A047225 -id:A047225 - cp's OEIS frontend

A175887 Numbers that are congruent to {1, 14} mod 15.

1, 14, 16, 29, 31, 44, 46, 59, 61, 74, 76, 89, 91, 104, 106, 119, 121, 134, 136, 149, 151, 164, 166, 179, 181, 194, 196, 209, 211, 224, 226, 239, 241, 254, 256, 269, 271, 284, 286, 299, 301, 314, 316, 329, 331, 344, 346, 359, 361, 374, 376, 389, 391, 404

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 15).

Bruno Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.

Cf. A132240 (primes).

a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..450] | n mod 15 in [1,14]];

[(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Select[Range[1, 450], MemberQ[{1,14}, Mod[#, 15]]&]
CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)

a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).
a(n) = (30*n+11*(-1)^n-15)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.
a(n) = A047209(A047225(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - David Lovler, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)

A047247 Numbers that are congruent to {2, 3, 4, 5} (mod 6).

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 64, 65, 68, 69, 70, 71, 74, 75, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 95, 98, 99

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047235 with A047270. - Guenther Schrack, Feb 10 2019

Numbers k for which A276076(k) and A276086(k) are multiples of three. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

Guenther Schrack, Table of n, a(n) for n = 1..10015
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for two-way infinite sequences.

Cf. A047227, A047235, A047246, A047270, A047257.
Cf. A007623, A049345, A276076, A276086.
Complement: A047225.

[n : n in [0..100] | n mod 6 in [2, 3, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

A047247:=n->(6*n-1-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047247(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-1-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,4,5,8},70] (* Harvey P. Dale, May 25 2024 *)

my(x='x+O('x^70)); Vec(x*(2+x+x^2+x^3+x^4)/((1-x)*(1-x^4))) \\ G. C. Greubel, Feb 16 2019

a=(x*(2+x+x^2+x^3+x^4)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x*(2+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(1-x)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 1 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i = sqrt(-1).
a(2*n) = A047270(n), a(2*n-1) = A047235(n).
a(n) = A047227(n) + 1, a(1-n) = - A047227(n). (End)
From Guenther Schrack, Feb 10 2019: (Start)
a(n) = (6*n - 1 - (-1)^n -2*(-1)^(n*(n+1)/2))/4.
a(n) = a(n-4) + 6, a(1)=2, a(2)=3, a(3)=4, a(4)=5, for n > 4.
a(n) = A047227(n) + 1. a(n) = A047246(n) + 2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 - 2*log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021

More terms from Wesley Ivan Hurt, May 21 2016

A095894 a(2n) = 6n^2 + 7n + 1; a(2n+1) = 6n^2 + 13n + 7.

Original entry on oeis.org

1, 7, 14, 26, 39, 57, 76, 100, 125, 155, 186, 222, 259, 301, 344, 392, 441, 495, 550, 610, 671, 737, 804, 876, 949, 1027, 1106, 1190, 1275, 1365, 1456, 1552, 1649, 1751, 1854, 1962, 2071, 2185, 2300, 2420, 2541, 2667, 2794, 2926, 3059, 3197, 3336, 3480

Offset: 0

Gary W. Adamson, Jun 11 2004

From Omar E. Pol, Jul 18 2012: (Start)
Positive terms of A051866 and positive terms of A049453 interleaved.
Also sequence found by reading the line from 1, in the direction 1, 14, ..., and the line from 7, in the direction 7, 26, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. (End)

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

Cf. A047225 (first differences), A049453, A051866.

LinearRecurrence[{2,0,-2,1},{1,7,14,26},60] (* Harvey P. Dale, Oct 13 2016 *)

x='x+O('x^50); Vec((-1-5*x)/((1+x)*(x-1)^3)) \\ G. C. Greubel, Jun 19 2017

G.f.: ( -1-5*x ) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Oct 26 2011

Edited by Don Reble, Nov 16 2005

A115731 Permutation of natural numbers generated by 3-rowed array shown below.

Original entry on oeis.org

1, 6, 2, 7, 5, 3, 12, 8, 4, 13, 11, 9, 18, 14, 10, 19, 17, 15, 24, 20, 16, 25, 23, 21, 30, 26, 22, 31, 29, 27, 36, 32, 28, 37, 35, 33, 42, 38, 34, 43, 41, 39, 48, 44, 40, 49, 47, 45, 54, 50, 46, 55, 53, 51, 60, 56, 52, 61, 59, 57, 66, 62, 58, 67, 65, 63, 72, 68, 64, 73, 71, 69

Offset: 1

Giovanni Teofilatto, Mar 19 2006

6 7 12 13 18 19 24 25 ... a(n)=congruent to {0, 1} mod 6, Cf. A047225.
5 8 11 14 17 20 23 26 ... a(n)= 3n+2= A016789.
4 9 10 15 16 21 22 27 ... a(n)=congruent to {3, 4} mod 6, Cf. A047230.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

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

Cf. A115302.

Vec(x*(1 + 4*x - 8*x^2 + 13*x^3 - 15*x^4 + 13*x^5 - 5*x^6 - 4*x^7 + 4*x^8) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)) + O(x^80)) \\ Colin Barker, Apr 01 2018

Starting with the term a(3), a(n+6k) = a(n) + 6k, with k>=1.
From Colin Barker, Apr 01 2018: (Start)
G.f.: x*(1 + 4*x - 8*x^2 + 13*x^3 - 15*x^4 + 13*x^5 - 5*x^6 - 4*x^7 + 4*x^8) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>9.
(End)

A131231 3A130296 - 2A128174.

Original entry on oeis.org

1, 6, 1, 7, 3, 1, 12, 1, 3, 1, 13, 3, 1, 3, 1, 18, 1, 3, 1, 3, 1, 19, 3, 1, 3, 1, 3, 1, 24, 1, 3, 1, 3, 1, 3, 1, 25, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Jun 20 2007

Left column = A047225, numbers congruent to {0,1} mod 6: (1, 6, 7, 12, 13, 18, 19, ...).

Row sums = A131229, numbers congruent to {1,7} mod 10: (1, 7, 11, 17, ...).

			First few rows of the triangle:
   1;
   6, 1;
   7, 3, 1;
  12, 1, 3, 1;
  13, 3, 1, 3, 1;
  ...

Cf. A131229, A047225, A130296, A128174.

3*A130296 - 2*A128174 as infinite lower triangular matrices.

A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

Original entry on oeis.org

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64

Offset: 0

Franck Maminirina Ramaharo, Aug 01 2018

Write n in base 8, then apply the following substitution to the rightmost digit: '0'->'2, '1'->'3', and vice versa. Convert back to decimal.
A self-inverse permutation: a(a(n)) = n.
Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.

			a(25) = a('3'1') = '3'3' = 27.
a(26) = a('3'2') = '3'0' = 24.
a(27) = a('3'3') = '3'1' = 25.
a(28) = a('3'4') = '3'4' = 28.
a(29) = a('3'5') = '3'5' = 29.
The sequence as array read by rows:
  A047463, A047621, A047451, A047522;
        2,       3,       0,       1;
        4,       5,       6,       7;
       10,      11,       8,       9;
       12,      13,      14,      15;
       18,      19,      16,      17;
       20,      21,      22,      23;
       26,      27,      24,      25;
       28,      29,      30,      31;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
OEIS wiki, Lodumo transform.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1)))); // G. C. Greubel, Sep 25 2018

Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
f[n_] := Block[{id = IntegerDigits[n, 8]}, FromDigits[ Join[Most@ id /. {{} -> {0}}, {id[[-1]] /. {0 -> 2, 1 -> 3, 2 -> 0, 3 -> 1}}], 8]]; Array[f, 67, 0] (* or *)
CoefficientList[ Series[(x^7 + x^5 + 3x^3 - 2x^2 - x + 2)/((x - 1)^2 (x^6 + x^4 + x^2 + 1)), {x, 0, 70}], x] (* or *)
LinearRecurrence[{2, -2, 2, -2, 2, -2, 2, -1}, {2, 3, 0, 1, 4, 5, 6, 7}, 70] (* Robert G. Wilson v, Aug 01 2018 *)

makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);

my(x='x+O('x^100)); Vec((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1))) \\ G. C. Greubel, Sep 25 2018

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8), n > 7.
a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.
a(n) = lod_4(A047247(n+1)).
a(4*n) = A047463(n+1).
a(4*n+1) = A047621(n+1).
a(4*n+2) = A047451(n+1).
a(4*n+3) = A047522(n+1).
a(A042948(n)) = A047596(n+1).
a(A042964(n+1)) = A047551(n+1).
G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).
E.g.f.: x*exp(x) + cos(x) + sin(x) + cos(x/sqrt(2))*cosh(x/sqrt(2)) + (sqrt(2)*cos(x/sqrt(2)) - sin(x/sqrt(2)))*sinh(x/sqrt(2)).
a(n+8) = a(n) + 8 . - Philippe Deléham, Mar 09 2023
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - Amiram Eldar, Mar 12 2023

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A175887 Numbers that are congruent to {1, 14} mod 15.

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A047247 Numbers that are congruent to {2, 3, 4, 5} (mod 6).

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A095894 a(2n) = 6*n^2 + 7*n + 1; a(2n+1) = 6*n^2 + 13*n + 7.

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A115731 Permutation of natural numbers generated by 3-rowed array shown below.

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A131231 3*A130296 - 2*A128174.

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A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

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A095894 a(2n) = 6n^2 + 7n + 1; a(2n+1) = 6n^2 + 13n + 7.

A131231 3A130296 - 2A128174.