A002001 -id:A002001 - cp's OEIS frontend

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

0, 1, 1, 3, 4, 5, 9, 12, 16, 21, 27, 36, 48, 64, 85, 81, 108, 144, 192, 256, 341, 243, 324, 432, 576, 768, 1024, 1365, 729, 972, 1296, 1728, 2304, 3072, 4096, 5461, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 21845, 6561

Offset: 0

Paul Curtz, Oct 26 2009

			The array starts:
0,   1,   5,  21,  85, 341,1365,5461,21845,87381,349525,    A002450
1,   4,  16,  64, 256,1024,4096,16384,65536,262144,1048576, A000302
3,  12,  48, 192, 768,3072,12288,49152,196608,786432,       A002001, A164346, A110594
9,  36, 144, 576,2304,9216,36864,147456                     A002063, A055841

A002450 := proc(n) (4^n-1)/3 ; end proc:
A166976 := proc(n,k) option remember; if n = 0 then A002450(k) else procname(n-1,k+1)-procname(n-1,k) ; end if; end proc: # R. J. Mathar, Jul 02 2011

T(0,k) = A002450(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.

A196512 Expansion of g.f. (1-9x)/(1-28x).

Original entry on oeis.org

1, 19, 532, 14896, 417088, 11678464, 326996992, 9155915776, 256365641728, 7178237968384, 200990663114752, 5627738567213056, 157576679881965568, 4412147036695035904, 123540117027461005312, 3459123276768908148736, 96855451749529428164608, 2711952648986823988609024

Offset: 0

Philippe Deléham, Oct 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (28).

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665, A196666, A196676.

[1] cat [19*28^(n-1) : n in [1..15]]; // Vincenzo Librandi, Oct 06 2011

Join[{1},19*28^Range[0,20]] (* Harvey P. Dale, Dec 06 2015 *)

a(n)=if(n,19*28^(n-1),1)  \\ M. F. Hasler, Oct 06 2011

a(n) = 19*28^(n-1) for n > 0, a(0)=1.
a(n) = Sum_{k=0..n} A193722(n,k)*9^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (19*exp(28*x) + 9)/28.
a(n) = 28*a(n-1) for n > 1. (End)

A196676 Expansion of g.f. (1-8x)/(1-25x).

Original entry on oeis.org

1, 17, 425, 10625, 265625, 6640625, 166015625, 4150390625, 103759765625, 2593994140625, 64849853515625, 1621246337890625, 40531158447265625, 1013278961181640625, 25331974029541015625, 633299350738525390625, 15832483768463134765625, 395812094211578369140625, 9895302355289459228515625

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (25).

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665, A196666.

CoefficientList[Series[(1-8x)/(1-25x),{x,0,30}],x] (* Harvey P. Dale, Apr 29 2014 *)

a(0) = 1, a(n) = 17*25^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*8^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (17*exp(25*x) + 8)/25.
a(n) = 25*a(n-1) for n > 1. (End)

More terms from Elmo R. Oliveira, Mar 25 2025

A337218 The positive integers uniquely represented by the ternary form x^2 + 2y^2 + 2z^2, with integers x <= 0, and 0 <= y <= z.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 13, 14, 21, 22, 30, 37, 42, 46, 48, 58, 70, 78, 93, 133, 142, 190, 192, 253, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888

Offset: 1

Wolfdieter Lang, Aug 20 2020

This sequence gives Theorem 2.2. of Kaplansky, p. 88, with a proof on p. 90.
This sequence is composed of two finite ones and an infinite one: (i) 2*A337217 = {2, 6, 10, 14, 22, 30, 42, 46, 58, 70, 78, 142, 190}, the even members of A094739, (ii) {1, 5, 13, 21, 37, 93, 133, 253}, the 1 (mod 4) members of A094739, and (iii) A002001(k+1) = 4^k*3, for integer k >= 0. Beginning with a(26) = 768 only the powers 4^k*3, for k >= 4 appear.
See eq. (2.2), (2,4), p. 87, of Kaplansky for the two finite sequences with 13 and 8 members, respectively.
The positive integers which have no such solution (x, y, z) are given by 4^k*(7+8*m) = A002001(k+1)*A004771(m), for k >= 0 and m >= 0. See Kaplansky, p. 88. The other missing positive integers have more than 1 solution.

			4 is not a member because (x, y, z) = (0, 1, 1) and (2, 0, 0) give both 4.
3 is a member with one solution (1, 0, 1).
5 is a member with one solutuion (1, 1, 1).
7 is not a member because there is no solution.
11 is not a member because there are two solutions (1, 1, 2) and (3, 0, 1).

Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.

Cf. A002001, A004771, A094739.

See the comment for the union of the three sequences (i), (ii) and (iii).

A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.

Original entry on oeis.org

1, 1, 2, 1, 8, 3, 1, 18, 25, 4, 1, 32, 98, 56, 5, 1, 50, 270, 336, 105, 6, 1, 72, 605, 1320, 891, 176, 7, 1, 98, 1183, 4004, 4719, 2002, 273, 8, 1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9, 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10, 1, 200, 5415

Offset: 0

Philippe Deléham, Nov 25 2013

Sum_{k=0..n}T(n,k)*x^k = A164111(n), A000012(n), A002001(n), A001653(n+1), A001019(n), A166965(n) for x =-1, 0, 1, 2, 4, 9 respectively.

			Triangle begins:
1
1, 2
1, 8, 3
1, 18, 25, 4
1, 32, 98, 56, 5
1, 50, 270, 336, 105, 6
1, 72, 605, 1320, 891, 176, 7
1, 98, 1183, 4004, 4719, 2002, 273, 8
1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9

Cf. A086645, A091042
Cf. Columns : A000012, A001105, A180324 ; Diagonals: A000027, A131423
Cf. T(2*n,n): A228329, Row sums : A002001

T := (n,k) -> binomial(2*n, 2*k)*(2*n+1-k)/(2*n+1-2*k);
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Nov 26 2013

Flatten[Table[(Binomial[2n,2k]+Binomial[2n+1,2k])/2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 05 2015 *)

G.f.: (1-x)/(1-2*x*(1+y)+x^2*(1-y)^2).
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)+2*T(n-2,k-1)-T(n-2,k)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
T(n,k) = (A086645(n,k) + A091042(n,k))/2.
T(n,k) = binomial(2*n,2*k)*(2*n+1-k)/(2*n+1-2*k). - Peter Luschny, Nov 26 2013

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144

Offset: 0

Wolfdieter Lang, Aug 01 2022

This is Boethius's triangle, with rows read as columns. See the link and reference.

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...

Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.

Columns: A000244, A003946, A257970, ...
Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
Row sums: A005061(n+1).

T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).

cp's OEIS Frontend

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

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A196512 Expansion of g.f. (1-9*x)/(1-28*x).

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A196676 Expansion of g.f. (1-8*x)/(1-25*x).

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A337218 The positive integers uniquely represented by the ternary form x^2 + 2*y^2 + 2*z^2, with integers x <= 0, and 0 <= y <= z.

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A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.

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Maple

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A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

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Comments

Examples

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Mathematica

Formula

A196512 Expansion of g.f. (1-9x)/(1-28x).

A196676 Expansion of g.f. (1-8x)/(1-25x).

A337218 The positive integers uniquely represented by the ternary form x^2 + 2y^2 + 2z^2, with integers x <= 0, and 0 <= y <= z.

A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.