A017077 -id:A017077 - cp's OEIS frontend

A372293 Odd numbers that do not occur in the odd bisection of A371094.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Odd numbers that do not occur in array A371100.

Antti Karttunen, Table of n, a(n) for n = 1..14930

Setwise difference A005408 \ A372290.
Cf. A371094, A371100.
Subsequences: A004767, A017077.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372293(n) = if(!(n%2),0,forstep(k=1,n,2,if(A371094(k)==n,return(0))); (1));

A047468 Numbers that are congruent to {1, 2} mod 8.

Original entry on oeis.org

1, 2, 9, 10, 17, 18, 25, 26, 33, 34, 41, 42, 49, 50, 57, 58, 65, 66, 73, 74, 81, 82, 89, 90, 97, 98, 105, 106, 113, 114, 121, 122, 129, 130, 137, 138, 145, 146, 153, 154, 161, 162, 169, 170, 177, 178, 185, 186, 193, 194, 201, 202, 209, 210, 217, 218, 225, 226, 233

Offset: 1

N. J. A. Sloane

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

Union of A017077 and A017089.

Cf. A047467.

Flatten[#+{1,2}&/@(8Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{1,2,9},60] (* Harvey P. Dale, Mar 26 2013 *)

a(n)=(n-1)\2*8+2-n%2 \\ Charles R Greathouse IV, May 14 2012

a(n) = 8*n - a(n-1) - 13 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(1+x+6*x^2)/((1-x)^2*(1+x)). - Colin Barker, May 13 2012
a(n) = 1 + 8*floor((n-1)/2) + ((n-1) mod 2). - Alois P. Heinz, May 13 2012
a(n) = (-3*(3 + (-1)^n) + 8*n)/2. - Colin Barker, May 14 2012
a(1)=1, a(2)=2, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Mar 26 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021
E.g.f.: 6 + ((8*x - 9)*exp(x) - 3*exp(-x))/2. - David Lovler, Sep 02 2022

More terms from Vincenzo Librandi, Aug 06 2010

A050479 a(n) = C(n)(9n + 1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 10, 38, 140, 518, 1932, 7260, 27456, 104390, 398684, 1528436, 5878600, 22673308, 87662200, 339653880, 1318498920, 5126862150, 19965297660, 77855108100, 303969268680, 1188105796020, 4648590733800, 18205030164360, 71356399639200, 279909199969308, 1098799886728152

Offset: 0

Barry E. Williams, Dec 24 1999

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=9 of A330965.

Cf. A017077, A000108, A051945.

[Catalan(n)*(9*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (4-7*x-4*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020

A050479[n_] := CatalanNumber[n]*(9*n + 1);
Array[A050479, 30, 0] (* Paolo Xausa, Aug 24 2025 *)

4*(n+1)*a(n) + (-23*n-1)*a(n-1) + 14*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 13 2015
-(n+1)*(9*n-8)*a(n) + 2*(9*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Feb 13 2015
G.f.: (4 - 7*x - 4*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 8*binomial(2*n, n-1) = A000984(n) + 8*A001791(n).
a(n) ~ 4^n * 9/sqrt(Pi*n). (End)

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A157337 a(n) = 128n^2 + 32n + 1.

Original entry on oeis.org

161, 577, 1249, 2177, 3361, 4801, 6497, 8449, 10657, 13121, 15841, 18817, 22049, 25537, 29281, 33281, 37537, 42049, 46817, 51841, 57121, 62657, 68449, 74497, 80801, 87361, 94177, 101249, 108577, 116161, 124001, 132097, 140449, 149057

Offset: 1

Vincenzo Librandi, Feb 27 2009

The identity (128*n^2+32*n+1)^2 - (4*n^2+n)*(64*n+8)^2 = 1 can be written as a(n)^2 - A007742(n)*A157336(n)^2 = 1 (see also second part of the comment in A157336). - Vincenzo Librandi, Jan 29 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007742, A157336.

I:=[161, 577, 1249]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012

LinearRecurrence[{3,-3,1},{161,577,1249},50] (* Vincenzo Librandi, Jan 29 2012 *)

for(n=1, 40, print1(128*n^2 + 32*n + 1", ")); \\ Vincenzo Librandi, Jan 29 2012

G.f.: x*(x^2 + 94*x + 161)/(1-x)^3. - Vincenzo Librandi, Jan 29 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 29 2012
a(n) = 2*A017077(n)^2 - 1. - Bruno Berselli, Jan 29 2012
E.g.f.: (1 + 160*x + 128*x^2)*exp(x) - 1. - G. C. Greubel, Feb 01 2018

A185377 Product of exactly two distinct primes congruent to 1 mod 8 (A007519).

Original entry on oeis.org

697, 1241, 1513, 1649, 1921, 2329, 2993, 3281, 3649, 3961, 3977, 4097, 4369, 4633, 4777, 5321, 5617, 5729, 6001, 6497, 6817, 6953, 7081, 7361, 7633, 7769, 7913, 8249, 8633, 8857, 9553, 9673, 9809, 9881, 10001, 10057, 10081, 10217, 10489, 10537

Offset: 1

Jonathan Vos Post, Feb 20 2011

Subset of semiprimes A001358. Subset of {d = p_1 * p_2 * ... * p_m where p_i == 1 (mod 8), 1 <= i <= m are distinct primes} as occurs in Wei, p. 2.

			10001 is in this sequence because 10001 = 73 * 137 = A007519(3) * A007519(7).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dasheng Wei, On the equation x^2-Dy^2=n, Feb 18, 2011.

Cf. A000040, A001358, A007519, A017077.

p = Select[Prime[Range[200]], Mod[#, 8] == 1 &]; Sort[Reap[Do[n=p[[i]] p[[j]]; If[n <= p[[1]]p[[-1]], Sow[n]], {i, 2, Length[p]}, {j, i - 1}]][[2,1]]]

list(lim)=my(v=List(),P=List(),t); forprime(p=2,lim\17, if(p%8==1, listput(P,p))); for(i=2,#P, my(p=P[i]); for(j=1,i-1, t=p*P[j]; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jul 03 2016

{A007519(i) * A007519(j) for i < j}.

{A000040(i) * A000040(j) for i < j, and A000040(i) in A017077 and A000040(j) in A017077}.

A281334 Triangle read by rows: T(n, k) = (n - k)*(k + 1)^3 + k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 9, 2, 3, 17, 29, 3, 4, 25, 56, 67, 4, 5, 33, 83, 131, 129, 5, 6, 41, 110, 195, 254, 221, 6, 7, 49, 137, 259, 379, 437, 349, 7, 8, 57, 164, 323, 504, 653, 692, 519, 8, 9, 65, 191, 387, 629, 869, 1035, 1031, 737, 9, 10, 73, 218, 451, 754, 1085, 1378, 1543, 1466, 1009, 10

Offset: 1

Juri-Stepan Gerasimov, Jan 23 2017

			Triangle begins:
   0;
   1,    1;
   2,    9,    2;
   3,   17,   29,    3;
   4,   25,   56,   67,    4;
   5,   33,   83,  131,  129,    5;
   6,   41,  110,  195,  254,  221,    6;
   7,   49,  137,  259,  379,  437,  349,    7;
   8,   57,  164,  323,  504,  653,  692,  519,    8;
   9,   65,  191,  387,  629,  869, 1035, 1031,  737,    9;
  10,   73,  218,  451,  754, 1085, 1378, 1543, 1466, 1009,   10;
  ...

Cf. Triangle read by rows: T(n,k) = (n-k)*(k+1)^m+k: A003056 (m = 0), A059036 (m = 1), A274602 (m = 2), this sequence (m = 3).

Cf. A001477 (column 0), A017077 (column 1), A281546 (column 2), A242604 (middle diagonal).

/* As triangle */ [[(n-k)*(k+1)^3+k: k in [1..n]]: n in [0..10]];

t[n_, k_] := (n - k)*(k + 1)^3 + k; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Feb 09 2017 *)

for(n=0,10,for(k=0,n,print1((n-k)*(k+1)^3+k,", "))) \\ Derek Orr, Feb 26 2017

Row sums sum_{k>=0} T(n,k) = n*(n+1)*(3*n^3+12*n^2+13*n+32)/60. - R. J. Mathar, Mar 19 2017

A330983 Alternatively add and multiply pairs of the nonnegative integers.

Original entry on oeis.org

1, 6, 9, 42, 17, 110, 25, 210, 33, 342, 41, 506, 49, 702, 57, 930, 65, 1190, 73, 1482, 81, 1806, 89, 2162, 97, 2550, 105, 2970, 113, 3422, 121, 3906, 129, 4422, 137, 4970, 145, 5550, 153, 6162, 161, 6806, 169, 7482, 177, 8190, 185, 8930, 193, 9702, 201, 10506, 209

Offset: 1

George E. Antoniou, Jan 05 2020

In groups of two, add and multiply the integers: 0+1, 2*3, 4+5, 6*7, ....

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

Cf. A330987.

Interspersion of A017077 and A256833. - Michel Marcus, Jan 06 2020

a[n_]:=If[OddQ[n],4n-3,2(n-1)(2n-1)]; Array[a,53] (* Stefano Spezia, Jan 05 2020 *)

Vec(x*(1 + 6*x + 6*x^2 + 24*x^3 - 7*x^4 + 2*x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 07 2020

From Colin Barker, Jan 05 2020: (Start)
G.f.: x*(1 + 6*x + 6*x^2 + 24*x^3 - 7*x^4 + 2*x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = (1/2)*(-1 + 5*(-1)^n - 2*(1 + 5*(-1)^n)*n + 4*(1+(-1)^n)*n^2).
(End)
E.g.f.: (2 + 4*x*(1 + x))*cosh(x) - (3 + 2*x)*sinh(x) - 2. - Stefano Spezia, Jan 05 2020 after Colin Barker

A330987 Alternatively add and half-multiply pairs of the nonnegative integers.

Original entry on oeis.org

1, 3, 9, 21, 17, 55, 25, 105, 33, 171, 41, 253, 49, 351, 57, 465, 65, 595, 73, 741, 81, 903, 89, 1081, 97, 1275, 105, 1485, 113, 1711, 121, 1953, 129, 2211, 137, 2485, 145, 2775, 153, 3081, 161, 3403, 169, 3741, 177, 4095, 185, 4465, 193, 4851, 201, 5253, 209

Offset: 1

George E. Antoniou, Jan 05 2020

In groups of two, add and half-multiply the integers: 0+1, (2*3)/2, 4+5, (6*7)/2, ....
From Bernard Schott, Jan 06 2020: (Start)
The bisection of this sequence gives:
For n odd = 2*k+1, k >= 0: a(2*k+1) = 8*k+1 = A017077(k),
For n even = 2*k, k >= 1: a(2*k) = T(4*k-2) = A000217(4*k-2) = (2*k-1)*(4*k-1) = A033567(k) where T(j) is the j-th triangular number. (End)

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

Cf. A330983.

Interspersion of A017077 and A033567 (excluding first term). - Michel Marcus, Jan 06 2020

a[n_]:=If[OddQ[n],4n-3,(n-1)(2n-1)]; Array[a,53] (* Stefano Spezia, Jan 05 2020 *)

Vec(x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 06 2020

From Colin Barker, Jan 05 2020: (Start)
G.f.: x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = -1 + 2*(-1)^n - (1/2)*(-1+7*(-1)^n)*n + (1+(-1)^n)*n^2.
(End)
E.g.f.: (1 + 4*x + 2*x^2)*cosh(x) - (3 + x)*sinh(x) - 1. - Stefano Spezia, Jan 05 2020 after Colin Barker

A348845 Part two of the trisection of A017101: a(n) = 11 + 24*n.

Original entry on oeis.org

11, 35, 59, 83, 107, 131, 155, 179, 203, 227, 251, 275, 299, 323, 347, 371, 395, 419, 443, 467, 491, 515, 539, 563, 587, 611, 635, 659, 683, 707, 731, 755, 779, 803, 827, 851, 875, 899, 923, 947, 971, 995, 1019, 1043, 1067

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017101 = {3 + 8*k}A017077%20=%20%7B3*(1%20+%2012*n)%7D">{k>=0} gives 3*A017077 = {3*(1 + 12*n)}{n>=0}, {a(n)}A350051%20=%20%7B19%20+%2024*n%7D">{n >= 0} and A350051 = {19 + 24*n}{n>=0}. These three sequences are congruent to 3 modulo 8 and to 3, 5, and 1 modulo 6, respectively.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008606, 3*A017077, A017101, A350051.

24 * Range[0, 44] + 11 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 11 + 24*n = 11 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -13, a(0) = 11.
G.f.: (11 + 13*x)/(1-x)^2.
E.g.f.: (11 + 24*x)*exp(x).

A350051 Part three of the trisection of A017101: a(n) = 19 + 24*n.

Original entry on oeis.org

19, 43, 67, 91, 115, 139, 163, 187, 211, 235, 259, 283, 307, 331, 355, 379, 403, 427, 451, 475, 499, 523, 547, 571, 595, 619, 643, 667, 691, 715, 739, 763, 787, 811, 835, 859, 883, 907, 931, 955, 979, 1003, 1027, 1051, 1075

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017101 = {3 + 8*k}A017077%20=%20%7B3*(1%20+%2012*n)%7D">{k>=0} gives 3*A017077 = {3*(1 + 12*n)}{n>=0}, {A348845(n)}{n >= 0} and {a(n)}{n>=0}. These three sequences are congruent to 3 modulo 8 and to 3, 5, and 1 modulo 6, respectively.

Winston de Greef, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008606, 3*A017077, A017101, A348845.

24 * Range[0, 44] + 19 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 19 + 24*n \\ Winston de Greef, Jan 28 2024

a(n) = 19 + 24*n = 19 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -5, a(0) = 19.
G.f.: (19 + 5*x)/(1-x)^2.
E.g.f.: (19 + 24*x)*exp(x).

cp's OEIS Frontend

A372293 Odd numbers that do not occur in the odd bisection of A371094.

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A047468 Numbers that are congruent to {1, 2} mod 8.

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A050479 a(n) = C(n)*(9*n + 1) where C(n) = Catalan numbers (A000108).

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A157337 a(n) = 128*n^2 + 32*n + 1.

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A185377 Product of exactly two distinct primes congruent to 1 mod 8 (A007519).

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A281334 Triangle read by rows: T(n, k) = (n - k)*(k + 1)^3 + k, 0 <= k <= n.

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Magma

Mathematica

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A330983 Alternatively add and multiply pairs of the nonnegative integers.

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Mathematica

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Formula

A050479 a(n) = C(n)(9n + 1) where C(n) = Catalan numbers (A000108).

A157337 a(n) = 128n^2 + 32n + 1.