A307666 -id:A307666 - cp's OEIS frontend

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

1, 10, 2180, 10053736, 13291443468940

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have
a(1) = S(1, 1);
a(2) = S(4, 1) = S(1, 3);
a(3) = S(31, 4) = S(27, 5) = S(9, 15);
a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);
a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

a(5) from Giovanni Resta, Apr 13 2020

A309783 Numbers that are sums of one or more consecutive positive triangular numbers in more than one way.

Original entry on oeis.org

10, 36, 55, 64, 100, 120, 136, 164, 210, 276, 361, 435, 460, 514, 560, 596, 676, 760, 1176, 1225, 1320, 1326, 1460, 1484, 1485, 1505, 1540, 1684, 1736, 1770, 1891, 1936, 2014, 2080, 2145, 2180, 2314, 2485, 2596, 2890, 3156, 3244, 3275, 3364, 3486, 3570, 3710, 3916

Offset: 1

Ilya Gutkovskiy, Aug 17 2019

nonn

The first number that is the sum in three ways is 2180. The first that is the sum in four ways is 10053736. - Robert Israel, Aug 20 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (first 2759 terms from Robert Israel)

Cf. A000217, A034706, A130052, A307666.

N:= 10000: # for terms <= N
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
  s:= i*(i+1)*(i+2)/6;
  for j from i-1 to 0 by -1 do
    t:= j*(j+1)*(j+2)/6;
    if s-t > N then break fi;
    V[s-t]:= V[s-t]+1
  od;
od:
select(t -> V[t]>1, [$1..N]); # Robert Israel, Aug 20 2019

A307666(a(n)) > 1.

A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

Original entry on oeis.org

1, 3, 3, 4, 5, 4, 3, 6, 7, 4, 4, 4, 5, 9, 4, 4, 5, 5, 7, 9, 4, 4, 4, 6, 4, 7, 7, 4, 7, 5, 3, 6, 6, 11, 9, 4, 4, 6, 4, 4, 6, 4, 5, 11, 5, 4, 4, 6, 6, 6, 5, 4, 7, 12, 8, 6, 4, 4, 6, 4, 4, 8, 5, 8, 9, 4, 4, 7, 8, 4, 5, 4, 5, 8, 4, 8, 9, 4, 5, 8, 4, 6, 10, 7, 4, 6

Offset: 2

Peter Kagey, Apr 08 2020

nonn

a(n) >= A001227(n) + A307666(n).
a(n) >= A003016(n) + A003016(n+1) - 2.
Records occur at indices 2, 3, 5, 6, 9, 10, 15, 35, 55, 105, 210, 1365, 2925, 3003,...

			The a(9) = 6 ways to write 9 as the difference of k-simplex numbers for k > 2 are:
C(5,  2) - C(2, 2) = 10 -  1,
C(6,  2) - C(4, 2) = 15 -  6,
C(10, 2) - C(9, 2) = 45 - 36,
C(5,  3) - C(3, 3) = 10 -  1,
C(9,  8) - C(7, 8) =  9 -  0, and
C(10, 9) - C(9, 9) = 10 -  1,
where C(n,k) = binomial(n,k) = A007318(n,k).

Peter Kagey, Table of n, a(n) for n = 2..5000

Cf. A001227, A003016, A007318, A307666.
Cf. A333822, A333836.
The k-simplex numbers for 2 <= k <= 6 are A000217 (k=2), A000292 (k=3), A000332 (k=4), A000389 (k=5), and A000579 (k=6).

A368072 Number of representations of n as the difference of two positive tetrahedral numbers.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 10 2023

nonn

			a(36) = 2: 36 = 56 - 20 = 120 - 84.

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A136107, A307666, A368073, A368076.

cp's OEIS Frontend

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

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A309783 Numbers that are sums of one or more consecutive positive triangular numbers in more than one way.

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A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

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A368072 Number of representations of n as the difference of two positive tetrahedral numbers.

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