A042948 -id:A042948 - cp's OEIS frontend

A190796 Number of digits in the minimal base-phi representation of n.

1, 4, 5, 5, 8, 8, 9, 9, 9, 9, 9, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

T. D. Noe, May 20 2011

See A130600(n) for the digits in the minimal base phi representation of n.

a(n) <= 2 * ceiling( log(n) / log(phi) ) for n > 1.

Michel Dekking and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." Fib. Quart. 61:2 (May 2023), 105-118.

T. D. Noe, Table of n, a(n) for n = 1..1000
Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.

Cf. A130600, A055778, A133775, A042948.

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; last1, {n, 1, nn}]

a(n) = A055778(n) + A133775(n).
From Michel Dekking, Jun 19 2024: (Start)
Let (L(n)) = (2, 1, 3, 4, 7, 11, 18, 29, 47, ...) = A000032 be the Lucas numbers.
If L(2n) <= i <= L(2n+1), then a(i) = 4n+1; if L(2n+1)+1 <= i < L(2n+2), then a(i) = 4n+4.
This formula follows from Proposition 4.2. in "On the representation of the natural numbers by powers of the golden mean".
For example if n=1: L(2)=3, L(3)=4, L(4)=7, so a(3) = a(4) = 5, and a(5) = a(6) = 8.
Let (v(n)) = 1,4,5,8,9,12,... be the sequence of values taken by (a(n)). Then it follows directly from the Lucas formula for (a(n)) that v(n) = A042948(n) (where A042948 has been given offset 1, as it should; see also the comment by Jianing Song in A042948).
(End)

A204877 Continued fraction expansion of 3*tanh(1/3).

Original entry on oeis.org

0, 1, 27, 5, 63, 9, 99, 13, 135, 17, 171, 21, 207, 25, 243, 29, 279, 33, 315, 37, 351, 41, 387, 45, 423, 49, 459, 53, 495, 57, 531, 61, 567, 65, 603, 69, 639, 73, 675, 77, 711, 81, 747, 85, 783, 89, 819, 93, 855, 97, 891, 101, 927, 105, 963, 109, 999, 113

Offset: 0

Bruno Berselli, Jan 23 2012

The continued fraction expansions of tanh(1) and 2*tanh(1/2) are in A004273 and A110185, respectively.

Bruno Berselli, Table of n, a(n) for n = 0..1000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A004273, A110185.

I:=[0,1,27,5,63]; [n le 5 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..58]];

ContinuedFraction[3 Tanh[1/3], 158]
CoefficientList[Series[x (1 + 27 x + 3 x^2 + 9 x^3) / ((1 - x)^2 (1 + x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 14 2013 *)

makelist(coeff(taylor(x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2), x, 0, n), x, n), n, 0, 57);

PARI
```
\p232;
       contfrac(3*tanh(1/3))
```

G.f.: x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2).
E.g.f.: 9-4*exp(-x)*(1+2*x)+5*exp(x)*(-1+2*x).
a(n) = (5+4*(-1)^n)*(2*n-1), with a(0)=0.
a(n) = 2*a(n-2)-a(n-4) for n>4.
a(n) = a(n-2)+A040314(n-2) for n>2.
a(n)*a(n+1) = a(2*n^2).
Sum(a(i), i=0..n) = A195162(A042948(n)).

A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 5, 6, 1, 1, 4, 9, 1, 1, 7, 8, 1, 6, 10, 1, 1, 1, 9, 10, 1, 1, 5, 16, 1, 7, 15, 1, 11, 12, 1, 1, 9, 16, 1, 1, 13, 14, 1, 1, 8, 21, 1, 1, 6, 10, 15, 16, 21, 25, 1, 1, 1, 12, 22, 1, 17, 18, 1, 15, 21, 1, 9, 28, 1, 1, 19, 20, 1, 13

Offset: 2

Eric M. Schmidt, Aug 05 2012

The n-th row has length A034444(n) - 1.
If m appears in row n, then gcd(n,m) appears in the n-th row of A077610. Moreover, if m', distinct from m, also appears in row n, then gcd(n, m) does not equal gcd(n, m').
For odd n and any integer m, m^2 == m (mod n) iff m^2 == m (mod 2n).
Let P(1)={1} and for integers x > 1, let P(x) be the set of distinct prime divisors of x. We can define an equivalence relation ~ on the set of elements in the ring (Z_n, +mod n,*mod n): for all a,b in Z_n (where a,b are the least nonnegative residues modulo n) a ~ b iff P(gcd(a,n)) intersect P(n) is equal to P(gcd(b,n)) intersect P(n). If we include 0 in each row then these elements can represent the equivalence classes. They form a commutative monoid. - Geoffrey Critzer, Feb 13 2016

			Triangle begins:
1;
1;
1;
1;
1, 3, 4;
1;
1;
1;
1, 5, 6;
1;
1, 4, 9;
1;
1, 7, 8;
1, 6, 10;
1;
1;
1, 9, 10; etc.  - _Bruno Berselli_, Aug 06 2012

Eric M. Schmidt, Rows 2..2000, flattened

For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13: A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984; n=100: A008852; n=1000: A008853.

[m: m in [1..n-1], n in [2..40] | m^2 mod n eq m]; // Bruno Berselli, Aug 06 2012

Table[Select[Range[n], Mod[#^2, n] == # &], {n, 2, 30}] // Grid (* Geoffrey Critzer, May 26 2015 *)

def A215202(n) : return [m for m in range(1, n) if m^2 % n == m];

A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

Original entry on oeis.org

0, 1, 10, 15, 36, 45, 78, 91, 136, 153, 210, 231, 300, 325, 406, 435, 528, 561, 666, 703, 820, 861, 990, 1035, 1176, 1225, 1378, 1431, 1596, 1653, 1830, 1891, 2080, 2145, 2346, 2415, 2628, 2701, 2926, 3003, 3240, 3321, 3570, 3655, 3916, 4005, 4278, 4371, 4656

Offset: 0

Daniel Poveda Parrilla, Sep 22 2016

All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
a(A276915(n)) is a triangular pentagonal number.
a(A079291(n)) is a triangular square number, as A275496 is a subsequence of this.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..10000
Daniel Poveda Parrilla, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A004526, A016813, A042948, A052928, A079291, A168277, A275496, A276915.

[n*(2*n+(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 19 2022

Table[n (2 n + (-1)^n), {n, 0, 48}] (* Michael De Vlieger, Sep 23 2016 *)

concat(0, Vec(x*(1+9*x+3*x^2+3*x^3)/((1-x)^3*(1+x)^2) + O(x^50))) \\ Colin Barker, Sep 23 2016

[n*(2*n+(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 19 2022

a(n) = n^2 + 2*A000217(A052928(n)).
a(n) = A000217(A042948(n)).
a(n) = n*(2*n + (-1)^n).
a(n) = n*A168277(n + 1).
a(n) = n*A016813(A004526(n)).
From Colin Barker, Sep 23 2016: (Start)
G.f.: x*(1 + 9*x + 3*x^2 + 3*x^3) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = n*(2*n+1) for n even.
a(n) = n*(2*n-1) for n odd. (End)
E.g.f.: x*( 2*(1+x)*exp(x) - exp(-x) ). - G. C. Greubel, Aug 19 2022
Sum_{n>=1} 1/a(n) = 2 - log(2). - Amiram Eldar, Aug 21 2022

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

3, 1, 2, 4, 7, 5, 6, 8, 11, 9, 10, 12, 15, 13, 14, 16, 19, 17, 18, 20, 23, 21, 22, 24, 27, 25, 26, 28, 31, 29, 30, 32, 35, 33, 34, 36, 39, 37, 38, 40, 43, 41, 42, 44, 47, 45, 46, 48, 51, 49, 50, 52, 55, 53, 54, 56, 59, 57, 58, 60, 63, 61, 62

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

Original entry on oeis.org

1, 4, 3, 5, 4, 1, 8, 7, 4, 3, 9, 8, 5, 4, 1, 12, 11, 8, 7, 4, 3, 13, 12, 9, 8, 5, 4, 1, 16, 15, 12, 11, 8, 7, 4, 3, 17, 16, 13, 12, 9, 8, 5, 4, 1, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1, 24, 23, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 25, 24, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 1

Gary W. Adamson, Mar 16 2005

			The first few rows of the triangle are:
  1;
  4, 3;
  5, 4, 1;
  8, 7, 4, 3;
  9, 8, 5, 4, 1;
  ...

Cf. A035608, A074377, A104570.

Row sums yield A074377. Columns 1, 3, 5, ... (starting at the diagonal entry) yield A042948. Columns 2, 4, 6, ... (starting at the diagonal entry) yield A014601. The product R*Q yields A104570.

T:=proc(i,j) if j>i then 0 elif i+j mod 2 = 1 then 2*(i-j)+2 elif i mod 2 = 1 and j mod 2 = 1 then 2*(i-j)+1 elif i mod 2 = 0 and j mod 2 = 0 then 2*(i-j)+3 else fi end: for i from 1 to 13 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 23 2005

Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
R[i_, j_] := If[j <= i, 1, 0];
T[i_, j_] := Sum[Q[i, k]*R[k, j], {k, 1, 13}];
Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jul 24 2024 *)

For 1<=j<=i: T(i, j)=2(i-j+1) if i and j are of opposite parity; T(i, j)=2(i-j)+1 if both i and j are odd; T(i, j)=2(i-j)+3 if both i and j are even. - Emeric Deutsch, Mar 23 2005

More terms from Emeric Deutsch, Mar 23 2005

A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 8, 10, 8, 1, 1, 9, 18, 18, 9, 1, 1, 12, 27, 40, 27, 12, 1, 1, 13, 39, 67, 67, 39, 13, 1, 1, 16, 52, 112, 134, 112, 52, 16, 1, 1, 17, 68, 164, 246, 246, 164, 68, 17, 1, 1, 20, 85, 240, 410, 504, 410, 240, 85, 20, 1

Offset: 0

Gary W. Adamson, Jan 01 2008

			First few rows of the triangle are:
  1;
  1,   1;
  1,   4,   1;
  1,   5,   5,   1;
  1,   8,  10,   8,   1;
  1,   9,  18,  18,   9,   1;
  1,  12,  27,  40,  27,  12,   1;
  1,  13,  39,  67,  67,  39,  13,   1;
  1,  16,  52, 112, 134, 112,  52,  16,   1;
  1,  17,  68, 164, 246, 246, 164,  68,  17,   1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A034851, A042948, A077957, A122746 (row sums).

A136489:= func< n,k | 2*Binomial(n,k) - Binomial(n mod 2, k mod 2)*Binomial(Floor(n/2), Floor(k/2)) >;
[A136489(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2023

T[n_, k_]:= 2*Binomial[n,k] -Binomial[Mod[n,2], Mod[k,2]]*Binomial[Floor[n/2], Floor[k/2]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2023 *)

def A136489(n,k): return 2*binomial(n,k) - binomial(n%2, k%2)*binomial(n//2, k//2)
flatten([[A136489(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 01 2023

T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).
Sum_{k=0..n} T(n, k) = A122746(n).
From G. C. Greubel, Aug 01 2023: (Start)
T(n, k) = 2*A007318(n, k) - A051159(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) if k is even.
T(n, n-k) = T(n, k).
T(n, n-1) = A042948(n).
Sum_{k=0..n} (-1)^k * T(n, k) = 2*[n=0] - A077957(n). (End)

A171507 a(n) = (52^(n+1)-9-(-1)^n)/6-2n.

Original entry on oeis.org

0, 0, 1, 6, 17, 42, 93, 198, 409, 834, 1685, 3390, 6801, 13626, 27277, 54582, 109193, 218418, 436869, 873774, 1747585, 3495210, 6990461, 13980966, 27961977, 55924002, 111848053, 223696158, 447392369, 894784794, 1789569645, 3579139350, 7158278761, 14316557586

Offset: 0

Paul Curtz, Dec 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -3, 2).

Cf. A042948, A084640.

[(5*2^(n+1)-9-(-1)^n)/6 -2*n: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011

A171507:=n->(5*2^(n+1)-9-(-1)^n)/6 -2*n: seq(A171507(n), n=0..50); # Wesley Ivan Hurt, May 03 2017

a(n) = 3*a(n-1)-a(n-2)-3*a(n-3)+2*a(n-4). G.f.: x^2*(1+3*x)/((1+x)*(1-2*x)*(1-x)^2).
a(n) = A084640(n) - A042948(n).
a(n+1)-2*a(n) = A042948(n+1).
First differences: a(n+1)-a(n) = A084640(n).
Last digits: a(n) == a(n+10) (mod 10), n>=1.

Edited and extended by R. J. Mathar, Dec 15 2009

A267089 T(n,k) is decimal conversion of 1's in an n X n table that lie on its principal diagonals.

Original entry on oeis.org

1, 3, 3, 5, 2, 5, 9, 6, 6, 9, 17, 10, 4, 10, 17, 33, 18, 12, 12, 18, 33, 65, 34, 20, 8, 20, 34, 65, 129, 66, 36, 24, 24, 36, 66, 129, 257, 130, 68, 40, 16, 40, 68, 130, 257, 513, 258, 132, 72, 48, 48, 72, 132, 258, 513, 1025, 514, 260, 136, 80, 32, 80, 136, 260, 514

Offset: 0

Kival Ngaokrajang, Jan 10 2016

Inspired by A137932 and A042948.
Conjectures:
(i) The first column is A083318.
(ii) T(n,k) = A086066(m) where m >= 10, n = m - 9*k, k = floor(m/10).

			See the "Illustration of initial terms" link for explicit examples.
Triangle begins:
n\k 0   1  2  3  4  5  6   7   8 ...
0   1
1   3   3
2   5   2  5
3   9   6  6  9
4  17  10  4 10 17
5  33  18 12 12 18 33
6  65  34 20  8 20 34 65
7 129  66 36 24 24 36 66 129
8 257 130 68 40 16 40 68 130 257
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..15, k = 0..10

Cf. A042948, A083318, A086066, A137932.

A267489 a(n) = n^2 - 4*floor(n^2/6).

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 12, 17, 24, 29, 36, 41, 48, 57, 68, 77, 88, 97, 108, 121, 136, 149, 164, 177, 192, 209, 228, 245, 264, 281, 300, 321, 344, 365, 388, 409, 432, 457, 484, 509, 536, 561, 588, 617, 648, 677, 708, 737, 768, 801, 836, 869, 904

Offset: 0

Kival Ngaokrajang, Jan 16 2016

Inspired by A137932 and A042948.
The pattern is generated by adding subdiagonals parallel to principal diagonals at a spacing of at least 1 box in any direction from the previous generation.
Conjectures:
(i) a(n) is the total number of boxes (or 1's) at the n-th iteration.
(ii) The total number of left boxes (or 0's) is 4*A056827.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A042948, A056827, A137932.

[0] cat [n^2-4*Floor(n^2/6): n in [1..70]]; // Vincenzo Librandi, Jan 16 2016

A267489:=n->n^2-4*floor(n^2/6): seq(A267489(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2017

Table[n^2 - 4 Floor[n^2 / 6], {n, 0, 70}] (* Vincenzo Librandi, Jan 16 2016 *)

for (n = 0, 100, a = n^2-4*floor(n^2/6); print1(a, ", "))

concat(0, Vec(x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6)/((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 16 2016

a(n)=n^2 - n^2\6*4 \\ Charles R Greathouse IV, Mar 22 2017

a(n) = n^2 - 4*floor(n^2/6) for n >= 0.
From Colin Barker, Jan 16 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>7.
G.f.: x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)

cp's OEIS Frontend

A190796 Number of digits in the minimal base-phi representation of n.

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A204877 Continued fraction expansion of 3*tanh(1/3).

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A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).

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A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

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A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

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MATLAB

PARI

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A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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Maple

Mathematica

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Extensions

A136489 Triangle T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).

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A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).

A171507 a(n) = (52^(n+1)-9-(-1)^n)/6-2n.