A004771 -id:A004771 - cp's OEIS frontend

A383414 Array read by ascending antidiagonals: A(n,k) = 4^n(8k + 7).

7, 28, 15, 112, 60, 23, 448, 240, 92, 31, 1792, 960, 368, 124, 39, 7168, 3840, 1472, 496, 156, 47, 28672, 15360, 5888, 1984, 624, 188, 55, 114688, 61440, 23552, 7936, 2496, 752, 220, 63, 458752, 245760, 94208, 31744, 9984, 3008, 880, 252, 71, 1835008, 983040, 376832, 126976, 39936, 12032, 3520, 1008, 284, 79

Offset: 0

Stefano Spezia, Apr 26 2025

			The array begins as:
      7,    15,    23,     31,     39,     47, ...
     28,    60,    92,    124,    156,    188, ...
    112,   240,   368,    496,    624,    752, ...
    448,   960,  1472,   1984,   2496,   3008, ...
   1792,  3840,  5888,   7936,   9984,  12032, ...
   7168, 15360, 23552,  31744,  39936,  48128, ...
  28672, 61440, 94208, 126976, 159744, 192512, ...
  ...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 12.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 246-247.

Cf. A000302, A004215, A383415 (antidiagonal sums).
Row n=0 gives A004771.
Column k=0 gives A002042.

A[n_,k_]:=4^n(8k+7); Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

A(n,k) = A000302(n)*A004771(k).
G.f.: (7 + y)/((1 - 4*x)*(1 - y)^2).
E.g.f.: exp(4*x+y)*(7 + 8*y).

A133655 a(n) = 2*A016777(n) + A016777(n-1) - (n+1).

Original entry on oeis.org

1, 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 359, 367, 375, 383, 391, 399, 407, 415

Offset: 0

Gary W. Adamson, Sep 20 2007

			a(3) = 23 = 2*A016777(3) + A016777(2) - 4 = 2*10 + 7 - 4.
a(3) = 23 = (1, 3, 3, 1) dot (1, 6, 2, -2) = (1, 18, 6, -2).

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004771, A016777.

CoefficientList[Series[(2 x^2+5 x+1)/(x-1)^2,{x,0,60}],x] (* Harvey P. Dale, Sep 13 2011 *)

Equals "1" followed by A004771.
Binomial transform of [1, 6, 2, -2, 2, -2, 2, ...].
G.f.: (2*x^2+5*x+1)/(x-1)^2. - Harvey P. Dale, Sep 13 2011

More terms and corrected definition from R. J. Mathar, Jun 08 2008

A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

Original entry on oeis.org

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

Offset: 0

XU Pingya, Feb 09 2018

nonn

4^i(8j + 7) - 1^7 - 1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.

More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative k-th power and a nonnegative m-th power.

			7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.

Wikipedia, Legendre's three-square theorem
Wikipedia, Lagrange's four-square theorem

Cf. A000164, A002635, A004215, A004771, A297970.

a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {u,0,(n-x^2-y^2-z^2)^(1/7)}, {v,u,(n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n,0,86}]

A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

Original entry on oeis.org

0, 1, 7, 2, 15, 26, 3, 23, 53, 63, 4, 31, 80, 127, 124, 5, 39, 107, 191, 249, 215, 6, 47, 134, 255, 374, 431, 342, 7, 55, 161, 319, 499, 647, 685, 511, 8, 63, 188, 383, 624, 863, 1028, 1023, 728, 9, 71, 215, 447, 749, 1079, 1371, 1535, 1457, 999, 10, 79, 242, 511, 874, 1295, 1714, 2047, 2186, 1999, 1330

Offset: 2

Stefano Spezia, Jul 11 2020

T(n, k) is a sharp upper bound of the tree width of a graph G that does not contain a clique on n vertices nor a minimal separator of size larger than k (see Theorem 2.1 in Pilipczuk et al.).

All the square matrices starting at top left of the table T are singular except for the 2 X 2 submatrix: det([0, 7; 1, 15]) = -7.

			The table starts at row n = 2 and column k = 1 as:
0   7   26   63  124   215 ...
1  15   53  127  249   431 ...
2  23   80  191  374   647 ...
3  31  107  255  499   863 ...
4  39  134  319  624  1079 ...
5  47  161  383  749  1295 ...
...

Marcin Pilipczuk, Ni Luh Dewi Sintiari, Stéphan Thomassé and Nicolas Trotignon, (Theta, triangle)-free and (even hole, K4)-free graphs. Part 2 : bounds on treewidth, arXiv:2001.01607 [cs.DM], 2020. See p. 7.
Index entries for sequences related to trees.

Cf. A000578, A001093, A001477 (k = 1), A004771 (k = 2), A068601 (n = 2), A085537, A109129, A123865 (main diagonal), A325543, A325612.

T[n_,k_]:=(n-1)*k^3-1; Flatten[Table[T[n+1-k,k],{n,2,12},{k,1,n-1}]]

PARI
```
T(n, k) = (n - 1)*k^3 - 1
```

O.g.f.: x^2*y*(y*(7 - 2*y + y^2) + x*(1 - y)^3)/((1 - x)^2*(1 - y)^4).
E.g.f.: -1 + exp(x) - x + exp(y)*x + exp(y)*(1 + y + 3*y^2 + y^3) + exp(x + y)*(-1 +(-1 + x)*y*(1 + 3*y + y^2)).
T(n, k) = n*A000578(k) - A001093(k).
T(n, n) = A085537(n) - 1 for n > 1.
T(n, k) = T(n+1, 1)*T(2, k) + T(n, 1).

cp's OEIS Frontend

A383414 Array read by ascending antidiagonals: A(n,k) = 4^n*(8*k + 7).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A133655 a(n) = 2*A016777(n) + A016777(n-1) - (n+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383414 Array read by ascending antidiagonals: A(n,k) = 4^n(8k + 7).