007
a(n) = ceiling(2*sqrt(2n+3)) - floor(2*sqrt(2n+2)) - 1. - Hieronymus Fischer, Aug 06 2007
From Franklin T. Adams-Watters, Jun 29 2009: (Start)
G.f.: (1/2 x^{-1/8}theta_2(0,x^{1/2}) - 1)/x, where theta_2 is a Jacobi theta function.
G.f. for triangle: Sum T(n,k) x^n y^k = 1/(1-x*y). Sum T(n,k) x^n y^k / n! = Sum T(n,k) x^n y^k / k! = exp(x*y). Sum T(n,k) x^n y^k / (n! k!) = I_0(2*sqrt(x*y)), where I is the modified Bessel function of the first kind. (End)
a(n) = A000007(m), where m=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 11 2013
The row polynomials are p(n,x) = x^n = (-1)^n n!Lag(n,-n,x), the normalized, associated Laguerre polynomials of order -n. As the prototypical Appell sequence with e.g.f. exp(x*y), its raising operator is R = x and lowering operator, L = d/dx, i.e., R p(n,x) = p(n+1,x), and L p(n,x) = n * p(n-1,x). - Tom Copeland, May 10 2014
a(n) = A010054(n+1) if n >= 0. - Michael Somos, May 17 2014
a(n) = floor(sqrt(2*(n+1)+1/2)-1/2) - floor(sqrt(2*n+1/2)-1/2). - Mikael Aaltonen, Jan 18 2015
a(n) = A003057(n+3) - A003057(n+2). - Robert Israel, Jan 18 2015
a(A000096(n)) = 1; a(A007701(n)) = 0. - Reinhard Zumkeller, Feb 14 2015
Characteristic function of A000096. - M. F. Hasler, Apr 12 2018
Sum_{k=1..n} a(k) ~ sqrt(2*n). - Amiram Eldar, Jan 13 2024

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).

Original entry on oeis.org

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

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)

A024315 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = floor(n/2), s = (natural numbers >= 3), t = (Fibonacci numbers).

Original entry on oeis.org

3, 6, 17, 27, 59, 96, 185, 299, 540, 874, 1518, 2456, 4163, 6736, 11239, 18185, 30029, 48588, 79685, 128933, 210490, 340580, 554332, 896928, 1456915, 2357338, 3824013, 6187383

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

Cf. A000032, A000045.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 16 2022

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+4] +F[n+3] -F[(n+10)/2] -((n+ 4)/2)*F[(n+6)/2], LucasL[n+4] +F[n+3] -F[(n+7)/2] -((n+7)/2)*F[(n+5)/2]]];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 16 2022 *)

def A024315_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024315_list(41); a[1:] # G. C. Greubel, Jan 16 2022

G.f.: x*(3 +3*x +2*x^2 -2*x^3 -4*x^4 -x^5 -2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
From G. C. Greubel, Jan 16 2022: (Start)
a(2*n) = L(2*n+4) + F(2*n+3) - F(n+5) - (n+2)*F(n+3), n >= 1.
a(2*n-1) = L(2*n+3) + F(2*n+2) - F(n+3) - (n+3)*F(n+2), n >= 1, where L(n) = A000032(n) and 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

cp's OEIS Frontend

A023531 a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.

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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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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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Formula

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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Formula

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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A024315 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = floor(n/2), s = (natural numbers >= 3), t = (Fibonacci numbers).

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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).

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).