A008598 -id:A008598 - cp's OEIS frontend

A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 2, 2, 3, 0, 3, 3, 2, 1, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 0

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).

A. Karttunen, Table of n, a(n) for n = 1..65536

a(n) = A163540(A008598(n)) = A004442(A163541(n)). See also A163542.

HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
   R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
   R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
   L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
   F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
   F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = F[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] :=
    Part[Flatten[a[(n + 16 - #)/16] /. HC /. HC], #]) &, Range[16]];
Part[FoldList[Mod[Plus[#1, #2], 4] &, 0,
  a[#] & /@ Range[4^4] /. {F[n_] :> 0, L[n_] :> 1, R[n_] :> -1}],
2 ;; -1] (* Bradley Klee, Aug 07 2015 *)

(define (A163540 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163539 n))) 4))

a(n) = A010873(A163538(n)+A163539(n)+abs(A163539(n))+3).

A174312 a(n) = 32*n.

Original entry on oeis.org

0, 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1056, 1088, 1120, 1152, 1184, 1216, 1248, 1280, 1312, 1344, 1376, 1408, 1440, 1472, 1504, 1536, 1568, 1600

Offset: 0

Paul Curtz, Nov 27 2010

Subsequence of squares is A017066 (see 2nd formula). - Bernard Schott, Mar 03 2023

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

Cf. A001105, A008598, A017066, A135628, A135631, A152691.

[32*n: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

Range[0, 3000, 32] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

a(n)=n<<5 \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 32*x/(1-x)^2.
a(A001105(n)) = A017066(n). - Bernard Schott, Mar 05 2023
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 32*x*exp(x).
a(n) = 2*A008598(n) = A152691(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A106839 Numbers congruent to 11 mod 16.

Original entry on oeis.org

11, 27, 43, 59, 75, 91, 107, 123, 139, 155, 171, 187, 203, 219, 235, 251, 267, 283, 299, 315, 331, 347, 363, 379, 395, 411, 427, 443, 459, 475, 491, 507, 523, 539, 555, 571, 587, 603, 619, 635, 651, 667, 683, 699, 715, 731, 747, 763, 779, 795, 811, 827, 843

Offset: 0

Ralf Stephan, May 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Differs from A044072.

Cf. A008598, A017101, A051062, A119413.

[[ n : n in [1..1000] | n mod 16 eq 11]]; // Vincenzo Librandi, Oct 10 2011

Range[11, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{11,27},60] (* Harvey P. Dale, Aug 12 2021 *)

a(n)=16*n+11 \\ Charles R Greathouse IV, Oct 16 2015

def a(n): return 11 + 16*n
print([a(n) for n in range(53)]) # Michael S. Branicky, Nov 27 2021

G.f.: x*(11+5*x)/(x-1)^2. - R. J. Mathar, Oct 08 2011
From Vincenzo Librandi, Oct 10 2011: (Start)
a(n) = 11 + 16*n.
a(n) = 32*n - a(n-1) + 6, a(0)=11. (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(11 + 16*x).
a(n) = 2*a(n-1) - a(n-2).
a(n) = A017101(2*n+1). (End)

A266956 Numbers m such that 9*m+7 is a square.

Original entry on oeis.org

1, 2, 18, 21, 53, 58, 106, 113, 177, 186, 266, 277, 373, 386, 498, 513, 641, 658, 802, 821, 981, 1002, 1178, 1201, 1393, 1418, 1626, 1653, 1877, 1906, 2146, 2177, 2433, 2466, 2738, 2773, 3061, 3098, 3402, 3441, 3761, 3802, 4138, 4181, 4533, 4578, 4946, 4993, 5377, 5426

Offset: 1

Bruno Berselli, Jan 07 2016

Equivalently, numbers of the form h*(9*h+8)+1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...
Also, integer values of k*(k+8)/9 plus 1.
It is easy to see that the Diophantine equation 9*x+3*j+1 = y^2 has infinitely many solutions in integers (x,y) for any j in Z. It follows a table with j = -5..5:
...
j = -5, x:  2, 7, 15, 30, 46, 71,  95, 130, 162, 207, 247, ...
j = -4, x:  3, 4, 20, 23, 55, 60, 108, 115, 179, 188, 268, ...
j = -3, x:  1, 8, 12, 33, 41, 76,  88, 137, 153, 216, 236, ...
j = -2, x:  1, 6, 14, 29, 45, 70,  94, 129, 161, 206, 246, ...
j = -1, x:  2, 3, 19, 22, 54, 59, 107, 114, 178, 187, 267, ...
j =  0, x:  0, 7, 11, 32, 40, 75,  87, 136, 152, 215, 235, ... (A132355)
j =  1, x:  0, 5, 13, 28, 44, 69,  93, 128, 160, 205, 245, ... (A185039)
j =  2, x:  1, 2, 18, 21, 53, 58, 106, 113, 177, 186, 266, ... (A266956)
j =  3, x: -1, 6, 10, 31, 39, 74,  86, 135, 151, 214, 234, ... (A266957)
j =  4, x: -1, 4, 12, 27, 43, 68,  92, 127, 159, 204, 244, ... (A266958)
j =  5, x:  0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, ... (A218864)
...
The general closed form of these sequences is:
b(n,j) = (18*(n-1)*n + s(j)*(2*n-1)*(-1)^n + t(j))/8, where s(j) = 6*(-j) + 18*floor(j/3) - (-1)^floor(2*(j+1)/3) + 6 and t(j) = 4*(-j) + 4*floor((j+1)/3) + 5.
a(2m) - a(2m-1) gives the odd numbers (A005408); a(2m+1) - a(2m) gives the multiples of 16 (A008598).

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

Cf. numbers m such that 9*m+i: A132355 (i=1), A185039 (i=4), this sequence (i=7), A005563 (i=9), A266957 (i=10), A266958 (i=13), A218864 (i=16), A008865 (i=18, without -2).
Cf. A156638: square roots of 9*a(n)+7.
Cf. A005408, A008598.

Magma
```
[n: n in [0..6000] | IsSquare(9*n+7)];
```

[(18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8: n in [1..50]];

Select[Range[0, 6000], IntegerQ[Sqrt[9 # + 7]] &]
Table[(18 (n - 1) n - 7 (2 n - 1) (-1)^n + 1)/8, {n, 1, 50}]

for(n=0, 6000, if(issquare(9*n+7), print1(n, ", ")))

vector(50, n, n; (18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8)

from gmpy2 import is_square
[n for n in range(6000) if is_square(9*n+7)]

[(18*(n-1)*n-7*(2*n-1)*(-1)**n+1)/8 for n in range(1, 60)]

[n for n in (0..6000) if is_square(9*n+7)]

[(18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8 for n in (1..50)]

G.f.: x*(1 + x + 14*x^2 + x^3 + x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*(n-1)*n - 7*(2*n-1)*(-1)^n + 1)/8.
a(n) = A218864(n) + 1.

A056753 Only odd numbers occur and for all k there are k numbers between any two successive occurrences of k.

Original entry on oeis.org

1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 13, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 17, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 19, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 21, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 23, 1, 3, 1, 7, 1, 3, 1

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 19 2001

Only the numbers 2^m - 1 occur more than once.
a(A005843(n)) = 1; a(A016813(n)) = 3; a(A004771(n)) = 7; a(A008598(n) + 35) = 15; a(A008598(n) + 155) = 31. - Reinhard Zumkeller, Aug 23 2009
A181497(n) = smallest m such that A056753(m) = 2*n + 1. - Reinhard Zumkeller, Oct 24 2010

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004771, A005843, A008598, A016813, A164632, A181497.

import Data.List (intercalate, group)
a056753 n = a056753_list !! n
a056753_list = [1] ++ odds [] where
   odds xs = xs ++ (intercalate xs' $ group [y+2,y+4..2*y+1]) ++ odds xs'
        where y = 2 * length xs + 1
              xs' = xs ++ [y] ++ xs
-- Reinhard Zumkeller, Feb 25 2012, Oct 24 2010

S:=[ 0: n in [1..100] ]; k:=1; p:=Position(S, 0, 1); while p gt 0 do for j in [p..#S by k+1] do if S[j] eq 0 then S[j]:=k; else break; end if; end for; f:=p; p:=Position(S, 0, f); k+:=2; end while; S; // Klaus Brockhaus, Oct 25 2010

a[n_] := a[n] = (ClearAll[f]; f[i_, x_, y_, z_] := f[i, x, y, z] = If[i == n, If[x == 1, a[n-z] + 2, a[n-z]], If[x == 1, If[y == 1, f[i+1, 2z, z, 2z], f[i+1, z, y-1, z]], f[i+1, x-1, y, z]]]; If[n == 0, 1, f[1, 1, 1, 1]]); Table[a[n], {n, 0, 98}] (* Jean-François Alcover, Dec 14 2011, after Reinhard Zumkeller *)

Let x = a(n - A164632(n)), a(n) = if (x occurred exactly once so far) then x+2 else x. - Reinhard Zumkeller, Aug 23 2009

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A212773 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

Original entry on oeis.org

0, 6, 11, 12, 15, 16, 18, 22, 24, 26, 30, 31, 32, 33, 35, 36, 40, 41, 42, 44, 45, 48, 51, 52, 54, 55, 56, 60, 61, 62, 64, 65, 66, 70, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 93, 96, 99, 102, 104, 105, 108, 110, 112, 114, 120, 121, 122, 123, 124

Offset: 1

Rick L. Shepherd, May 26 2012

nonn

Nonnegative multiples of each of 6, 11, 15, 16, 26, 31, 35, 40, 41, 51, 56, 61, 65, 76, 81, 85, 86, and 91.

All products of terms are terms.

			4 is not a term because it is not an appropriate multiple. Also 4 = 4*1 cannot be represented with more than one denomination of coin. Similarly 7 is not a term; although 7 = 7*1 = 2*1 + 1*5 does have a representation in terms of two denominations, 1 and 5, there are unequal numbers of each.
a(11) = 30 is a term because it is a multiple of 6. 30 = 5*1 + 5*5 = 2*5 + 2*10 = 1*5 + 1*25, so five coins each of denominations 1 and 5, two each of 5 and 10, or one each of 5 and 25 totals 30.
The term 34924118340711600 (5 times the LCM of the numbers in the first comment, so also divisible by 75) is the smallest which can be expressed in 26 such ways, one for each possible combination of two or more of these five coin denominations. (It also can be expressed as a multiple of each of these five alone of course.)

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
Index entries for sequences related to making change.

Cf. A212774, A008588, A008593, A008597, A008598, A135631.

{c = 0; n = -1; until(c==10000, n++;
if(n%6==0 || n%11==0 || n%15==0 || n%16==0 || n%26==0 ||
  n%31==0 || n%35==0 || n%40==0 || n%41==0 || n%51==0 ||
  n%56==0 || n%61==0 || n%65==0 || n%76==0 || n%81==0 ||
  n%85==0 || n%86==0 || n%91==0,
  c++; write("b212773.txt", c, " ", n)))}

A082285 a(n) = 16*n + 13.

Original entry on oeis.org

13, 29, 45, 61, 77, 93, 109, 125, 141, 157, 173, 189, 205, 221, 237, 253, 269, 285, 301, 317, 333, 349, 365, 381, 397, 413, 429, 445, 461, 477, 493, 509, 525, 541, 557, 573, 589, 605, 621, 637, 653, 669, 685, 701, 717, 733, 749, 765, 781, 797, 813, 829, 845

Offset: 0

Cino Hilliard, May 10 2003

Solutions to (7^x + 11^x) mod 17 = 13.

a(n-2), n>=2, gives the second column in triangle A238476 related to the Collatz problem. - Wolfdieter Lang, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Leo Tavares, Illustration: Bounded Star Crosses.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004770, A008594, A008598, A016813, A098502, A106839, A119413.

[[ n : n in [1..1000] | n mod 16 eq 13]]; // Vincenzo Librandi, Oct 10 2011

Range[13, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{13,29},60] (* Harvey P. Dale, Jan 28 2023 *)

\\ solutions to 7^x+11^x == 13 mod 17
anpbn(n) = { for(x=1,n, if((7^x+11^x-13)%17==0,print1(x" "))) }

a(n) = 16*n + 13.
a(n) = 32*n - a(n-1) + 10; a(0)=13. - Vincenzo Librandi, Oct 10 2011
From Stefano Spezia, Dec 27 2019: (Start)
O.g.f.: (13 + 3*x)/(1 - x)^2.
E.g.f.: exp(x)*(13 + 16*x). (End)
a(n) = A008594(n+1) + A016813(n+1) - 4. - Leo Tavares, Sep 22 2022
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A004770(2*n+2). (End)

A163541 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

Original entry on oeis.org

1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 3, 3, 2, 1, 2, 2, 3, 0, 3, 2, 3, 0, 0, 1, 0, 3, 0, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 1, 0, 3, 0, 0, 1

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).

Antti Karttunen, Table of n, a(n) for n = 1..4096

a(n) = A163541(A008598(n)) = A004442(A163540(n)). See also A163543.

HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = L[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] := Part[Flatten[a[(n + 16 - #)/16]/.HC/.HC],#]) &, Range[16]];
Part[FoldList[Mod[Plus[#1, #2], 4] &, 0, a[#] & /@ Range[4^4]/.{F[n_]:>0,L[n_]:>1,R[n_]:>-1}], 2 ;; -1] (* Bradley Klee, Aug 07 2015 *)

(define (A163541 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163538 n))) 4))

a(n) = A010873(A163538(n) + A163539(n) + abs(A163538(n)) + 3).

A317316 Multiples of 16 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 16, 3, 32, 5, 48, 7, 64, 9, 80, 11, 96, 13, 112, 15, 128, 17, 144, 19, 160, 21, 176, 23, 192, 25, 208, 27, 224, 29, 240, 31, 256, 33, 272, 35, 288, 37, 304, 39, 320, 41, 336, 43, 352, 45, 368, 47, 384, 49, 400, 51, 416, 53, 432, 55, 448, 57, 464, 59, 480, 61, 496, 63, 512, 65, 528, 67, 544, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 20-gonal numbers (A218864).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 20-gonal numbers.

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

Cf. A008598 and A005408 interleaved.
Column 16 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A218864.

a[n_] := If[OddQ[n], n, 8*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 16*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 2^(e+3), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 7*2^(1-s)). - Amiram Eldar, Oct 25 2023

cp's OEIS Frontend

A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

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A174312 a(n) = 32*n.

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A106839 Numbers congruent to 11 mod 16.

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A266956 Numbers m such that 9*m+7 is a square.

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A056753 Only odd numbers occur and for all k there are k numbers between any two successive occurrences of k.

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A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

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