A023361 -id:A023361 - cp's OEIS frontend

A339416 Number of compositions (ordered partitions) of n into an even number of triangular numbers.

1, 0, 1, 0, 3, 0, 6, 2, 13, 6, 28, 20, 61, 56, 135, 148, 308, 380, 707, 950, 1654, 2340, 3897, 5714, 9252, 13858, 22055, 33492, 52735, 80744, 126313, 194376, 302906, 467506, 726862, 1123830, 1744947, 2700682, 4190016, 6488824, 10062649, 15588714, 24168232, 37447884

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(9) = 6 because we have [6, 3], [3, 6], [6, 1, 1, 1], [1, 6, 1, 1], [1, 1, 6, 1] and [1, 1, 1, 6].

Cf. A000217, A023361, A034008, A106507, A339373, A339417.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) + 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) + 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
a(n) = (A023361(n) + A106507(n)) / 2.
a(n) = Sum_{k=0..n} A023361(k) * A106507(n-k).

A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 9, 3, 19, 12, 41, 33, 91, 92, 203, 238, 466, 602, 1080, 1493, 2536, 3661, 6001, 8902, 14278, 21554, 34094, 52013, 81602, 125297, 195582, 301475, 469193, 724881, 1126161, 1742206, 2703888, 4186276, 6493192, 10057553, 15594636, 24161364, 37455851

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [1, 1, 6].

Cf. A000217, A023361, A106507, A166444, A339374, A339416.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) - 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) - 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
a(n) = (A023361(n) - A106507(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023361(k) * A106507(n-k).

A224678 L.g.f.: -log(1 - Sum_{n>=1} x^(n(n+1)/2)) = Sum_{n>=1} a(n)x^n/n.

Original entry on oeis.org

1, 1, 4, 5, 6, 16, 22, 29, 49, 86, 122, 188, 300, 456, 714, 1117, 1718, 2653, 4124, 6390, 9916, 15368, 23806, 36884, 57181, 88622, 137344, 212896, 329934, 511316, 792516, 1228285, 1903598, 2950334, 4572602, 7086833, 10983562, 17022956, 26382984, 40889694, 63373086, 98218920

Offset: 1

Paul D. Hanna, Apr 14 2013

nonn

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 6*x^5/5 + 16*x^6/6 + 22*x^7/7 + 29*x^8/8 + 49*x^9/9 + 86*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 11*x^7 + 16*x^8 + 25*x^9 + 40*x^10 + 61*x^11 + 94*x^12 +...+ A023361(n)*x^n +...
exp(-L(x)) = 1 - x - x^3 - x^6 - x^10 - x^21 - x^28 +...+ -x^(n*(n+1)/2) +...

Cf. A023361, A219331, A224680.

{a(n)=n*polcoeff(-log(1-sum(r=1, sqrtint(2*n+1), x^(r*(r+1)/2)+x*O(x^n))), n)}
for(n=1, 50, print1(a(n), ", "))

Logarithmic derivative of A023361, where A023361(n) = number of compositions of n into positive triangular numbers.

A224680 a(n) = A224678(n^2).

Original entry on oeis.org

1, 5, 49, 1117, 57181, 7086833, 2109733585, 1508630963069, 2591308566579217, 10691434112980070315, 105957942450483004330197, 2522387398320711543274084153, 144235039901139444727535460625985, 19811186631607253937472121882634566325

Offset: 1

Paul D. Hanna, Apr 14 2013

nonn

A224678 is the logarithmic derivative of A023361, where A023361(n) = number of compositions of n into positive triangular numbers.

			L.g.f.: L(x) = x + 5*x^2/2 + 49*x^3/3 + 1117*x^4/4 + 57181*x^5/5 + 7086833*x^6/6 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 19*x^3 + 300*x^4 + 11768*x^5 + 1193594*x^6 +...+ A224681(n)*x^n +...

Cf. A224681, A224678, A224607.

{a(n)=n^2*polcoeff(-log(1-sum(r=1, 2*n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
for(n=1, 20, print1(a(n), ", "))

Logarithmic derivative of A224681.

A301334 a(n) = [x^n] 1/(1 + n(1 - theta_2(sqrt(x))/(2x^(1/8)))), where theta_2() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 30, 288, 3500, 51882, 908705, 18376192, 421518897, 10815546010, 306954846231, 9547629128208, 322979502072591, 11805623386524688, 463679308850798265, 19474458473055138816, 870962008703995217038, 41324081662873427484240, 2073203796753598883831150, 109655938011610286565760400

Offset: 0

Ilya Gutkovskiy, Mar 18 2018

nonn

Number of compositions (ordered partitions) of n into triangular numbers of n kinds.

Cf. A000217, A023361, A301335.

Table[SeriesCoefficient[1/(1 + n (1 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - n Sum[x^(k (k + 1)/2), {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k*(k+1)/2)).

a(n) ~ n^n * (1 + 1/n - 3/(2*n^2) - 13/(3*n^3) + 181/(24*n^4) + 2251/(120*n^5) - 34949/(720*n^6) - 221539/(2520*n^7) + 13489169/(40320*n^8) + ...). - Vaclav Kotesovec, Mar 19 2018

A348529 Number of compositions (ordered partitions) of n into two or more triangular numbers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 11, 16, 25, 39, 61, 94, 147, 227, 350, 546, 846, 1309, 2030, 3147, 4875, 7558, 11715, 18154, 28136, 43609, 67586, 104747, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487152, 5404570, 8376292, 12982016, 20120202, 31183350, 48329596, 74903735

Offset: 0

Ilya Gutkovskiy, Oct 21 2021

nonn

Cf. A000217, A010054, A023361, A347805, A348526, A348528.

b:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(issqr(8*j+1), b(n-j), 0), j=1..n))
    end:
a:= n-> b(n)-`if`(issqr(8*n+1), 1, 0):
seq(a(n), n=0..43);  # Alois P. Heinz, Oct 21 2021

b[n_] := b[n] = If[n == 0, 1, Sum[
     If[IntegerQ@ Sqrt[8*j + 1], b[n - j], 0], {j, 1, n}]];
a[n_] := b[n] - If[IntegerQ@ Sqrt[8*n + 1], 1, 0];
Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

a(n) = A023361(n) - A010054(n). - Alois P. Heinz, Oct 21 2021

A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 08 2006

nonn

This problem has the optimal substructure property.

			a(1) = 1 because "1" has a single 1.
a(2) = 2 because "1+1" has two 1's.
a(3) = 2 because 3 = T(1+1) has two 1's.
a(6) = 2 because 6 = T(T(1+1)).
a(10) = 3 because 10 = T(T(1+1)+1).
a(12) = 4 because 12 = T(T(1+1)) + T(T(1+1)).
a(15) = 4 because 15 = T(T(1+1)+1+1).
a(21) = 2 because 21 = T(T(T(1+1))).
a(28) = 3 because 28 = T(T(T(1+1))+1).
a(55) = 3 because 55 = T(T(T(1+1)+1)).

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, 1971.
R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Ed Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
Wikipedia, Optimal substructure.

See also A023361 = number of compositions into sums of triangular numbers, A053614 = numbers that are not the sum of triangular numbers. Iterated triangular numbers: A050536, A050542, A050548, A050909, A007501.

Cf. A000217, A005245, A005520, A003313, A076142, A076091, A061373, A005421, A023361, A053614, A064097, A025280, A003037, A099129, A050536, A050542, A050548, A050909, A007501.

a:= proc(n) option remember; local m; m:= floor (sqrt (n*2));
      if n<3 then n
    elif n=m*(m+1)/2 then a(m)
    else min (seq (a(i)+a(n-i), i=1..floor(n/2)))
      fi
    end:
seq (a(n), n=1..110);  # Alois P. Heinz, Jan 05 2011

a[n_] := a[n] = Module[{m = Floor[Sqrt[n*2]]}, If[n < 3, n, If[n == m*(m + 1)/2, a[m], Min[Table[a[i] + a[n - i], {i, 1, Floor[n/2]}]]]]];
Array[a, 110] (* Jean-François Alcover, Jun 02 2018, from Maple *)

I do not know how many of these entries have been proved to be minimal. - N. J. A. Sloane, Apr 15 2006

Corrected and extended by Alois P. Heinz, Jan 05 2011

A208061 G.f. 1/sum(k>=0, (-1)^k * x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 1, 4, 5, 2, -5, -12, -13, -3, 17, 34, 32, -1, -54, -93, -72, 28, 169, 248, 152, -147, -510, -646, -282, 582, 1484, 1627, 375, -2045, -4195, -3927, 110, 6716, 11544, 9002, -3458, -20996, -30921, -19123, 17974, 63154, 80435, 35553, -71525, -183969

Offset: 0

Franklin T. Adams-Watters, Feb 23 2012

sign

			G.f. = 1 + x + x^2 - x^4 - 2*x^5 - x^6 + x^7 + 4*x^8 + 5*x^9 + 2*x^10 - 5*x^11 + ...

Cf. A006950, A023361, A106507, A197870.

al(n)=Vec(1/(sum(k=0,sqrtint(2*n),(-1)^k*x^(k*(k+1)\2))+x*O(x^n)))

G.f.: 1 / (1 - x*(1 - x^2*(1 - x^3*(1 - x^4*(1 - ...))))). - Michael Somos, Mar 03 2014

Convolution inverse of A197870. - Michael Somos, Mar 03 2014

A224681 G.f.: exp( Sum_{n>=1} A224678(n^2) * x^n/n ).

Original entry on oeis.org

1, 1, 3, 19, 300, 11768, 1193594, 302611474, 188884066846, 288112683033594, 1069431906358800731, 9633610233639395592895, 210208585613243673600527636, 11095213297186302234251136888284, 1415095855034367649056280021793496073, 435753686684779400844511781608578944222819

Offset: 0

Paul D. Hanna, Apr 14 2013

nonn

A224678 is the logarithmic derivative of A023361, where A023361(n) = number of compositions of n into positive triangular numbers.

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 300*x^4 + 11768*x^5 + 1193594*x^6 +...
where
log(A(x)) = x + 5*x^2/2 + 49*x^3/3 + 1117*x^4/4 + 57181*x^5/5 + 7086833*x^6/6 +...+ A224678(n^2)*x^n/n +...

Cf. A224678, A224680, A224608.

{A224678(n)=n*polcoeff(-log(1-sum(r=1, sqrtint(2*n+1), x^(r*(r+1)/2)+x*O(x^n))), n)}
{a(n)=polcoeff(exp(sum(m=1, n, A224678(m^2)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Logarithmic derivative yields A224680.

A281810 Expansion of Sum_{i>=1} x^(i(i+1)/2) / (1 - Sum_{j>=1} x^(j(j+1)/2))^2.

Original entry on oeis.org

1, 2, 4, 8, 14, 25, 45, 77, 131, 224, 377, 629, 1049, 1738, 2863, 4708, 7716, 12598, 20524, 33363, 54102, 87567, 141489, 228216, 367538, 591098, 949372, 1522917, 2440190, 3905747, 6245198, 9976535, 15923083, 25392755, 40462155, 64426278, 102510580, 162997910, 259010672, 411328655, 652842792, 1035591110

Offset: 1

Ilya Gutkovskiy, Jan 30 2017

nonn

Total number of parts in all compositions (ordered partitions) of n into nonzero triangular numbers (A000217).

			a(6) = 25 because we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3], [1, 1, 1, 1, 1, 1] and 1 + 2 + 4 + 4 + 4 + 4 + 6 = 25.

Cf. A000217, A023361.

b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+[0, p[1]])(b(n-j*(j+1)/2)), j=1..isqrt(2*n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 07 2019

nmax = 42; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2), {i, 1, nmax}]/(1 - Sum[x^(j (j + 1)/2), {j, 1, nmax}])^2, {x, 0, nmax}], x]]
nmax = 42; Rest[CoefficientList[Series[(2 x^(1/8) EllipticTheta[2, 0, Sqrt[x]] - 4 x^(1/4))/(4 x^(1/8) - EllipticTheta[2, 0, Sqrt[x]])^2, {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^(i*(i+1)/2) / (1 - Sum_{j>=1} x^(j*(j+1)/2))^2.

cp's OEIS Frontend

A339416 Number of compositions (ordered partitions) of n into an even number of triangular numbers.

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A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

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A224678 L.g.f.: -log(1 - Sum_{n>=1} x^(n*(n+1)/2)) = Sum_{n>=1} a(n)*x^n/n.

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PARI

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A224680 a(n) = A224678(n^2).

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PARI

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A301334 a(n) = [x^n] 1/(1 + n*(1 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function.

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A348529 Number of compositions (ordered partitions) of n into two or more triangular numbers.

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Maple

Mathematica

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A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

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Maple

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A208061 G.f. 1/sum(k>=0, (-1)^k * x^(k*(k+1)/2)).

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A224678 L.g.f.: -log(1 - Sum_{n>=1} x^(n(n+1)/2)) = Sum_{n>=1} a(n)x^n/n.

A301334 a(n) = [x^n] 1/(1 + n(1 - theta_2(sqrt(x))/(2x^(1/8)))), where theta_2() is the Jacobi theta function.

A281810 Expansion of Sum_{i>=1} x^(i(i+1)/2) / (1 - Sum_{j>=1} x^(j(j+1)/2))^2.