A004770 -id:A004770 - cp's OEIS frontend

A168392 a(n) = 5 + 8*floor((n-1)/2).

5, 5, 13, 13, 21, 21, 29, 29, 37, 37, 45, 45, 53, 53, 61, 61, 69, 69, 77, 77, 85, 85, 93, 93, 101, 101, 109, 109, 117, 117, 125, 125, 133, 133, 141, 141, 149, 149, 157, 157, 165, 165, 173, 173, 181, 181, 189, 189, 197, 197, 205, 205, 213, 213, 221, 221, 229, 229

Offset: 1

Vincenzo Librandi, Nov 24 2009

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

Cf. A004770, A168379.

[5+8*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

RecurrenceTable[{a[1]==5,a[n]==8n-a[n-1]-6},a,{n,80}] (* or *) LinearRecurrence[{1,1,-1},{5,5,13},80] (* or *) With[{c= LinearRecurrence[ {2,-1},{5,13},40]}, Riffle[c,c]] (* Harvey P. Dale, Jan 27 2013 *)
Table[5 + 8 Floor[(n - 1)/2], {n, 60}] (* Bruno Berselli, Sep 18 2013 *)
CoefficientList[Series[(5 + 3 x^2)/((1 + x) (x - 1)^2),{x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)

a(n) = 8*n - a(n-1) - 6 with n>1, a(1)=5.
a(1) = 5, a(2)=5, a(3)=13; for n>3, a(n) = a(n-1) +a(n-2) -a(n-3). - Harvey P. Dale, Jan 27 2013
G.f.: x*(5 + 3*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
E.g.f.: (-2 + 3*exp(x) + (4*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 19 2016
a(n) = A168379(n) - 2. - Filip Zaludek, Nov 01 2016

New definition by Vincenzo Librandi, Sep 18 2013

A175462 Number of divisors of integers of the form 5 + 8n.

Original entry on oeis.org

2, 2, 4, 2, 2, 6, 2, 2, 4, 4, 4, 4, 2, 2, 6, 4, 4, 4, 2, 2, 8, 2, 2, 8, 2, 4, 4, 4, 2, 4, 6, 4, 6, 2, 2, 8, 2, 4, 4, 2, 6, 6, 4, 2, 8, 4, 2, 4, 2, 2, 10, 4, 2, 8, 4, 4, 4, 2, 4, 6, 4, 4, 4, 2, 4, 12, 4, 2, 6, 2, 4, 4, 4, 4, 4, 6, 2, 8, 4, 6, 8, 2, 2, 4, 2, 4, 12, 2, 2, 4, 6, 2, 8, 4, 2, 12, 2, 4, 4, 2, 8, 4, 2

Offset: 0

Zak Seidov, May 23 2010

All terms are even.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004770 (Numbers of form 8n + 5), A007521 (Primes of form 8n + 5). A000005 (d(n) : number of divisors of n), A001620.

map(numtheory:-tau,[seq(i,i=5..1000,8)]); # Robert Israel, Mar 20 2020

Table[DivisorSigma[0, 8*n + 5], {n, 0, 100}] (* Amiram Eldar, Jan 14 2024 *)

a(n) = numdiv(5+8*n); \\ Michel Marcus, Oct 15 2013

a(n) = d(5 + 8*n).
a(n) = A000005(A004770(n)).
Sum_{k=1..n} a(k) ~ (n/2) * (log(n) + 2*gamma - 1 + 5*log(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A177065 a(n) = (8n+3)(8*n+5).

Original entry on oeis.org

15, 143, 399, 783, 1295, 1935, 2703, 3599, 4623, 5775, 7055, 8463, 9999, 11663, 13455, 15375, 17423, 19599, 21903, 24335, 26895, 29583, 32399, 35343, 38415, 41615, 44943, 48399, 51983, 55695, 59535, 63503, 67599, 71823, 76175, 80655, 85263, 89999, 94863, 99855

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - Bruno Berselli, Aug 24 2010

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

Cf. A002061, A004770, A016754, A017101, A125169, A177059.

[(8*n+3)*(8*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

A177065:=n->(8*n+3)*(8*n+5): seq(A177065(n), n=0..100); # Wesley Ivan Hurt, Apr 24 2017

Table[(8n+3)(8n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{15,143,399},40] (* Harvey P. Dale, Mar 13 2013 *)
CoefficientList[Series[(15 + 98 x + 15 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)

a(n)=(8*n+3)*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 128*n + a(n-1) with n > 0, a(0)=15.
a(n) = A125169(A016754(n) - 1). - Reinhard Zumkeller, Jul 05 2010
a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 13 2013
G.f.: (15+98*x+15*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 08 2013
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017101(n)*A004770(n).
Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)
E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - Elmo R. Oliveira, Oct 25 2024

Edited by N. J. A. Sloane, Jun 22 2010

A265228 Interleave the even numbers with the numbers that are congruent to {1, 3, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 9, 8, 11, 10, 15, 12, 17, 14, 19, 16, 23, 18, 25, 20, 27, 22, 31, 24, 33, 26, 35, 28, 39, 30, 41, 32, 43, 34, 47, 36, 49, 38, 51, 40, 55, 42, 57, 44, 59, 46, 63, 48, 65, 50, 67, 52, 71, 54, 73, 56, 75, 58, 79, 60, 81, 62, 83, 64, 87, 66

Offset: 0

Paul Curtz, Dec 06 2015

b(n) denotes the sequence:
0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 2, -2, 2, -2, 2, -2, 2, 3, -3, 3, -3, 3, -3, 3, 4, -4, ..., and
c(n) = n + b(n) = n + floor((n+1)/7)*(-1)^((n+1) mod 7) provides:
0, 1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 10, 13, 15, 12, 17, 14, 19, 16, 21, 23, 18, 25, 20, 27, 22, 29, ..., which is a permutation of A001477.
a(n) differs from c(n) because c(n) contains the terms of the form 8*k+5.

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

Cf. A001477, A004770, A005843, A047529, A079979, A260160, A260699, A260708.

lim = 11; Riffle[Range[0, 6 lim, 2], Select[Range[8 lim], MemberQ[{1, 3, 7}, Mod[#, 8]] &]] (* Michael De Vlieger, Dec 06 2015 *)

concat(0, Vec(x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6)/((1-x)^2 *(1+x)^2*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 06 2015

vector(100, n, n--; n+(1-(-1)^n)*floor(n/6+1/3)) \\ Altug Alkan, Dec 09 2015

a(n) = n + 2*A260160(n) = n + (1-(-1)^n)*floor(n/6+1/3). Therefore, for odd n, a(n) = A047529((n+1)/2); otherwise, a(n) = n.
a(n) = a(n-6) - (-1)^n + 7.
a(n) = A260708(n) - A260699(n-1) - A079979(n+3), with A260699(-1) = 0.
From Colin Barker, Dec 06 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n > 7.
G.f.: x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6) / ((1-x)^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)). (End)

A137194 Lucky numbers (A000959) which are congruent to 5 mod 8.

Original entry on oeis.org

13, 21, 37, 69, 93, 133, 141, 189, 205, 237, 261, 285, 349, 357, 421, 429, 477, 517, 541, 613, 621, 645, 685, 693, 717, 741, 781, 805, 885, 925, 933, 957, 981, 997, 1021, 1029, 1053, 1093, 1101, 1117, 1189, 1197, 1245, 1261, 1285, 1309, 1357, 1365, 1389, 1485

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A004770.

A146302 a(n) = (8n+5)(8*n+9).

Original entry on oeis.org

45, 221, 525, 957, 1517, 2205, 3021, 3965, 5037, 6237, 7565, 9021, 10605, 12317, 14157, 16125, 18221, 20445, 22797, 25277, 27885, 30621, 33485, 36477, 39597, 42845, 46221, 49725, 53357, 57117, 61005, 65021, 69165, 73437, 77837, 82365

Offset: 0

Miklos Kristof, Oct 29 2008

From Miklos Kristof, Nov 03 2008: (Start)
f(y) = y^4*(1 + y^4) = y^4 - y^8 + y^12 - y^16 + y^20 - y^24 + ...
Integral_{y} f(y) dy = y^5/5 - y^9/9 + y^13/13 - y^17/17 + y^21/21 - y^25/25 + ...
Integral_{y=0..1} f(y) dy = 1/5 - 1/9 + 1/13 - 1/17 + 1/21 - 1/25 + ...
                  = (9 - 5)/(5*9) + (17 - 13)/(13*17) + (25 - 21)/(21*25) + ...
                  = 4/(5*9) + 4/(13*17) + 4/(21*25) + ...
Integral_{y=0..1} f(y) dy = Sum_{m>=0} 4/((8*m+5)*(8*m+9))
                  = -(1/8)*sqrt(2)*Pi + 1 - (1/4)*sqrt(2)*log(1+sqrt(2))
                  = 0.13302701266008896241... (End)

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

seq((8*m+5)*(8*m+9),m=0..40); # Miklos Kristof, Nov 03 2008

Table[(8n+5)(8n+9),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{45,221,525},40] (* Harvey P. Dale, Oct 10 2015 *)

a(n)=(8*n+5)*(8*n+9) \\ Charles R Greathouse IV, Jun 17 2017

G.f: (45 + 86*x - 3*x^2)/(1-x)^3.
E.g.f.: (45 + 176*x + 64*x^2)*exp(x).
a(n) = A004770(n) * A004768(n). - Reinhard Zumkeller, Oct 30 2008

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A047436 Numbers that are congruent to {5, 6} mod 8.

Original entry on oeis.org

5, 6, 13, 14, 21, 22, 29, 30, 37, 38, 45, 46, 53, 54, 61, 62, 69, 70, 77, 78, 85, 86, 93, 94, 101, 102, 109, 110, 117, 118, 125, 126, 133, 134, 141, 142, 149, 150, 157, 158, 165, 166, 173, 174, 181, 182, 189, 190

Offset: 1

N. J. A. Sloane

nonn

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

Union of A004770 and A017137.

Select[Range[200], MemberQ[{5, 6}, Mod[#, 8]] &] (* Amiram Eldar, Dec 19 2021 *)

From Vincenzo Librandi, Aug 06 2010: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 8*n - a(n-1) - 5, n > 1. (End)
G.f. x*(5+x+2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 19 2021

A175486 Composite numbers of form 8n+5 with all prime factors of form 8m+5.

Original entry on oeis.org

125, 325, 725, 845, 925, 1325, 1525, 1885, 2197, 2405, 2525, 2725, 3125, 3445, 3725, 3925, 3965, 4205, 4325, 4525, 4901, 4925, 5365, 5725, 6253, 6565, 6725, 6845, 6925, 7085, 7325, 7685, 7925, 8125, 8725, 8845, 8957, 9325, 9685, 9725, 9805, 9925, 10205

Offset: 1

Zak Seidov, May 27 2010

nonn

There are no squares and no semiprimes in the sequence.

			125=5^3, 325=5^2*13, 725=5^2*29.

Cf. A004770 Numbers of form 8n+5, A175461 Semiprimes of form 8n+5, A007521 Primes of form 8n+5.

Do[nn=8n+5;If[ !PrimeQ[nn]&&{5}==Union[Mod[(fi=First/@FactorInteger[nn]),8]],Print[nn]],{n,2*10^3}]

A360033 Table T(n,k), n >= 1 and k >= 0, read by antidiagonals, related to Jacobsthal numbers A001045.

Original entry on oeis.org

1, 2, 1, 3, 3, 3, 4, 5, 7, 5, 5, 7, 11, 13, 11, 6, 9, 15, 21, 27, 21, 7, 11, 19, 29, 43, 53, 43, 8, 13, 23, 37, 59, 85, 107, 85, 9, 15, 27, 45, 75, 117, 171, 213, 171, 10, 17, 31, 53, 91, 149, 235, 341, 427, 341, 11, 19, 35, 61, 107, 181, 299, 469

Offset: 1

Philippe Deléham, Jan 22 2023

			The array T(n,k), for n <= 1 and k >= 0, begins:
n = 1: 1,  1,  3,  5,  11,  21,  43, ... -> A001045(k+1)
n = 2: 2,  3,  7, 13,  27,  53, 107, ... -> A048573(k)
n = 3: 3,  5, 11, 21,  43,  85, 171, ... -> A001045(k+3)
n = 4: 4,  7, 15, 29,  59, 117, 235, ... -> ?
n = 5: 5,  9, 19, 37,  75, 149, 299, ... -> A062092(k+1)
n = 6: 6, 11, 23, 45,  91, 181, 363, ... -> ?
n = 7: 7, 13, 27, 53, 107, 213, 427, ... -> A048573(k+2)

Cf. A001045, A048573, A062092.

Columns: A000027, A005408, A004767, A004770, A106839 for k = 0, 1, 2, 3, 4.

T(n,k) = T(1,k) + (n-1)*2^k.
T(n,k) = 2*T(n, k-1) + (-1)^k.
T(n,k) = T(n-1,k) + 2^k.
T(n,k) = 2^k * n - A001045(k).
T(n,k) = T(n,k-1) +2*T(n,k-2).

cp's OEIS Frontend

A168392 a(n) = 5 + 8*floor((n-1)/2).

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A175462 Number of divisors of integers of the form 5 + 8n.

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A177065 a(n) = (8*n+3)*(8*n+5).

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A265228 Interleave the even numbers with the numbers that are congruent to {1, 3, 7} mod 8.

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A137194 Lucky numbers (A000959) which are congruent to 5 mod 8.

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A146302 a(n) = (8*n+5)*(8*n+9).

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A177065 a(n) = (8n+3)(8*n+5).

A146302 a(n) = (8n+5)(8*n+9).