A053603 -id:A053603 - cp's OEIS frontend

A340955 Number of ways to write n as an ordered sum of 10 nonzero triangular numbers.

1, 0, 10, 0, 45, 10, 120, 90, 210, 370, 297, 930, 570, 1620, 1480, 2220, 3375, 2940, 6085, 4590, 8981, 8370, 11430, 15100, 13890, 23832, 19155, 31940, 30195, 38520, 46890, 46440, 66550, 59400, 86355, 81532, 104220, 114390, 122410, 153450, 149490, 193440, 188010, 235350, 238840

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..10000

Cf. A000217, A010054, A053603, A053604, A226254, A319820, A340949, A340950, A340951, A340952, A340953, A340954.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..54);  # Alois P. Heinz, Jan 31 2021

nmax = 54; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^10, where theta_2() is the Jacobi theta function.

A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

Original entry on oeis.org

1, 3, 5, 8, 10, 14, 15, 17, 19, 23, 26, 28, 32, 33, 35, 40, 41, 44, 45, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 78, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 105, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

A053603(a(n)) = 0. - Reinhard Zumkeller, Jun 28 2013

T. D. Noe, Table of n, a(n) for n=1..1000

Integers not in the sequence A051533. Cf. A002097, A000217, A007294, A051611, A053603.

a051611 n = a051611_list !! (n-1)
a051611_list = filter ((== 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

notSumQ[n_] := Reduce[ 0 < x <= y && n == x*(x + 1)/2 + y*(y + 1)/2, {x, y}, Integers] === False; Select[ Range[150], notSumQ] (* Jean-François Alcover, Jun 27 2012 *)
With[{trnos=Accumulate[Range[100]]},Complement[Range[150],Total/@ Tuples[ trnos,2]]] (* Harvey P. Dale, Jun 01 2016 *)

A347730 Number of compositions (ordered partitions) of n into at most 2 triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 2, 0, 1, 4, 0, 2, 0, 1, 3, 2, 0, 2, 2, 0, 2, 1, 2, 1, 4, 0, 0, 2, 0, 3, 2, 2, 2, 0, 0, 3, 2, 0, 1, 4, 0, 2, 2, 0, 4, 0, 0, 0, 3, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 3, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 1, 2

Offset: 0

Ilya Gutkovskiy, Sep 11 2021

Cf. A000217, A010054, A023361, A053603, A347731, A347732, A347733, A347734.

a(n) = c(n) + Sum_{k=1..n-1} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

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A340955 Number of ways to write n as an ordered sum of 10 nonzero triangular numbers.

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A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

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A347730 Number of compositions (ordered partitions) of n into at most 2 triangular numbers.

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