A073577 -id:A073577 - cp's OEIS frontend

A214345 Interleaved reading of A073577 and A053755.

5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839, 901, 959, 1025, 1087, 1157, 1223, 1297, 1367, 1445, 1519, 1601, 1679, 1765, 1847, 1937, 2023, 2117, 2207, 2305, 2399, 2501

Offset: 0

Yasir Karamelghani Gasmallah, Jul 13 2012

The elements of this sequence satisfy the property that for every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2. In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2 : in the case of this sequence 7^2, 17^2, and 23^2 is such a triple (i.e. 15-8 =7, 17, 8+15=23, and 8^2+15^2=17^2) .

The first differences of such a sequence is always an interleaved sequence; in this case the interleaved sequence is 2,10,6,14,10,... (A142954).

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79

Guenther Schrack, Table of n, a(n) for n = 0..10001
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A053755, A073577, A178218.

First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - Guenther Schrack, Oct 25 2018

a:=[7,17];; for n in [3..50] do a[n]:=4*(n+1)+a[n-2]; od; Concatenation([5],a); # Muniru A Asiru, Oct 26 2018

I:=[5, 7, 17, 23];[n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

seq(coeff(series((x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 26 2018

LinearRecurrence[{2,0,-2,1},{5,7,17,23},50] (* Harvey P. Dale, Apr 02 2018 *)

A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
makelist(A214345(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

a(2n+1) = A073577(n+1); a(2n) = A053755(n+1).
a(n+1)-a(n) = A142954(n+1).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)).
a(n) = (2*n*(n+4)+3*(-1)^n+7)/2.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
a(n) = 4*(n+1) + a(n-2) for n > 1; a(-n) = a(n-4). - Guenther Schrack, Oct 24 2018
E.g.f.: (5 + 5*x + x^2)*cosh(x) + (2 + 5*x + x^2)*sinh(x). - Stefano Spezia, Feb 22 2024

A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

Original entry on oeis.org

1, 2, 16, 58, 149, 177, 534, 681, 954, 1045, 1052, 1255, 1367, 1563, 2046, 2074, 2515, 2557, 2564, 2788, 3586, 3593, 3908, 4062, 4552, 5252, 5371, 5385, 6400, 6729, 7443, 7478, 9305, 9375, 9942, 10355, 10411, 10726, 10740, 11286, 11545, 11559, 11832, 11965

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite.
Sierpiński showed that the only quadruple of consecutive primes of the form (2k+1)^2 - 2 are for k = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence).
Numbers k such that the 3 consecutive integers k, k+1 and k+2 belong to A088572. - Michel Marcus, Oct 13 2017

			The first triples are: k = 1: (7, 23, 47), k = 2: (23, 47, 79), k = 16: (1087, 1223, 1367).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Schinzel's hypothesis H.

Cf. A088572, A008865, A028870, A028871, A073577, A088572.

Select[Range[10^4], AllTrue[{(2#+1)^2-2, (2#+3)^2-2, (2#+5)^2-2},PrimeQ] &]
SequencePosition[Table[If[PrimeQ[(2k+1)^2-2],1,0],{k,12000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 09 2025 *)

f(n) = 4*n^2 + 4*n - 1;
isok(n) = isprime(f(n)) && isprime(f(n+1)) && isprime(f(n+2)); \\ Michel Marcus, Oct 13 2017

A049598 12 times triangular numbers.

Original entry on oeis.org

0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792, 936, 1092, 1260, 1440, 1632, 1836, 2052, 2280, 2520, 2772, 3036, 3312, 3600, 3900, 4212, 4536, 4872, 5220, 5580, 5952, 6336, 6732, 7140, 7560, 7992, 8436, 8892, 9360, 9840, 10332, 10836, 11352

Offset: 0

Joe Keane (jgk(AT)jgk.org)

a(n-1) is the Wiener index of the helm graph H(n) (n>=3). The graph H(n) is obtained from an n-wheel graph (on n+1 nodes) by adjoining a pendant edge at each node of the cycle. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of H(n) is (1/2)*n*t*((n-3)t^3 + 2(n-2)t^2 + (n+3)t + 6). - Emeric Deutsch, Sep 28 2010
Also sequence found by reading the line from 0, in the direction 0, 12, ..., and the same line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Axis perpendicular to A195158 in the same spiral. - Omar E. Pol, Sep 29 2011
Also the Wiener index of the (n+1)-gear graph. - Eric W. Weisstein, Sep 08 2017

			a(1) = 12*1 + 0 = 12;
a(2) = 12*2 + 12 = 36;
a(3) = 12*3 + 36 = 72.

G. C. Greubel, Table of n, a(n) for n = 0..5000
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60 (1996), pp. 959-969.
Leo Tavares, Illustration: Centroid Stars.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Helm Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001844, A003154, A027468, A035008, A073577.

12 * Accumulate[Range[0, 50]] (* Harvey P. Dale, Feb 05 2013 *)
(* Start from Eric W. Weisstein, Sep 08 2017 *)
Table[6 n (n + 1), {n, 0, 20}]
12 PolygonalNumber[3, Range[0, 20]]
12 Binomial[Range[20], 2]
LinearRecurrence[{3, -3, 1}, {12, 36, 72}, {0, 20}]
(* End *)

a(n)=6*n*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 6*n*(n+1).
G.f.: 12*x/(1-x)^3.
a(n) = 12*A000217(n). - Omar E. Pol, Dec 11 2008
a(n) = 12*n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n) = A003154(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A032528(2*n+1) - 1. - Adriano Caroli, Jul 19 2013
a(n) = A001844(n) + A073577(n). - Bruce J. Nicholson, Aug 06 2017
E.g.f.: 6*x*(x+2)*exp(x). - G. C. Greubel, Aug 23 2017
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 1/6. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(6/Pi)*cos(sqrt(5/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (6/Pi)*cos(Pi/(2*sqrt(3))). (End)

A088572 Numbers n such that (2n+1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 13, 14, 16, 17, 18, 21, 23, 24, 27, 30, 31, 34, 35, 37, 38, 44, 46, 51, 53, 58, 59, 60, 63, 65, 67, 69, 72, 77, 80, 84, 86, 88, 91, 95, 102, 105, 108, 111, 115, 116, 118, 119, 123, 126, 128, 129, 132, 133, 136, 139, 142, 146, 149, 150, 151, 154, 156, 157

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Nov 17 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A028870, A073577.

q:= n-> isprime((2*n+1)^2-2):
select(q, [$0..157])[];  # Alois P. Heinz, Nov 13 2024

is(n)=isprime((2*n+1)^2-2) \\ Charles R Greathouse IV, Jun 12 2017

A030440 Values of Newton-Gregory forward interpolating polynomial (1/3)(n-1)(2n+3)(2*n-1).

Original entry on oeis.org

1, 0, 7, 30, 77, 156, 275, 442, 665, 952, 1311, 1750, 2277, 2900, 3627, 4466, 5425, 6512, 7735, 9102, 10621, 12300, 14147, 16170, 18377, 20776, 23375, 26182, 29205, 32452, 35931, 39650, 43617, 47840, 52327, 57086, 62125, 67452, 73075, 79002, 85241, 91800, 98687, 105910

Offset: 0

Ilias.Kotsireas(AT)lip6.fr (Ilias Kotsireas)

Starting at a(2)=7, partial sums of A073577. - J. M. Bergot, Apr 20 2016

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; http://www.scirp.org/journal/am; http://dx.doi.org/10.4236/am.2014.515216

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A073577, A000384 (trinomial k=2 column), A106734, A127672, A027907.

[(1/3)*(n-1)*(2*n-1)*(2*n+3):n in [0..50]]; // Vincenzo Librandi, Apr 20 2018

LinearRecurrence[{4, -6, 4, -1}, {1, 0, 7, 30}, 40] (* Vincenzo Librandi, Apr 20 2018 *)

a(n) = (n-1)*(2*n-1)*(2*n+3)/3; \\ Altug Alkan, Apr 19 2018

G.f.: (1+13*x^2-2*x^3-4*x)/(1-x)^4. - R. J. Mathar, May 18 2014
a(n) = (1/6) * (A106734(2n) - 1), n > 0. - Mathew Englander, Jun 06 2014
E.g.f.: (3 - 3*x + 12*x^2 + 4*x^3)*exp(x)/3. - Ilya Gutkovskiy, Apr 20 2016
a(n+1) = trinomial(2*n+1, 3) = binomial(2*n+1, 3) + (2*n+1)*(2*n) = n*(2*n+1)*(2*n+5)/3, for n >= 0, with the trinomial irregular triangle A027907. a(n+1) = trinomial(2*n+1, 4*n-1), for n >= 1 (symmetry). a(n+1) = Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n+1)*R(4*(n-1), x)/Pi with the R polynomial coefficients given in A127672. [Comtet, p. 77, the integral formula for q=3, n -> 2*n+1, k = 3, rewritten with x = 2*cos(phi)]. The g.f. of {a(n+1)}{n>= 0} is x*(7 + 2*x - x^2)/(1 - x)^4. - _Wolfdieter Lang, Apr 19 2018

A227786 Take squares larger than 1, subtract 3 from even squares and 2 from odd squares; a(n) = a(n-1) + A168276(n+1) (with a(1) = 1).

Original entry on oeis.org

1, 7, 13, 23, 33, 47, 61, 79, 97, 119, 141, 167, 193, 223, 253, 287, 321, 359, 397, 439, 481, 527, 573, 623, 673, 727, 781, 839, 897, 959, 1021, 1087, 1153, 1223, 1293, 1367, 1441, 1519, 1597, 1679, 1761, 1847, 1933, 2023, 2113, 2207, 2301, 2399, 2497, 2599, 2701

Offset: 1

Antti Karttunen, Jul 31 2013

nonn

Conjecture: from n>=2 onward, a(n) gives the positions of 2's in A227761.

a(29) = 897 = 3*13*23 is the first term which is neither prime nor semiprime, that is, has more than two prime divisors.

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

Bisections: A082109, A073577. Cf. also A227761.

a(n) = A000290(n+1) - 2 - (n mod 2).
a(1)=1, and for n>1, a(n) = a(n-1)+A168276(n+1).
a(n) = (1/2) * (2*n^2 + 4*n -3 + (-1)^n) = 2*A116940(n-1) + 1. a(n-1) = 2*ceiling(n^2/2) - 3 = 2*A000985(n) - 3. G.f.: x*(-x^3 - x^2 + 5*x + 1)/((1-x)^3 * (1+x)). - Ralf Stephan, Aug 10 2013

A145009 Array read by antidiagonals: array of odd integers found in |A144912| with axes b = {4, 6, 8, ...} and n = {b^2, b^4, b^6, ...}.

Original entry on oeis.org

7, 13, 13, 19, 23, 19, 25, 33, 33, 25, 31, 43, 47, 43, 31, 37, 53, 61, 61, 53, 37, 43, 63, 75, 79, 75, 63, 43, 49, 73, 89, 97, 97, 89, 73, 49, 55, 83, 103, 115, 119, 115, 103, 83, 55, 61, 93, 117, 133, 141, 141, 133, 117, 93, 61

Offset: 0

Reikku Kulon, Sep 28 2008

The complete array can be defined as 6(x + y) + 4xy + 7.
Values along the edges are given by 6x + 7 and thus include the larger member of every twin prime pair except 5. The smaller member, 6x + 5, is adjacent in |A144912|.
Taking the origin to be z = 1, the main diagonal is given by 4z^2 + 4z - 1 (A073577).
Sums along antidiagonals are given by z(2z^2 + 12z + 7) / 3.
From Reikku Kulon, Sep 29 2008: (Start)
Any entry in the triangle can be produced from the two terms diagonally above or below and the edges can be found by taking the odd numbers as the "missing" values, starting from 1. If the terms are denoted:
.. a0 .. ...
a1 .. a2 ...
.. a3 .. ...
then:
a0 = (a1 + a2)/2 - 4*(a1 + a2 + 4)/(a2 - a1);
a3 = (a1 + a2)/2 + 4*(a1 + a2 + 4)/(a2 - a1). [Corrected by Jinyuan Wang, Jul 29 2020]
(End)

			Array A(n,k) begins:
7,  13, 19, 25,  31,  37,  43,  ...
13, 23, 33, 43,  53,  63,  73,  ...
19, 33, 47, 61,  75,  89,  103, ...
25, 43, 61, 79,  97,  115, 133, ...
31, 53, 75, 97,  119, 141, 163, ...
37, 63, 89, 115, 141, 167, 193, ...
...

Cf. A000040, A006512, A073577, A144912.

A(n, k) = |A144912(2*n+4, (2*n+4)^(2*k+2))| = 6*(n+k) + 4*n*k + 7.

A153167 Numbers n such that n+2 is not a Chen prime.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 90

Offset: 1

Vincenzo Librandi, Dec 20 2008

nonn

Contains all strictly positive even numbers A005843.
For each odd k>1 we can accumulate the numbers == k^2-2 (mod 2k) in a row, the last entry equal to A073577(k):
7; (k=3)
13, 23; (k=5)
19, 33, 47; (k=7)
25, 43, 61, 79; (k=9)
31, 53, 75, 97, 119; (k=11)
7, 63, 89, 115, 141, 167; (k=13)
43, 73, 103, 133, 163, 193,223; (k=17)
49, 83, 17, 151,185, 219, 253, 287; (k=19)
Each element T of this table has the format T= k^2-2-j*2*k, so T+2 is of the form k*(k-2*j), therefore not prime, and consequently all elements T are in the sequence.

Cf. A109611, A040976, A073577, A067076.

Edited, 41, 59 (see A102540) etc. inserted by R. J. Mathar, Oct 16 2009

A338284 a(n) is the smallest nonsquare m such that the second partial quotient in the continued fraction for sqrt(m) equals n.

Original entry on oeis.org

7, 2, 23, 5, 47, 10, 79, 17, 119, 26, 167, 37, 223, 50, 287, 65, 359, 82, 439, 101, 527, 122, 623, 145, 727, 170, 839, 197, 959, 226, 1087, 257, 1223, 290, 1367, 325, 1519, 362, 1679, 401, 1847, 442, 2023, 485, 2207, 530, 2399, 577, 2599, 626, 2807, 677, 3023, 730

Offset: 1

Max Alekseyev, Oct 20 2020

			a(3) = 23, since sqrt(23) = [4; 1, 3, ...] and m=23 is the smallest integer such that sqrt(m) has with second partial quotient equal 3.

Max Alekseyev, Table of n, a(n) for n = 1..100

Interweaving of A073577 and A002522.

Cf. A013945 (first partial quotient = n).

CoefficientList[Series[(7 + 2*x + 2*x^2 - x^3 - x^4 + x^5) / ((1-x)^3 * (1+x)^3),{x,0,20}], x] (* Georg Fischer, Aug 18 2021 *)

For even n, a(n) = A013945(n) = A002522(n/2) = (n/2)^2 + 1.
For odd n, a(n) = A073577((n+1)/2) = n^2 + 4*n + 2.
O.g.f.: (7 + 2*x + 2*x^2 - x^3 - x^4 + x^5) / ((1-x)^3 * (1+x)^3).

cp's OEIS Frontend

A214345 Interleaved reading of A073577 and A053755.

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A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

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A049598 12 times triangular numbers.

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A088572 Numbers n such that (2n+1)^2 - 2 is prime.

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A030440 Values of Newton-Gregory forward interpolating polynomial (1/3)*(n-1)*(2*n+3)*(2*n-1).

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A227786 Take squares larger than 1, subtract 3 from even squares and 2 from odd squares; a(n) = a(n-1) + A168276(n+1) (with a(1) = 1).

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A145009 Array read by antidiagonals: array of odd integers found in |A144912| with axes b = {4, 6, 8, ...} and n = {b^2, b^4, b^6, ...}.

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A153167 Numbers n such that n+2 is not a Chen prime.

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A030440 Values of Newton-Gregory forward interpolating polynomial (1/3)(n-1)(2n+3)(2*n-1).