A010036 -id:A010036 - cp's OEIS frontend

A173464 When regarded as a triangle, the rows of A162797 converge to this sequence.

1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129, 21, 53, 81, 133, 81, 165, 289, 321

Offset: 1

Omar E. Pol, Feb 22 2010

			From _Omar E. Pol_, Apr 01 2011: (Start)
If written as a triangle begins:
1,
5,
5,17,
5,17,21,49,
5,17,21,49,21,53,81,129,
5,17,21,49,21,53,81,129,21,53,81,133,81,165,289,321,
...
It appears that the final terms of rows give A000337.
It appears that row sums give A010036.
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A162795, A162796, A162797.

Cf. A010036, A162793, A162794. - Omar E. Pol, Apr 01 2011

More terms a(20)-a(32) from Omar E. Pol, Apr 01 2011

A291658 a(n) is the sum of all integers from 5^n to 5^(n+1)-1.

Original entry on oeis.org

10, 290, 7450, 187250, 4686250, 117181250, 2929656250, 73242031250, 1831053906250, 45776363281250, 1144409160156250, 28610229394531250, 715255736816406250, 17881393430175781250, 447034835803222656250, 11175870895324707031250, 279396772384338378906250

Offset: 0

Enrique Navarrete, Aug 28 2017

a(n) is the sum of all (positive) numbers having exactly (n+1) digits when written in base 5. - Alois P. Heinz, Sep 25 2017

			For n=0, the sum is from 1 to 4, so a(0)=10;
for n=1, the sum is from 5 to 24, so a(1)=290, etc.

Colin Barker, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (30,-125).

Cf. A010036, A226508, A121544, A101291, A162729.

a:= unapply(sum(i, i=5^n..5^(n+1)-1), n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 25 2017

Vec(10*(1 - x) / ((1 - 5*x)*(1 - 25*x)) + O(x^30)) \\ Colin Barker, Sep 12 2017

a(n) = ((5^n)/2)*(5^(n+2) - 5^n - 4), n >= 0.
From Colin Barker, Sep 12 2017: (Start)
G.f.: 10*(1 - x) / ((1 - 5*x)*(1 - 25*x)).
a(n) = 30*a(n-1) - 125*a(n-2) for n>1.
(End)
a(n) = A162729(n+1) - A162729(n). - Alois P. Heinz, Sep 25 2017

A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.

Original entry on oeis.org

3, 4, 5, 21, 20, 29, 105, 88, 137, 465, 368, 593, 1953, 1504, 2465, 8001, 6080, 10049, 32385, 24448, 40577, 130305, 98048, 163073, 522753, 392704, 653825, 2094081, 1571840, 2618369, 8382465, 6289408, 10479617, 33542145, 25161728, 41930753, 134193153, 100655104, 167747585, 536821761, 402636800, 671039489, 2147385345, 1610579968, 2684256257

Offset: 0

Ralf Steiner, May 07 2020

Let [h21] = {{1, 3}, {0, 2}} be the matrix [h_2]*[h_1] in Firstov's notation, from eqs. (24) and (39). Then primitive Pythagorean triples (pPT) (x(n), y(n), z(n)) = (u(n)^2 - v(n)^2, 2*u(n)*v(n), u(n)^2 + v(n)^2), with u(n) and v(n) of different parity, gcd(u(n), v(n)) = 1, and u(n) > v(n) > 0, are generated by (u(n), v(n))^T = [h21]^n*(2,1)^T (T for transpose).
For n > 0: (x(n), y(n), z(n)) = (1, 0, 1) (mod 4). Thus some z are Pythagorean primes (A002144).
The triples converge to the proportion (4:3:5) with:
lim_{n->infinity} x(n)/y(n) = 4/3, lim_{n->infinity} y(n)/z(n) = 3/5.
Altitude h(n) = x(n)*y(n)/z(n) is an irreducible fraction because of primitivity.
From Wolfdieter Lang, Jun 13 2020: (Start)
[h21]^n = sqrt(2)^n*(S(n, 3/sqrt(2))*[1_3] + S(n-1, 3/sqrt(2))*(1/sqrt(2))*([h21] - 3*[1_3])) with the Chebyshev S polynomials (A049310).
u(n) = sqrt(2)^n*(2*S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A153893(n),
v(n) = sqrt(2)^n*(S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A000079(n). Proof from the recurrence, using the Cayley-Hamilton theorem.
With the monic Chebyshev T polynomials, called R in A127672:
x(n)/3 = 2^(n+1)*(R(2*(n+1), 3/sqrt(2)) - (sqrt(2)/3)*R(2*n+1,3/sqrt(2)) - 1) = A171477(n),
y(n)/4 = 3*2^(n-1)*(sqrt(2)*R(2*n+1,3/sqrt(2)) - R(2*n,3/sqrt(2)) - 1/3)
= A010036(n),
z(n) = 3*2^(n+1)*((3/sqrt(2))*R(2*n+1, 3/sqrt(2)) - (4/3)*R(2*n,3/sqrt(2)) - 1).
Using 2^n*Rnx(2*n, 3/sqrt(2)) = A052539(n) = 2^(2*n) + 1, and
  2^(n)*(sqrt(2)/3)*Rnx(2*n+1, 3/sqrt(2)) = A007583(n) = (2^(2*n + 1) + 1)/3,
produces the explicit formulas given by the author in the formula section.
G.f.s for {x(n)} G0(x) = 3/((1 - 4*x)*(1 - 2*x)*(1 - x)), for {y(n)} G1(x) =  4*(1-x)/((1 - 4*x)*(1 - 2*x)), and for {z(n)} = (5 - 6*x + 4*x^2)/((1 - 4*x)*(1 - 2*x)*(1 - x)). This produces the g.f. for the array, read as sequence {a(n)}: G(x) = G0(x^3) + x*G1(x^3) + x^2*G2(x^3) given in the formula section by Colin Barker.
(End)

			The three-column array pPT(n,k) begins:
n\k        0        1         2
-------------------------------
0:         3        4         5
1:        21       20        29
2:       105       88       137
3:       465      368       593
4:      1953     1504      2465
5:      8001     6080     10049
6:     32385    24448     40577
7:    130305    98048    163073
8:    522753   392704    653825
9:   2094081  1571840   2618369
10:  8382465  6289408  10479617
... - _Wolfdieter Lang_, Jun 13 2020

V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
Ralf Steiner, Weitere spezielle Folge primitiver pythagoraischer Dreiecke, ResearchGate.
Wikipedia, Tree of primitive Pythagorean triples.
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-14,0,0,8).

Cf. A000079, A010035, A049310, A083329, A093357, A010036, A134057, A153893, A171477.

h21={{1, 3}, {0, 2}}; l = {}; Do[v = MatrixPower[h21, n, {2, 1}]; p = v[[1]]; q = v[[2]];
a = p^2 - q^2; b = 2 p q; c = p^2 + q^2; l = AppendTo[l, {a, b, c}], {n, 0, 14}]; l // Flatten

Vec((3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)) + O(x^35)) \\ Colin Barker, Jun 12 2020

The three-column array PT(n, k) is for k = 0, 1, 2:  x(n), y(n), z(n), for n >= 0, with
x(n) = a(3*n + 0) = A153893(n)^2 - A000079(n)^2 = 1 - 3*2^(n+1) + 2^(2*n+3) = binomial(2^(n+2) - 1, 2) = 3*A171477(n),
y(n) = a(3*n + 1) = 2*A153893(n)*A000079(n) = 2^(n+1)*(-1 + 3*2^n) = 4*A010036(n),
z(n) = a(3*n + 2) = A153893(n)^2 + A000079(n)^2 = 1 - 6*2^n + 10*2^(2*n).
From Colin Barker, May 08 2020: (Start)
G.f. (read as sequence {a(n)}): (3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)).
a(n) = 7*a(n-3) - 14*a(n-6) + 8*a(n-9), for n > 8.
(End)

Edited, and corrected proportion by Wolfdieter Lang, Jun 13 2020

Minor grammatical edits. - N. J. A. Sloane, Sep 12 2020

A061298 Table by antidiagonals of rows of sequences where each row is binomial transform of preceding row and row 1 is (1,2,1,2,1,2,1,2,...).

Original entry on oeis.org

1, 1, 1, -2, 2, 1, 4, 1, 3, 1, -8, 2, 6, 4, 1, 16, 1, 12, 13, 5, 1, -32, 2, 24, 40, 22, 6, 1, 64, 1, 48, 121, 92, 33, 7, 1, -128, 2, 96, 364, 376, 174, 46, 8, 1, 256, 1, 192, 1093, 1520, 897, 292, 61, 9, 1, -512, 2, 384, 3280, 6112, 4566, 1816, 452, 78, 10, 1, 1024, 1, 768, 9841, 24512, 23073, 11152, 3289, 660, 97, 11, 1, -2048

Offset: 0

Henry Bottomley, Jun 05 2001

Rows include A011782 (but signed), A000034, A003945, A003462, A010036. Columns include A000012, A000027, A028872.

T(n, k) =(3n^k-(n-2)^k)/2. Coefficient of x^k in expansion of (1-(n-3)x)/((1-nx)(1-(n-2)x)).

A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

Original entry on oeis.org

1, 35, 770, 14260, 244776, 4053840, 65979040, 1064678720, 17107266176, 274296689920, 4393395202560, 70331527418880, 1125602147608576, 18012016334950400, 288211318352814080, 4611533554425610240, 73785756576381566976, 1180581862943988449280

Offset: 0

Ralf Steiner, May 16 2020

See A334638 for these triangles, and also for the Firstov reference.
For primitive Pythagorean triangle (x, y, z) = (u^2 - v^2, 2*u*v, u^2 + v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction.
From A334638 follows A(n)/6 = (x(n)/3)*(y(n)/4) = A171477(n)*A010036(n), for n >= 0. See the formula section.
Limit_{n->infinity} A(n)/(3*2^(4*n+3)) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for (3, 4, 5).

Index entries for linear recurrences with constant coefficients, signature (30,-280,960,-1024).

Cf. A010036, A171477, A334638.

Table[ 2^(-1 + n) (-1 + 3 2^n) (-1 + 2^(1 + n)) (-1 + 2^(2 + n))/3, {n, 0, 17}]

Vec((1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, May 17 2020

a(n) = 2^(n-1)*(3*2^n - 1)*(2^(n+1) - 1)*(2^(n+2) - 1)/3.
a(n) = 2^(4*n+2)*(1 - 13/(3*2^(n+2)) + 3/2^(2*n+3) - 1/(3*2^(3*(n+1)))), for n >= 0.
From Colin Barker: (Start)
G.f.: (1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)).
a(n) = 30*a(n-1) - 280*a(n-2) + 960*a(n-3) - 1024*a(n-4) for n > 3. (End)

Edited by Wolfdieter Lang, Jun 14 2020

A338131 Triangle read by rows, T(n, k) = k^(n - k) + Sum_{i = 1..n-k} k^(n - k - i)*2^(i - 1), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 8, 4, 1, 16, 16, 20, 14, 5, 1, 32, 32, 48, 46, 22, 6, 1, 64, 64, 112, 146, 92, 32, 7, 1, 128, 128, 256, 454, 376, 164, 44, 8, 1, 256, 256, 576, 1394, 1520, 828, 268, 58, 9, 1, 512, 512, 1280, 4246, 6112, 4156, 1616, 410, 74, 10, 1

Offset: 0

Werner Schulte, Oct 11 2020

This number triangle is case s = 2 of the triangles T(s; n,k) depending on some fixed integer s. Here are several (generalized) formulas and properties (attention: negative values are possible if s < 0):
  (1) T(s; n,k) = k^(n-k) + Sum_{i=1..n-k} k^(n-k-i)*s^(i-1) for 0<=k<=n;
  (2) T(s; n,n) = 1 for n >= 0, and T(s; n,n-1) = n for n > 0;
  (3) T(s; n+1,k) = k * T(s; n,k) + s^(n-k) for 0<=k<=n;
  (4) T(s; n,k) = (k+s) * T(s; n-1,k) - s*k * T(s; n-2,k) for 0<=k<=n-2;
  (5) G.f. of column k: Sum_{n>=k} T(s; n,k)*t^n = ((1-(s-1)*t)/(1-s*t))
      *(t^k/(1-k*t)) when t^k/(1-k*t) is g.f. of column k>=0 of A004248.

			The number triangle T(n, k) for 0 <= k <= n starts:
n\ k :    0     1      2      3      4      5      6     7    8    9   10
=========================================================================
   0 :    1
   1 :    1     1
   2 :    2     2      1
   3 :    4     4      3      1
   4 :    8     8      8      4      1
   5 :   16    16     20     14      5      1
   6 :   32    32     48     46     22      6      1
   7 :   64    64    112    146     92     32      7     1
   8 :  128   128    256    454    376    164     44     8    1
   9 :  256   256    576   1394   1520    828    268    58    9    1
  10 :  512   512   1280   4246   6112   4156   1616   410   74   10   1

Cf. A004248.

For columns k = 0, 1, 2, 3, 4 see A011782, A000079, A001792, A027649, A010036 respectively.

T := proc(n, k) if k = 0 then `if`(n = 0, 1, 2^(n-1)) elif k = 2 then n*2^(n-3)
else (k^(n-k)*(1-k) + 2^(n-k))/(2-k) fi end:
seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Oct 29 2020

T(n,k) = k^(n-k) + sum(i=1, n-k, k^(n-k-i) * 2^(i-1));
matrix(7,7, n, k, if(n>=k, T(n-1,k-1), 0)) \\ to see the triangle \\ Michel Marcus, Oct 12 2020

T(n,k) = ((k-1) * k^(n-k) - 2^(n-k)) / (k-2) if k <> 2, and T(n,2) = n * 2^(n-3) for n >= k.
T(n,n) = 1 for n >= 0, and T(n,n-1) = n for n > 0.
T(n+1,k) = k * T(n,k) + 2^(n-k) for 0 <= k <= n.
T(n,k) = (k+2) * T(n-1,k) - 2*k * T(n-2,k) for 0 <= k <= n-2.
T(n,k) = k * T(n-1,k) + T(n-1,k-1) - (k-1) * T(n-2,k-1) for 0 < k < n.
G.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n = ((1-t) / (1-2*t)) * (t^k / (1-k*t)) when t^k / (1-k*t) is g. f. of column k of A004248.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((1-t) / (1-2*t)) * (Sum_{k>=0} (x*t)^k / (1-k*t)).

cp's OEIS Frontend

A173464 When regarded as a triangle, the rows of A162797 converge to this sequence.

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A291658 a(n) is the sum of all integers from 5^n to 5^(n+1)-1.

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A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2*A153893*A000079, z = A153893^2 + A000079^2, ordered by increasing z.

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A061298 Table by antidiagonals of rows of sequences where each row is binomial transform of preceding row and row 1 is (1,2,1,2,1,2,1,2,...).

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A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

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A338131 Triangle read by rows, T(n, k) = k^(n - k) + Sum_{i = 1..n-k} k^(n - k - i)*2^(i - 1), for 0 <= k <= n.

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A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.