A290334 -id:A290334 - cp's OEIS frontend

A290335 Number of representations of n as a sum of four terms of A020330 (including 0), where order matters.

1, 0, 0, 4, 0, 0, 6, 0, 0, 4, 4, 0, 1, 12, 0, 4, 12, 0, 12, 4, 6, 12, 0, 12, 4, 12, 6, 0, 24, 0, 10, 12, 0, 16, 0, 12, 10, 0, 12, 12, 13, 0, 12, 12, 0, 16, 12, 0, 16, 24, 6, 24, 12, 0, 32, 16, 12, 24, 24, 12, 25, 36, 0, 32, 36, 12, 40, 24, 12, 36, 36, 12, 34, 36, 12, 40, 36, 12, 30, 36, 12, 40, 36, 12, 52, 24, 12, 36, 24, 12, 34, 48, 6, 52, 36, 0, 54, 12, 12

Offset: 0

Jeffrey Shallit, Jul 27 2017

			For n = 24 there are four representations, which are the distinct permutations of [15,3,3,3].

Amiram Eldar, Table of n, a(n) for n = 0..10000
Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, in: I. Potapov, P. Spirakis and J. Worrell (eds.), 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Schloss Dagstuhl, 2018, pp. 18:1-18:14; arXiv preprint, arXiv:1710.04247 [math.NT], 2017-2018.

Cf. A020330, A261132, A290334, A298731.

v = Table[k + k * 2^Floor[Log2[k] + 1], {k, 0, 8}]; a[n_] := If[(ip = IntegerPartitions[n, {4}, v]) == {}, 0, Plus @@ Length /@ (Permutations /@ ip)]; Table[a[n], {n, 0, v[[-1]]}] (* Amiram Eldar, Apr 09 2021 *)

A298731 Number of distinct representations of n as a sum of four terms of A020330 (including 0), where order does not matter.

Original entry on oeis.org

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

Offset: 0

Jeffrey Shallit, Jan 25 2018

nonn

			For n = 45, the a(45) = 4 solutions are 45 = 15+15+15 = 36+3+3+3 = 15+10+10+10.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, in: I. Potapov, P. Spirakis and J. Worrell (eds.), 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Schloss Dagstuhl, 2018, pp. 18:1-18:14; arXiv preprint, arXiv:1710.04247 [math.NT], 2017-2018.

Cf. A020330, A290334, A290335 (which is the same sequence where order matters).

v = Table[k + k * 2^Floor[Log2[k] + 1], {k, 0, 8}]; a[n_] := Length @ IntegerPartitions[n, {4}, v]; Table[a[n], {n, 0, v[[-1]]}] (* Amiram Eldar, Apr 09 2021 *)

A343268 Numbers that are not the sum of exactly four terms from A020330 (not necessarily distinct).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 34, 35, 37, 39, 41, 42, 44, 46, 47, 49, 51, 53, 56, 58, 62, 65, 67, 74, 83, 88, 95, 100, 104, 107, 109, 113, 116, 122, 125, 131, 134, 140, 143, 148, 149, 155

Offset: 1

Amiram Eldar, Apr 09 2021

nonn

Madhusudan et al. (2018) conjectured that a(112) = 1772 is the last term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..112
Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, in: I. Potapov, P. Spirakis and J. Worrell (eds.), 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Schloss Dagstuhl, 2018, pp. 18:1-18:14; arXiv preprint, arXiv:1710.04247 [math.NT], 2017-2018.

Cf. A020330, A290334, A290335, A298731, A343267.

v = Table[n + n * 2^Floor[Log2[n] + 1], {n, 1, 31}]; Complement[Range[0, 2000], Plus @@@ Tuples[v, 4]]

A343267 Numbers that are not the sum of four or fewer terms from A020330 (not necessarily distinct).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 14, 17, 22, 27, 29, 32, 34, 37, 41, 44, 47, 53, 62, 95, 104, 107, 113, 116, 122, 125, 131, 134, 140, 143, 148, 155, 158, 160, 167, 407, 424, 441, 458, 475, 492, 509, 526, 552, 560, 569, 587, 599, 608, 613, 620, 638, 653, 671, 686

Offset: 1

Amiram Eldar, Apr 09 2021

Madhusudan et al. (2018) proved that a(56) = 686 is the last term of this sequence.

Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, in: I. Potapov, P. Spirakis and J. Worrell (eds.), 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Schloss Dagstuhl, 2018, pp. 18:1-18:14; arXiv preprint, arXiv:1710.04247 [math.NT], 2017-2018.

Cf. A020330, A290334, A290335, A298731, A343268.

v = Table[n + n * 2^Floor[Log2[n] + 1], {n, 0, 20}]; Complement[Range[0, 700], Plus @@@ Tuples[v, 4]]

cp's OEIS Frontend

A290335 Number of representations of n as a sum of four terms of A020330 (including 0), where order matters.

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A298731 Number of distinct representations of n as a sum of four terms of A020330 (including 0), where order does not matter.

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A343268 Numbers that are not the sum of exactly four terms from A020330 (not necessarily distinct).

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Author

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Links

Crossrefs

Programs

Mathematica

A343267 Numbers that are not the sum of four or fewer terms from A020330 (not necessarily distinct).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica